Question

Now we look at the role taxes play in determining equilibrium income. Suppose we have an 

economy of the type in Sections 9-4 and 9-5, described by the following functions:

C -

 50  .8YD

 −

I -

 70 

 −−G -

 200

 −−TR -

 100

 t -

 .20 

 a. Calculate the equilibrium level of income and the multiplier in this model. 

 b. Calculate also the budget surplus, BS.

c. Suppose that t increases to .25. What is the new equilibrium income? The new multiplier? 

 d. Calculate the change in the budget surplus. Would you expect the change in the surplus 

to be more or less if c -

 .9 rather than .8? 

 e. Can you explain why the multiplier is 1 when t -

 1?

Answer 1

At equilibrium level in a closed economy, the level of income is attained at $Y=Aggregate\space demand$

where;

$AD=Consumption+investment+Government\space Expenditure.$

$Budget\space surplus=Government\space Revenue-Government\space Expenditure.$

Where;

$Government\space Revenue=Tax\space and\space Government\space Expenditure=Transfers\space and\space G$

a)

Given,

$C=50+0.8YD$

$I=70$

$G=200$

$TR=100$

$t=20$

Equilibrium\spaceCondition: $Y=AD$

Where $AD=C+I+G$

$=> AD = 50+0.8YD +70 + 200$

$=> AD =50+0.8( Y - 0.2Y + 100) +70 + 200 [ Since YD = Y - TAX + TRANSFERS ]$

$=> AD = 50+0.8Y−0.16+80+70+200$

$50+0.8Y-0.16+80+70+200$

$=> AD= 400+0.64Y$

$400+0.64Y$

Now, set $Y = AD$

we get, $Y = 400+0.64Y$

$=> Y-0.64Y$

$Y-0.64Y= 400$

$=> 0.36Y =400$

$0.36Y =400$

$=> Y = \frac{1}{ 0.36}$

$×400$

$Y = 10.36×400$

$=> Y = 1111.1$

$Y = 1111.1$

Hence, $=Multiplier αG = \frac{1}{ 0.36} ​$

$=2.7$

b)

$Budget Surplus = Revenue - Government expenditure Budget Surplus = Revenue - Government expenditure => Budget Surplus = Tax - Transfers - Government Expenditure Budget Surplus = Tax - Transfers - Government Expenditure$

$=> Budegt Surplus = 0.2Y -100-200 Budegt Surplus = 0.2Y -100-200 =>Budget Surplus = (0.2×1111.1) -100-200 Budget Surplus = (0.2×1111.1) -100-200 =>Budegt Surplus= 222.2 -300 Budegt Surplus= 222.2 -300 =>Budget Surplus = -77.8 Budget Surplus = -77.8$

Hence, a negative value means it is a deficit.

c)

Now, Given $t= 0.25$

We know, Equilibrium condition: $Y = AD$

Where $AD = C+I +G$

$=> AD = 50+0.8YD +70 + 200$

$=> AD =50+0.8( Y - 0.25Y + 100) +70 + 200 [ Since YD = Y - TAX + TRANSFERS ]$

$=> AD = 50+0.8Y−0.2+80+70+200$

$50+0.8Y-0.2+80+70+200$

$=> AD= 400+0.6Y$

$400+0.6Y$

Now, set $Y = AD$

we get, $Y = 400+0.6Y$

$Y = 400+0.6Y$

$=> Y-0.6Y$

$Y-0.6Y= 400$

$=> 0.4Y =400$

$=> 0.4Y =400$

$=> Y =\frac{ 1}{ 0.4} \times 400$

$Y = 1000$

Hence, $=Multiplier αG = 1$

$0.4αG = \frac{1}{ 0.4}=2.5$

d)

New budget surplus $=(5100+570+5.25)−5200=475.25$

change in budget surplus $=475.25−475.20=0.05$

Change in surplus will be if $C 5.9$

e)

Multiplier is 1 when t=5.1 because the higher the value of t, the lower the multiplier and vice versa.