is lm model1
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1
IS-LM Model
IS Function
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Outline Introduction Assumptions Investment Function I= f(r) Deriving the IS Function: Income-
Expenditure Approach (Y = E) Deriving the IS Function: Injection-
Withdrawal Approach (I + G = S + T) 4-quadrant diagram
3
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’
4
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
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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
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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
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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
8
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.
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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
10
Assumptions Investment is assumed to be
negatively related / correlated to the interest rate I/r = -b
Money supply is determined by the monetary authority.
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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
12
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 =
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Investment Function
I
Y
I1 = I’ - br1
I2 = I’ - br2
r I
The greater is the value of b,
the more interest elastic is the investment function
the greater will be the increase in investment Iin response to a fall in interest rate r
14
Investment Functionv.s. the one on slide 13the independent variable here is r (y-axis) instead of Y
I
r
I’
Slope = r/I = -1/b flatter r I
r
I
I
r
I’
I = I’ - br
r= 0 I =I’
I= 0 r =I’/b
This is only like a mirror image
15
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
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Deriving the IS FunctionOutput-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
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Deriving the IS FunctionOutput-Expenditure Approach
In equilibrium, Y = E Y = C’ – cT’ + I’ + G’ – br + cY Y = kE * E’
E = C’+I’+G’–br + cY- ctY if it’s a proportional tax system
In equilibrium, Y = E Y =kE * E’
18
Deriving the IS FunctionOutput-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
19
Deriving the IS FunctionOutput-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.
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Deriving the IS FunctionOutput-Expenditure Approach
E1 = C’ - cT’ + I’ - br1 + G’ + cY
y-intercept = E’ =
slope = c
Y
E, C, I, G
Y1
when Y = planned E
If r I E’ E
If b is large, r I
E2 = C’ - cT’ + I’ - br2 + G’ + cY
Y2
Y= kE I’
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Deriving the IS FunctionOutput-Expenditure Approach
r
YIS
r1
r2
Y1 Y2
Slope of the IS curve depends on 2 factors
b : If investment is interest elastic r I
kE:If expenditure multiplier is large I Y*
*
22
Deriving the IS FunctionInjection-Withdrawal Approach
C = C’ + cYd I = I’ - br G = G’ T = T’ if there’s only a lump sum tax
S = S’ + s( Y – T’) S = S’ – sT’ + sY
S = S’ + sY - stY If it’s a proportional tax system
23
Deriving the IS FunctionInjection-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]
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Deriving the IS FunctionInjection-Withdrawal Approach
Y
I, G, S, T
T = T’G = G’
I1 = I’-b r1
S + T
I1 + GI2 = I’-b r2
I2 + G
Y1
when S+T=I+G
Y2
The IS function derived here is the same as the one on slide 21
S =S’–sT’+sY
25
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
26
Investment Functionrefer slide 14
r r
II
I/r = -b =
I’
I/r = -b = 0r
I
I/r = -b
Slope = r/I = -1/b
I’
27
Government Expenditure Function
r
G G’
As G is independent of r
G = G’
28
Injection = I + Gr
J, I, G
I= I’- brG = G’At each interest rate r,
J = I + G
29
Saving Function
S
Y
S = S’ - sT’ + sY
30
Tax Function
Y
T
T’
As tax is independent of Y
T = T’
31
Withdrawal W = S + T
T = T’
At each income level Y,
W = S + T
S = S’ - sT’ + sY
Y
W, S, T
32
Equilibrium J = W
J
W
J = W
45
33
4-Quadrant diagram Quadrant 1 - IS function-
Equilibrium in goods market relationship between r & Y
Quadrant 2 (slide 28) relationship between r & J
Quadrant 3 (slide 32) Equilibrium condition: J = W
Quadrant 4 (slide 31) relationship between Y & W
34
4 - Quadrant Diagram
r
Y
W
J
r1
J1
W1
Y1
*
r2 *Y2J2
W2
IS
I + G
I+G=S+T S + T
(r1, Y1)
(r2, Y2)
35
Simple Algebra of the IS Curverefer slide 16 & 17
E = C’ - cT’ + I’ + G’ - br + cY In equilibrium, Y = E Y = [C’ - cT’ + I’ + G’ - br]
1
1 - c
36
Simple Algebra of the IS Curverefer slide 21 & 34
r = - Y r/Y =
C = 100 + 0.8Yd I = 40 - 10r G = 20 T = 10
Y = Y =
C’ - cT’ + I’ + G’
b
S
b
37
Slope of the IS Curveflatter = 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.
