Transcript Simple Keynesian Model

IS-LM Model
IS Function
1
Outline





Introduction
Assumptions
Investment Function I= f(r)
Deriving the IS Function: IncomeExpenditure Approach (Y = E)
Deriving the IS Function: InjectionWithdrawal Approach (I + G = S + T)

4-quadrant diagram
2
Outline



Simple Algebra of the IS function
Slope of the IS function
 Interest Elasticity of Investment b
 Marginal Propensity to Save s
Shift of the IS function
 T’ v.s. E’
3
Introduction







In the elementary Keynesian model,
investment I is independent of interest rate r.
The Paradox of Thrift
In a 2-sector model, at equilibrium,
planned I = planned S
I = I’ = S’ + sY = S  if S’  OR s  Y
However, when S’ OR s  r  I’
I  S  Y uncertain
4
Introduction

Sometimes, investment depends on income
Y and is an endogenous function



I = f(Y) e.g. I = I’ + iY
Marginal Propensity to Invest MPI: I/Y= i
However, in the IS-LM model, investment
depends on the interest rate


I = f (r ) e.g. I = I’ - br
Interest Elasticity to Invest: I/r = -b
5
Introduction



In the elementary Keynesian model, only
the goods market is considered.
In the IS-LM model, both the goods market
and the money market are considered.
In the goods market


Investment I = Saving S
In the money market

Liquidity Preference L = Money Supply M
6
Introduction


In the elementary Keynesian model,
equilibrium is attained
when income is equal to ex-ante aggregate
expenditure


Y
= C + I + G + (X - M)
OR ex-ante withdrawal is equal to ex-ante
injection

S+T+M =I+G+X
7
Introduction



In the IS-LM model, equilibrium is attained
when both the goods market and the money
market are in equilibrium.
Yet, the labour market may not be in
equilibrium at this moment.
There may be excess supply/ unemployment
OR excess demand / labour shortage.
8
Introduction



There is a similar relationship between the
goods market and the labour market in the
simple Keynesian model
Equilibrium is achieved but Ye can be less than,
equal to OR greater than Yf
Equilibrium is achieved when planned output
is enough to meet planned expenditure. Yet,
planned expenditure may not guarantee full
employment, especially in times of depression
9
Assumptions


Investment is assumed to be negatively
related / correlated to the interest rate
I/r = -b
Money supply is determined by the
monetary authority.
10
Assumptions





The level of employment Ye is far below the full
employment level Yf i.e. vast unemployment
 output can be raised by using currently idle
resources without bidding up prices
 price rigidity P’
 no difference between nominal income and
real income
 national income is demand-side determined
11
Investment Function






I= f(r )
I = I’ - br
b>0
I/r = -b
The coefficient b is the interest elasticity of
investment. It measures the responsiveness of
investment I to a change in the interest rate r
c=
i=
s=
m=
t=
kE =
kT =
12
Investment Function
I
The greater is the value of b,
the more interest elastic is the investment function
the greater will be the increase in investment
I
in response to a fall in interest rate r
I2 = I’ - br2
r  I
I1 = I’ - br1
Y
13
Investment Function
v.s. the one on slide 13
the independent variable here is r (y-axis) instead of Y
r
Slope = r/I = -1/b  flatter  r  I
r
I = I’ - br
r
r= 0  I =I’
I
This is only like a mirror image
I= 0  r =I’/b
I
I’
I
I’
14
IS Function




The IS curve is the loci of all the
combinations of r and Y at which the goods
market is in equilibrium, i.e.,
planned output equals planned expenditure /
planned saving equals planned investment /
planned withdrawal equals planned injection

You’ve learnt the method of deriving the
relationship between 2 variables in Micro, like
ICC, PCC, Demand Curve
15
Deriving the IS Function
Output-Expenditure Approach







C = C’ + cYd
I = I’ - br
G = G’
T = T’
if there’s only a lump sum tax
E=C+I+G
E = C’ + cYd + I’ – br + G’
E = C’ – cT’ + I’ + G’ – br + cY
16
Deriving the IS Function
Output-Expenditure Approach






In equilibrium, Y = E
Y = C’ – cT’ + I’ + G’ – br + cY
Y = kE * E’
E = C’+I’+G’–br + cY- ctY
In equilibrium, Y = E
Y =kE * E’
if it’s a proportional tax system
17
Deriving the IS Function
Output-Expenditure Approach



First of all, find out the planned aggregate
expenditure function E which corresponds
to a certain level of interest rate r1
Then, determine the equilibrium national
income Y1.
This combination of r1 and Y1 constitutes
the first locus of the IS function
18
Deriving the IS Function
Output-Expenditure Approach





