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Simple Keynesian Model

Simple Keynesian Model. National Income Determination Three-Sector National Income Model. Outline. Three-Sector Model Tax Function T = f (Y) Consumption Function C = f (Yd) Government Expenditure Function G=f(Y) Aggregate Expenditure Function E = f(Y)

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Simple Keynesian Model

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  1. Simple Keynesian Model National Income Determination Three-Sector National Income Model

  2. Outline • Three-Sector Model • Tax Function T = f (Y) • Consumption Function C = f (Yd) • Government Expenditure Function G=f(Y) • Aggregate Expenditure Function E = f(Y) • Output-Expenditure Approach: Equilibrium National Income Ye

  3. Outline • Factors affecting Ye • Expenditure Multipliers k E • Tax Multipliers k T • Balanced-Budget Multipliers k B • Injection-Withdrawal Approach: Equilibrium National Income Ye

  4. Outline • Fiscal Policy (v.s. Monetary Policy) • Recessionary Gap Yf - Ye • Inflationary Gap Ye - Yf • Financing the Government Budget • Automatic Built-in Stabilizers

  5. Three-Sector Model • With the introduction of the government sector (i.e. together with households C, firms I), aggregate expenditure E consists of one more component, government expenditure G. E = C + I+ G • Still, the equilibrium condition is Planned Y = Planned E

  6. Three-Sector Model • Consumption function is positively related to disposable income Yd [slide 37 of 2-sector model], C = f(Yd) C= C’ C= cYd C= C’ + cYd

  7. Three-Sector Model • National Income  Personal Income  Disposable Personal Income • w/ direct income tax Ta and transfer payment Tr • Yd  Y • Yd = Y - Ta + Tr

  8. Three-Sector Model • Transfer payment Tr can be treated as negative tax, T is defined as direct income tax Ta net of transfer payment Tr • T = Ta - Tr • Yd = Y - (Ta - Tr) • Yd = Y - T

  9. Three-Sector Model • The assumptions for the 2-sector Keynesian model are still valid for this 3-sector model [slide 24-25 of 2-sector model]

  10. Tax Function • T = f(Y) • T = T’ • T = tY • T = T’ + tY

  11. Tax Function T = T’ Y-intercept=T’ slope of tangent=0 T = tY Y-intercept=0 slope of tangent=t T = T’ +tY Y-intercept=T’ slope of tangent=t

  12. Tax Function • Autonomous Tax T’ • this is a lump-sum tax which is independent of income level Y • Proportional Income Tax tY • marginal tax rate t is a constant • Progressive Income Tax tY • marginal tax rate t increases • Regressive Income Tax tY • marginal tax rate t decreases

  13. Consumption Function • C = f(Yd) • C = C’ C = C’ • C = cYd C = c(Y - T) • C = C’ + cYd C = C’ + c(Y - T)

  14. Consumption FunctionC = C’ + c(Y - T) • T = T’ C = C’ + c(Y - T’)  C = C’- cT’ + cY  slope of tangent = c • T = tY C = C’ + c(Y - tY)  C = C’ + (c - ct)Y slope of tangent = c - ct • T = T’ + tY C = C’+c[Y-(T’+tY)]C = C’ - cT’ + (c - ct) Y slope of tangent = c - ct

  15. Consumption FunctionC = C’ + c (Y - T’) Y-intercept = C’ - cT’ slope of tangent = c = MPC slope of ray APC  when Y

  16. Consumption FunctionC = C’ + c (Y - tY) Y-intercept = C’ slope of tangent = c - ct = MPC (1-t) slope of ray APC  when Y

  17. Consumption Function C = C’ + c [Y - (T’ + tY)] Y-intercept = C’ -cT’ slope of tangent = c - ct = MPC (1-t) slope of ray APC  when Y

  18. Consumption Function C = C’ - cT’ + (c - ct)Y • C’ OR T’   y-intercept C’ - cT’   C shift upward • t   c(1-t)   C flatter • c   c(1-t)  C steeper  y-intercept C’ - cT’ C shift downward

  19. Government Expenditure Function • G only includes the part of government expenditure spending on goods and services, i.e. transfer payments Tr are excluded. • Usually, G is assumed to be an exogenous / autonomous function • G = G’

  20. Government Expenditure Function Y-intercept = G’ slope of tangent = 0 slope of ray  when Y

  21. Aggregate Expenditure Function • E = C + I + G given C = C’ + cYd T = T’ + tY I = I’ G = G’ • E = C’ + c[Y -(T’+tY)] + I’ + G’ • E = C’ - cT’ + I’+ G’ + (c-ct)Y • E = E’ + c(1-t) Y

  22. Aggregate Expenditure Function • E = C’ - cT’ + I’ + G’ + (c - ct)Y • E = E’ + (c - ct)Y given E’ = C’ - cT’ + I’ + G’ • E’ is the y-intercept of the aggregate expenditure function E • c - ct is the slope of the aggregate expenditure function E

  23. Aggregate Expenditure Function • Derive the aggregate expenditure function E if T = T’ • E = C’- cT’ + I’ + G’ + cY • y-intercept = C’- cT’ + I’ + G’ • slope of tangent = c

  24. Aggregate Expenditure Function • Derive the aggregate expenditure function E if T = tY • E = C’ + I’ + G’ + (c-ct)Y • y-intercept = C’ + I’ + G’ • slope of tangent = (c-ct)

  25. Aggregate Expenditure Function • Derive the aggregate expenditure function E if T = T’ and I = I’ + iY • E = C’- cT’ + I’ + G’ + (c + i)Y • y-intercept = C’- cT’ + I’ + G’ • slope of tangent = (c + i)