38
Slope of the IS Curver 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.
39
Slope of the IS Curveb =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
40
Slope of the IS Curveb =I/r is large IS flat slope = s/b small
r
YJ
r1
J1
W1
Y1
*
r2 *Y2J2
W2
Steeper ISJ = I + G
I+G=S+TW = S + T
(r1, Y1)
(r2, Y2)
* (r2, Y3)
Flatter IS
Y3
41
Relationship between MPC & MPS
Increase in MPC
Will lead to a
Decrease in MPS
Y
Suppose T = T’
Otherwise MPC is not the slope of the consumption function
C, S
42
Relationship between MPC & MPS
An Increase in MPC is the same as a Decrease in MPS
Y
S
43
Slope of the IS Curvek 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
44
Slope of the IS Curvek E = 1/s = Y/E’ is large
IS flat slope = s/b small MPS smallr
Y
W
J
r1
J1
W1
Y1
*
r2 *Y2J2
W2
Steeper ISJ = I + G
I+G=S+T W = S + T
(r1, Y1)
(r2, Y2)
Y3
* Flatter IS
45
Slope of the IS Curvek 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
46
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
Y
W
J
r1
J1
W1
Y1
*
r2 *Y2J2
W2
Steeper ISJ = I + G
I+G=S+T W = S + T
(r1, Y1)
(r2, Y2)
Y3
* Flatter IS
47
Shift of the IS Curverefer slide 36
r = - Y
Y = - r
Y/C’ = Y/T’ = Y/I’ = Y/G’ = Y/r = r/Y =
C’ – cT’ + I’ + G’
b
s
bC’ – cT’ + I’ + G’
s
b
s
48
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?
49
Shift of the IS CurveRelationship between C and S
Increase in Autonomous Consumption
will lead to a
Decrease in Autonomous Saving and vice versa Y
C, S
C’
-C’
50
Shift of the IS CurveRelationship between C and S
S = S’ – sT + sYS
Y
51
Shift of the IS Curve C’ S’
r
Y
W
J
r1
J1
W1
Y1
*
r2 *Y2J2
W2
ISI + G
I+G=S+TS + T
*
Y3
*
IS
52
Shift of the IS Curve I’
r
J, I, G
I= I’- brG = G’
At each interest rate r,
J = I + G
I’
53
Shift of the IS Curve I’
r
Y
W
J
r1
J1
W1
Y1
*
r2 *Y2J2
W2
ISI + G
I+G=S+T
S + T
54
Shift of the IS Curve G’
r
J, I, G
I= I’- brG = G’
At each interest rate r,
J = I + G
55
Shift of the IS Curve G’
r
Y
W
J
r1
J1
W1
Y1
*
r2 *Y2J2
W2
ISI + G
I+G=S+T
S + T
56
Shift of the IS Curve T’ T by T’ S by -sT’ W by c T’
T = T’
At each income level Y,
W = S + T
S = S’ - sT’ + sY
Y
W, S, T
57
Shift of the IS Curve T’
r
Y
W
J
r1
J1
W1
Y1
*
r2 *Y2J2
W2
ISI + G
I+G=S+T
S + T
58
Shift of the IS Curve T’ G’
r
Y
W
J
r1
J1
W1
Y1
*
r2 *Y2J2
W2
ISI + G
I+G=S+T
S + T
59
Disequilibrium in the goods market
r
Y
W
J
r1*
ISJ = I + G
I+G=S+TW = S + T
J is greater/ smaller than W
unplanned inventory
*
* *
* *
Y is greater / smaller than AD
unplanned inventory
Y will
G’
J = W
60
Two Extreme Cases
rr
Y Y
Slope larger
IS steeper
Slope = -s/b =
tan 90 =
Either s =
Or b = 0
Vertical ISSlope smaller
IS flatter
Slope = -s/b = 0
tan 0 = 0
Either s = 0
Or b =
Horizontal IS
Remember a horizontal demand curve has a Ed of infinity
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