If r  (from r1  r2)  I  E’  E 
Ye by a multiple k E (Y = k E E’)
It means that when r decreases (may be
due to an increase in money supply)
Y will increase in order to restore
equilibrium in the goods market.
What has happened before Y ?
That’s why r and Y are negatively related.
19
Deriving the IS Function
Output-Expenditure Approach
E2 = C’ - cT’ + I’ - br2 + G’ + cY
E, C, I, G
If r  I  E’ E
If b is large, r  I
E1 = C’ - cT’ + I’ - br1 + G’ + cY
y-intercept = E’ =
Y= kE I’ slope = c
Y1
Y2
when Y = planned E
Y
20
Deriving the IS Function
Output-Expenditure Approach
Slope of the IS curve depends on 2 factors
r
r1
b : If investment is interest elastic r  I
*
kE:If expenditure multiplier is large I Y
*
r2
IS
Y
Y1
Y2
21
Deriving the IS Function
Injection-Withdrawal Approach

C = C’ + cYd
I = I’ - br
G = G’
T = T’
S = S’ + s( Y – T’)
S = S’ – sT’ + sY

S = S’ + sY - stY





if there’s only a lump sum tax
If it’s a proportional tax system
22
Deriving the IS Function
Injection-Withdrawal Approach

In equilibrium, S + T = I + G

S’ – sT’ + sY + T’ = I’ – br + G’




sY = -S’ + sT’ – T’ + I’ + G’ – br
(1-c)Y = C’ + (1-c)T’ – T’ + I’ + G’ – br
(1-c)Y = C’ - cT’ + I’ + G’ – br
Y = kE * E’
[same as slide 17]
23
Deriving the IS Function
Injection-Withdrawal Approach
S+T
S =S’–sT’+sY
I, G, S, T
I2 + G
I2 = I’-b r2
I1 + G
I1 = I’-b r1
T = T’
G = G’
Y1
Y2
when S+T=I+G
Y
The IS function derived here is the same as the one on slide 21
24
4-Quadrant Diagram








Investment Function
Government Expenditure Function
The relationship between r & Injection J
Saving Function
Tax Function
The relationship between Y & Withdrawal W
J=W
[45 - line]
The IS Function
25
Investment Function
refer slide 14
I/r = -b = 
I
I/r = -b
I/r = -b = 0
r
I
Slope = r/I = -1/b
r
r
I
I’
I’
26
Government Expenditure Function
r
As G is independent of r
G = G’
G
G’
27
Injection = I + G
r
G = G’ I= I’- br
At each interest rate r,
J=I+G
J, I, G
28
Saving Function
Y
S = S’ - sT’ + sY
S
29
Tax Function
Y
T’
As tax is independent of Y
T = T’
T
30
Withdrawal W = S + T
Y
T = T’
S = S’ - sT’ + sY
W, S, T
At each income level Y,
W=S+T
31
Equilibrium J = W
J
45
J=W
W
32
4-Quadrant diagram

Quadrant 1 - IS function


Quadrant 2 (slide 28)


relationship between r & J
Quadrant 3 (slide 32)


Equilibrium in goods market
relationship between r & Y
Equilibrium condition: J = W
Quadrant 4 (slide 31)

relationship between Y & W
33
4 - Quadrant Diagram
r
I+G
J
J2
r1
(r1, Y1)
r2
*
J1
*
Y1
IS
(r2, Y2)
Y
Y2
W1
I+G=S+T
W2
W
S+T
34
Simple Algebra of the IS Curve
refer slide 16 & 17



E = C’ - cT’ + I’ + G’ - br + cY
In equilibrium, Y = E
1
Y=
[C’ - cT’ + I’ + G’ - br]
1-c
35
Simple Algebra of the IS Curve
refer slide 21 & 34

r=






C’ - cT’ + I’ + G’
b
-
S
b
Y
r/Y =
C = 100 + 0.8Yd
I = 40 - 10r
G = 20
T = 10
Y=
Y=
36
Slope of the IS Curve
flatter = slope smaller





slope of the IS curve =
 the curve is negatively sloped
The slope of the IS curve shows the responsiveness
of the equilibrium income Y to a change in interest
rate  r.
The greater the interest elasticity of investment b,
the flatter the IS curve
The smaller the MPS OR the greater the MPC, I.e.,
the greater the kE the flatter the IS curve.
37
Slope of the IS Curve
r  I  Y




When interest rate falls, investment will increase.
If investment is interest elastic b = I /r, the
increase in investment will be great.
When investment increase, income will increase
by a multiple.
If expenditure multiplier (s is small or c is large) is
great k E = Y /I , the increase in income will
also be great.
38
Slope of the IS Curve
b =I/r is large  IS flat  slope = s/b small




If investment is interest elastic, given any
reduction in interest rate, the increase in
investment I is large.
This leads to a larger increase in income
Y = k E I
That is, for any reduction in interest rate,
the increase in income is larger
 a flatter IS curve
39
Slope of the IS Curve
b =I/r is large  IS flat  slope = s/b small
r
J=I+G
r1
r2
J
J2
J1
Steeper IS
(r1, Y1)
*
IS
(r2, Y2) Flatter
(r2, Y3)
**
Y1 Y2
Y3
Y
W1
I+G=S+T
W2
W=S+T
40
Relationship between MPC & MPS
C, S
Increase in MPC
Will lead to a
Decrease in MPS
Y
Suppose T = T’
Otherwise MPC is not the slope of the consumption function
41
Relationship between MPC & MPS
An Increase in MPC is the same as a Decrease in MPS
Y
S
42
Slope of the IS Curve
k E = 1/s = Y/E’ is large
 IS flat  slope = s/b small  MPS small




