1. IS-LM Model
IS Function

2. 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

5. 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

6. 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

7. 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.

9. 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.

11. 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 =

13. Investment Function 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 I
I2 = I’ - br2
r  I
I1 = I’ - br1 Y

14. Investment Function v. s
Investment Function v.s. the one on slide 13 the independent variable here is r (y-axis) instead of Y r r
Slope = r/I = -1/b  flatter  r  I
I = I’ - br
r= 0  I =I’
I= 0  r =I’/b r
This is only like a mirror image I I I I’ I’

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

16. 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

17. 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 if it’s a proportional tax system
Y =kE * E’

18. 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

19. 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.

20. 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’ =
slope = c
Y= kE I’ Y Y1
when Y = planned E Y2

21. Deriving the IS Function Output-Expenditure Approach
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 r r1 * r2 * IS Y Y1 Y2

22. Deriving the IS Function Injection-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 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]

24. Deriving the IS Function Injection-Withdrawal Approach
S + T
I, G, S, T
S =S’–sT’+sY
I2 + G
I2 = I’-b r2
I1 + G
I1 = I’-b r1
T = T’
G = G’ Y1
when S+T=I+G Y2 Y
The IS function derived here is the same as the one on slide 21

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 Function refer slide 14
I/r = -b
Slope = r/I = -1/b
I/r = -b = 
I/r = -b = 0 r r r I I I I’ I’

27. Government Expenditure Function
As G is independent of r
G = G’ G G’

28. Injection = I + G r G = G’ I= I’- br At each interest rate r,
J = I + G
J, I, G

29. Saving Function Y
S = S’ - sT’ + sY S

30. Tax Function Y T’
As tax is independent of Y
T = T’ T

31. Withdrawal W = S + T T = T’ S = S’ - sT’ + sY W = S + T Y
At each income level Y,
W = S + T
W, S, T

32. Equilibrium J = W J
45
J = W W

33. 4-Quadrant diagram Quadrant 1 - IS function- Quadrant 2 (slide 28)
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 * * IS r I + G (r1, Y1) r1 (r2, Y2) r2 J Y J2 J1W1
S + T W2
I+G=S+T W

35. Simple Algebra of the IS Curve refer 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 Curve refer slide 21 & 34
C’ - cT’ + I’ + G’ b S b
r = - Y r/Y =
C = Yd
I = r
G = 20
T = 10
Y =

37. Slope of the IS Curve flatter = slope smaller
 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 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.

39. 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

40. Slope of the IS Curve b =I/r is large  IS flat  slope = s/b small
Steeper IS
J = I + G
(r1, Y1) r1 *
Flatter IS
(r2, Y2)
(r2, Y3) * * r2 J Y J2 J1 Y1 Y2 Y3 W1
W = S + T W2
I+G=S+T

41. Relationship between MPC & MPS
Increase in MPC
Will lead to a
Decrease in MPS
C, S Y
Suppose T = T’
Otherwise MPC is not the slope of the consumption function

42. Relationship between MPC & MPS
An Increase in MPC is the same as a Decrease in MPS Y S

43. 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

44. Slope of the IS Curve k E = 1/s = Y/E’ is large  IS flat  slope = s/b small  MPS small
Steeper IS
J = I + G
(r1, Y1) r1 *
(r2, Y2)
Flatter IS * * r2 J Y J2 J1 Y1 Y2 Y3 W1
W = S + T W2
I+G=S+T W

45. 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

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
Steeper IS
J = I + G
(r1, Y1) r1 *
(r2, Y2)
Flatter IS * * r2 J Y J2 J1 Y1 Y2 Y3 W1
W = S + T W2
I+G=S+T W

47. Shift of the IS Curve refer slide 36
C’ – cT’ + I’ + G’ b s b
r = Y
Y = r
Y/C’ =
Y/T’ =
Y/I’ =
Y/G’ =
Y/r =
r/Y =
C’ – 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 Curve Relationship between C and S
Increase in Autonomous Consumption
will lead to a
Decrease in Autonomous Saving and vice versa
C, S
C’ Y
-C’

50. Shift of the IS Curve Relationship between C and SY S
S = S’ – sT + sY

51. Shift of the IS Curve C’  S’
I + G r1 * * * * r2 J Y J2 J1 Y1 Y2 Y3 W1 W2
I+G=S+T W
S + T

52. Shift of the IS Curve I’ 
G = G’
I= I’- br r
At each interest rate r,
J = I + G I’
J, I, G

53. Shift of the IS Curve I’ 
I + G IS r1 * * r2 J Y J2 J1 Y1 Y2 W1
S + T W2
I+G=S+T W

54. Shift of the IS Curve G’
I= I’- br
G = G’ r
At each interest rate r,
J = I + G
J, I, G

55. Shift of the IS Curve G’ 
I + G IS r1 * * r2 J Y J2 J1 Y1 Y2 W1
S + T W2
I+G=S+T W

56. Shift of the IS Curve T’ T by T’ S by -sT’ W by c T’
S = S’ - sT’ + sY
At each income level Y,
W = S + T
W, S, T

57. Shift of the IS Curve T’ 
I + G IS r1 * * r2 J Y J2 J1 Y1 Y2 W1
S + T W2
I+G=S+T W

58. Shift of the IS Curve T’  G’ 
I + G IS r1 * * r2 J Y J2 J1 Y1 Y2 W1
S + T W2
I+G=S+T W

59. Disequilibrium in the goods market
Y is greater / smaller than AD
unplanned inventory
Y will IS r G’
J = I + G r1 * * * J Y
J is greater/ smaller than W
unplanned inventory * * *
J = W
W = S + T
I+G=S+T W

60. Two Extreme Cases Slope larger IS steeper Slope = -s/b =  tan 90 = 
Either s = 
Or b = 0
Slope smaller
IS flatter
Slope = -s/b = 0
tan 0 = 0
Either s = 0
Or b =  r
Vertical IS r
Horizontal IS Y Y
Remember a horizontal demand curve has a Ed of infinity