IS-LM Model

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 theISfunction • T’ v.s. E’

4. Introduction • In the elementary Keynesian model, investment Iis 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 ifS’ ORs  Y • However, when S’ORs  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 Functionv.s. the one on slide 13the 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 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

17. 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.

20. Deriving the IS FunctionOutput-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 Y2 Y1 when Y = planned E

21. Deriving the IS FunctionOutput-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 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]

24. Deriving the IS FunctionInjection-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 Functionrefer 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 r 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 Y T = T’ S = S’ - sT’ + sY 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- • 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 I + G (r1, Y1) r1 IS * * (r2, Y2) r2 J Y J2 J1 Y1 Y2 W1 S + T W2 I+G=S+T W

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 C’ - cT’ + I’ + G’ b S b • r = - Y r/Y = • C = 100 + 0.8Yd • I = 40 - 10r • G = 20 • T = 10 • Y = • Y =

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 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 CurvekE = 1/s = Y/E’ is large  IS flat  slope = s/b small  MPS small • If MPS S/Yis 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 kE 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 small 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

45. Slope of the IS CurvekE = 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 44k 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 Curverefer 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 CurveRelationship 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 CurveRelationship between C and S Y S S = S’ – sT + sY