If MPS S/Y is small, given any increase in
income, the increase in saving is small, i.e., the
increase in consumption is large, leading to a
larger multiplying effect on income.
When interest rate decreases, investment will
increase.
If k E is larger, the increase in income is larger
as well
 a flatter IS curve
43
Slope of the IS Curve
k E = 1/s = Y/E’ is large
 IS flat  slope = s/b small  MPS small
r
J=I+G
r1
Steeper IS
(r1, Y1)
*
* *
r2
J
J2
J1
(r2, Y2)
Y1
Flatter IS
Y2 Y3
Y
W1
I+G=S+T
W2
W
W=S+T
44
Slope of the IS Curve
k E = 1/(1 - c) = Y/E’ is large
IS flat slope = (1–




c )/b small  MPC large
If MPC C/Y is large, given any increase in
income, the increase in consumption is large,
leading to a larger multiplying effect on income.
When interest rate decreases, investment will
increase.
If k E is larger, the increase in income is larger as
well
 a flatter IS curve
45
Slope of the IS Curve refer slide 44
k E = 1/(1- c ) = Y/E’ is large
 IS flat  slope= (1– c)/b small  MPC large
r
J=I+G
r1
Steeper IS
(r1, Y1)
*
* *
r2
J
J2
J1
(r2, Y2)
Y1
Flatter IS
Y2 Y3
Y
W1
I+G=S+T
W2
W
W=S+T
46
Shift of the IS Curve
refer slide 36
C’ – cT’ + I’ + G’

r=

Y=






b
C’ – cT’ + I’ + G’
Y/C’
Y/T’
Y/I’
Y/G’
Y/r
r/Y
s
=
=
=
=
=
=
-
s
b
b
s
Y
r
47
Shift of the IS Curve




The X-intercept of the IS curve =
At each interest rate level, a rise in either one of
the autonomous expenditure E’ (i.e., C’, I’, G’) will
shift the IS curve outward by
At each interest rate level, a fall in the
autonomous tax T’ will shift the IS curve outward
by
But this does not mean Y will ultimately increase by
that amount. We have to consider the LM curve as
well. What will be the shape of the LM curve if Y
indeed increase by that amount?
48
Shift of the IS Curve
Relationship between
C
and
S
Increase in
C, S
Autonomous
Consumption
will lead to a
C’
Decrease in
Autonomous Saving
and vice versa
Y
-C’
49
Shift of the IS Curve
Relationship between C and S
Y
S
S = S’ – sT + sY
50
Shift of the IS Curve C’  S’
r
I+G
r1
IS
IS
*
J2
J1
*
*
r2
J
*
Y1
Y2
Y3
Y
W1
I+G=S+T
W2
W
S+T
51
Shift of the IS Curve I’ 
G = G’ I= I’- br
r
At each interest rate r,
J=I+G
J, I, G
I’
52
Shift of the IS Curve I’ 
r
I+G
r1
IS
*
*
r2
J
J2
J1
Y1
Y
Y2
W1
I+G=S+T
W2
W
S+T
53
Shift of the IS Curve G’
G = G’
I= I’- br
r
At each interest rate r,
J=I+G
J, I, G
54
Shift of the IS Curve G’ 
r
I+G
r1
IS
*
*
r2
J
J2
J1
Y1
Y
Y2
W1
I+G=S+T
W2
W
S+T
55
Shift of the IS Curve T’
T by T’ S by -sT’ W by c T’
Y
T = T’
S = S’ - sT’ + sY
W, S, T
At each income level Y,
W=S+T
56
Shift of the IS Curve T’ 
r
I+G
r1
IS
*
*
r2
J
J2
J1
Y1
Y
Y2
W1
I+G=S+T
W2
W
S+T
57
Shift of the IS Curve T’  G’ 
r
I+G
r1
IS
*
*
r2
J
J2
J1
Y1
Y
Y2
W1
I+G=S+T
W2
W
S+T
58
Disequilibrium in the goods market
J=I+G
r1
J
*
J=W
I+G=S+T
*
unplanned inventory
*
Y will
*
Y
J is greater/ smaller than W
unplanned inventory
Y is greater / smaller than AD
G’ r IS
* *
W
W=S+T
59
Two Extreme Cases
r Vertical IS
Slope larger
r
Slope smaller
IS steeper
IS flatter
Slope = -s/b = 
Slope = -s/b = 0
tan 90 = 
tan 0 = 0
Either s = 
Either s = 0
Or b = 0
Or b = 
Horizontal IS
Y
Y
Remember a horizontal demand curve has a Ed of infinity
60