  26. Aggregate Expenditure Function • Derive the aggregate expenditure function E if T = tY and I = I’ +iY • E = C’ + I’ + G’ + (c - ct+i )Y • y-intercept = C’ + I’ + G’ • slope of tangent = (c - ct+i )

  27. Aggregate Expenditure Function • Derive the aggregate expenditure function E if T = T’ + tY and I = I’ +iY • E = C’- cT’ + I’ + G’ + (c - ct+i)Y • y-intercept = C’- cT’ + I’ + G’ • slope of tangent = (c - ct+i)

  28. Output-Expenditure Approachw/ T = T’ + tYw/ C = C’ + cYd C 2-Sector C = C’ + cYd = C’ + cY Slope of tangent = c = MPC =C/Yd Slope of tangent = c (1-t) = (1-t)*MPC  MPC C = C’ - cT’ + c(1-t)Y 3-Sector C’ C’ -cT’ Y

  29. I, G, C, E, Y Y=E Y Planned Y = Planned E

  30. Output-Expenditure ApproachI = I’ exogenous function • E = E’ + (c - ct) Y [slide 21-22] • In equilibrium, planned Y = planned E • Y = E’+ (c - ct) Y • (1- c + ct) Y = E’ • Y = E’ E’ = C’ - cT’ + I’ + G’ k E = 1 1 - c + ct 1 1 - c + ct

  31. Output-Expenditure ApproachI= I’+iY endogenous function • E = E’ + (c - ct + i) Y [slide 27] • In equilibrium, planned Y = planned E • Y = E’ + (c - ct + i) Y • (1- c + ct - i) Y = E’ • Y = E’ E’ = C’ - cT’ + I’ + G’ k E = 1 1 - c - i + ct 1 1 - c - i + ct

  32. Output-Expenditure ApproachT = T’ exogenous functionI = I’ + iY • E = E’+ (c + i) Y [slide 25] • In equilibrium, planned Y = planned E • Y = E’+ (c + i) Y • (1 - c - i) Y = E’ • Y = E’ E’ = C’ - cT’ + I’ + G’ k E = 1 1 - c - i 1 1 - c - i

  33. Factors affecting Ye • Ye = k E * E’ • In the Keynesian model, aggregate expenditure E is the determinant of Ye since AS is horizontal and price is rigid. • In equilibrium, planned Y = planned E • E = C’ - cT’ + I’ + G’ + (c - ct + i) Y • Any change to the exogenous variables will cause the aggregate expenditure function to change and hence Ye

  34. Factors affecting Ye • Change in E’ • If C’I’G’  E’  E Y  • If T’C’ - cT’ E’ by- cT’E Y • Change in k E / slope of tangent of E • If c i   E steeper  Y • If c   C’ - cT’  E’  E  Y  • If t   E steeper  Y 

  35. I, G, C, E, Y Y=E Y

  36. I, E, Y I’ E’ = I’  I’ Y Ye = k E E’

  37. G, E, Y G’ Y

  38. C, E, Y C’ Y

  39. C, E, Y T’ C  by -cT’ Y

  40. I, E, Y  i Y

  41. Digression • Differentiation • y = c + mx • differentiate y with respect to x • dy/dx = m

  42. Expenditure Multiplier k E • Y = k E * E’ E’ = C’ - cT’ + I’ + G’ • k E = if I=I’ & T=T’+tY • k E = if I=I’+iY & T=T’+tY • k E = if I=I’+iY & T=T’ 1 1 - c + ct 1 1 - c + ct - i 1 1 - c - i

  43. Expenditure Multiplier k E • Whenever there is a change in the autonomous spending C’I’ or G’ the national income Ye will change by a multiple of k E. • It actually measures the ratio of the change in national income Ye to the change in the autonomous expenditure E’ • Ye/E’ = k E

  44. Tax Multiplier k T • Y = k E * ( C’- cT’ + I’ + G’) • k T = if I=I’ & T=T’+tY • k T = if I=I’+iY & T=T’+tY • k T = if I=I’+iY & T=T’ -c 1 - c + ct -c 1 - c + ct + i -c 1 - c - i

  45. Tax Multiplier k T • Any change in the lump-sum taxT’ will lead to a change in the national income Ye by a multiple of k T in the opposite direction since k T takes on a negative value • Besides, the absolute value of k T is less than the value of k E.

  46. Balanced-Budget Multiplier k B • G’  E’   E   Ye  by k E times • T’  E’   E   Ye  by k T times • If G’  = T’  , the change in Ye can be measured by k B • Y/ G’ = k E • Y/ T’ = k T • k B = k E + k T • k B = + = 1 1 1-c -c 1-c

  47. Balanced-Budget Multiplier k B • The balanced-budget multiplier k B = 1 when t=0 & i=0 • What is the value of k B if t  0 ? • If k B = 1 an increase in government expenditure of $1 which is financed by a $1 increase in the lump-sum income tax, the national income Ye will also increase by $1

  48. Injection-Withdrawal Approach • In a 3-sector model, national income is either consumed, saved or taxed by the government • Y = C + S + T • Given E = C + I + G • In equilibrium, Y = E • C + S + T = C + I + G •  S + T = I + G

  49. Injection-Withdrawal Approach • Since S + T = I + G • S  I • T  G • I > S  T > G • I < S  T < G • (Compare with 2-sector model) • In equilibrium S = I

  50. Injection-Withdrawal Approach • T = T’ + tY • S = -C’ + (1-c) Yd • S = -C’ + (1 - c)[Y -_(T’ + tY)] • S = -C’ + (1 - c)[Y - T’ - tY] • S = -C’ + Y - T’ - tY - cY + cT’ + ctY • S = -C’ + cT’ -T’ - tY + Y - cY + ctY • S = -C’ + cT’ - (T’ + tY) + Y - cY + ctY

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