1 ICU winter 2011, Principles of Macroeconomics February 11, 2012
Quiz #2 Solution
Problem 1 (10pts×6) (References: topic 8 and chapter 10) The Keynesian cross model
C = 140 + 0.8(Y – T) (1)
G = 100 (2)
T = 100 (3)
I = 120 (4)
PE = C + I + G (5)
Y = PE (6)
(a) Substitute (1)~(5) into the equilibrium condition (6), arrange terms and solve for the equilibrium level of income as follows:
T from (3) I from (4) G from (2) C from (1)
PE from (5)
Y = [140 + 0.8(Y - 100 )] + 120 + 100
Y-0.8Y = 140-80+120+100 0.2Y = 280 0.2Y = 280/0.2 or Y= Y1=1400
(b) When the actual level of output (which is also income) happens to be Y=1200, the planned expenditure PE= C + I + G = [140+0.8(Y - T)]+I+G = [140+0.8(1200 - 100)] +120+100 = 1240, while the actual expenditure = the level of output = 1200.
Unplanned investment = actual expenditure - planned expenditure = 1200 - 1240 = -40. Thus in this case the actual level of output falls short of the planned expenditure (which is also the aggregate demand), resulting in a decrease in unplaned inventories of firms (i.e. negative unplanned investment). Put differently, in this case the amount of output that firms expected to sell is smaller than the amount they actually sell, so there must be a fall in their inventories.
Actual investment = planned investment + unplanned inventory investment = 120 + (-40) = 80.
Facing a situation of decreasing inventories like this, the action would firms take is to increase output, and because of this the economy moves toward the equilibrium computed in (a)
(c) When there is a 10% tax cut, the new level of tax revenue Tnew = (1-0.1) T =0.9T = 0.9∗100 = 90. Use this new level of taxes to solve for the equilibrium level of income similarly as (a):
Y = [140 + 0.8(Y - 90)] + 120 +100 Y-0.8Y = 140-72+120+100
0.2Y = 288 0.2Y = 288/0.2 or Y=Y1 =1440
2
△Y= Y2 - Y1 = 1440-1400 = 40, and %△Y = △Y/ Y1 =40/1400 =2.9% Thus the tax cut raises income by 40 units or 2.9%.
(d) Draw a graph to explain briefly the effect in (c).
Problem 2 (10pts×4) (References: topic 8 and chapter 10)
With equation (3) changing to (3’), the new model now becomes C = 140 + 0.8(Y – T) (1)
G = 100 (2)
T =0.05Y (3’)
I = 120 (4)
PE = C + I + G (5)
Y = PE (6)
(a) Solve similarly as (a) in problem 1 to obtain the equilibrium level of income: Y = [140 + 0.8(Y – 0.05Y)] + 120 +100 Y-0.8Y+0.04Y = 140+120+100
0.24Y = 360 Y = 360/0.24 or Y= Y3=1500
(b) The government budget deficit = government expenditure - government revenue = government expenditure - tax revenue = G - T. According to (3’), T=0.05Y=0.05Y1 government budget deficit = G - 0.05Y1 = 100 - 0.05∗1500 = 25.
(c) When the government raises the tax rate to 10%, (3’) now becomes
T =0.1Y (3’’)
Use this equation (3’’) and the other unchanged equations to solve similarly as (a) for the new equilibrium level of income:
Y = [140 + 0.8(Y – 0.1Y)] + 120 +100 Y-0.8Y+0.08Y = 140+120+100
0.28Y = 360 Y = 360/0.28 or Y=Y4 = 1285.7
△Y= Y4 – Y3 = 1285.7-1500 = -214.3, and %△Y = △Y/ Y1 =-214.3/1500 =-14.3% Thus the increase in income tax rate reduces income by 214.3 units or 14.3%.
Y E'
E
PE=Y
Explanation:
T ⇒disposal income
⇒C ⇒PE and the PE line shifts upward (PE PE’) ⇒Y (Y Y’) PE
Y Y’
PE PE’
3
(d) According to (3’) & (3’’), the tax revenue before and after the increase of the tax rate are T= 0.05Y3 = 0.05∗1500 = 75, and Tnew = 0.1Y4 =0.1∗1285.7 = 128.6 the government deficit before and after the increase of the tax rate are G - T = 100 - 75 = 25, and G - Tnew = 100 - 128.6 = -28.6 (the negative sign means a surplus!). Thus by raising the income tax rate, the government can turn its budget balance from a deficit of 25 to a surplus of 28.6 (a change of -28.6 - 25 = -53.6)
