5 IS LM

IS – LM model

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5


Context
• This chapter develops the IS-LM model,
the theory that yields the aggregate demand curve.

• We focus on the short run and assume the price
level is fixed.
• This chapter focus on the closed-economy case.

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The Keynesian Cross
• A simple closed economy model in which income
is determined by expenditure.
(due to J.M. Keynes)

• Notation:
I = planned investment
E = C + I + G = planned expenditure
Y = real GDP = actual expenditure
• Difference between actual & planned expenditure:
unplanned inventory investment
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Elements of the Keynesian Cross
consumption function:
govt policy variables:
for now, planned
investment is exogenous:
planned expenditure:

C  C (Y T )
G  G , T T
I I
E  C (Y  T )  I  G

Equilibrium condition:

Actual expenditure  Planned expenditure
Y  E
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Graphing planned expenditure
E
planned
expenditure

E =C +I +G
MPC
1

income, output, Y

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Graphing the equilibrium condition
E

E =Y

planned
expenditure

45º
income, output, Y

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The equilibrium value of income
E

E =Y

planned
expenditure

E =C +I +G

income, output, Y

Equilibrium
income
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The IS curve
A graph of all combinations of r and Y that result in
goods market equilibrium,
i.e. actual expenditure (output) = planned expenditure

The equation for the IS curve is:

Y  C (Y  T )  I (r )  G

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Deriving the IS curve
E =Y

E

r  I
 E
 Y

E =C +I (r2 )+G
E =C +I (r1 )+G

I

r

Y1

Y

Y2

r1
r2

IS
Y1

Y2

Y

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Why the IS curve is negatively sloped
• A fall in the interest rate motivates firms to
increase investment spending, which drives up
total planned spending (E ).
• To restore equilibrium in the goods market,
output (a.k.a. actual expenditure, Y ) must
increase.
IS is negative: increase (decrease) r
 decrease (increase) I

 decrease (increase) Y
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The IS curve and the Loanable Funds model
(b) The IS curve

(a) The L.F. model

r

S2

r

S1

r2

r2

r1

r1

I (r )
S, I

IS

Y2

Y1

Y

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Fiscal Policy and the IS curve
• We can use the IS-LM model to see how
fiscal policy (G and T ) can affect
aggregate demand and output.
• Let’s start by using the Keynesian Cross to
see how fiscal policy shifts the IS curve…

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An increase in government purchases
At Y1,
there is now an
unplanned drop
in inventory…

E
E =C +I +G2

E =C +I +G1

G

…so firms
increase output,
and income
rises toward a
new equilibrium

Y
E1 = Y1

Y

E2 = Y2

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Solving for Y
Y  C  I  G
Y  C  I  G


C

 G

 MPC  Y  G
Collect terms with Y
on the left side of the

equals sign:

(1  MPC) Y  G

equilibrium condition
in changes
because I exogenous
because C = MPC Y

Finally, solve for Y :



1
Y  
  G
 1  MPC 

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The government purchases multiplier
Definition: the increase in income resulting from a
$1 increase in G.
In this model, the govt purchases multiplier equals
Y
1

G
1  MPC

Example: If MPC = 0.8, then
An increase in G
Y
1

 5
causes income to
G
1  0.8
increase by 5 times
as much!
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Why the multiplier is greater than 1
• Initially, the increase in G causes an equal increase in
Y: Y = G.
• But Y  C
 further Y
 further C
 further Y
• So the final impact on income is much bigger than the
initial G.

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An increase in taxes
Initially, the tax
increase reduces

consumption, and
therefore E:

E
E =C1 +I +G

E =C2 +I +G
At Y1, there is now
an unplanned
inventory buildup…

C = MPC T
…so firms
reduce output,
and income falls
toward a new
equilibrium

Y
E2 = Y2

Y

E1 = Y1

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Solving for Y
Y  C  I  G

eq’m condition in changes

 C

I and G exogenous

 MPC   Y  T
Solving for Y :

Final result:



(1  MPC) Y   MPC  T
  MPC 
Y  
  T
 1  MPC 

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The Tax Multiplier
The change in income resulting from a $1 increase in T :

Y
T

 MPC


1  MPC

If MPC = 0.8, then the tax multiplier equals

Y
T

 0.8
 0.8


 4
1  0.8
0.2

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The Tax Multiplier
…is negative:
A tax hike reduces
consumer spending,
which reduces income.
…is greater than one
(in absolute value):
A change in taxes has a
multiplier effect on income.

…is smaller than the govt spending multiplier:

Consumers save the fraction (1-MPC) of a tax cut,
so the initial boost in spending from a tax cut is
smaller than from an equal increase in G.
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Shifting the IS curve: G
At any value of r,

E =Y

E

E =C +I (r1 )+G1

G  E  Y
…so the IS curve
shifts to the right.
The horizontal
distance of the
IS shift equals

Y 

E =C +I (r1 )+G2

r

Y1

Y

Y2

r1

1
G
1  MPC

Y

Y1

IS1

Y2

IS2
Y

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The Theory of Liquidity Preference
• due to John Maynard Keynes.

• A simple theory in which the interest rate is
determined by money supply and money demand.

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Money Supply
r
interest
rate

M

P

s

The supply of
real money is fixed

M P

Md
real money
balances
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Equilibrium
The interest rate
adjusts
to equate the
supply and

demand for
money:

M P  L (r )

r
interest
rate

M

P

s

r1

L (r )
M P

real money
balances
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The LM curve
A graph of all combinations of r and Y that
result in money market equilibrium,

M P  L (r ,Y )

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