Transcript (AE) is given by

Chapter 10 Appendices
Outline
•
Finding equilibrium GDP algebraically.
•
Finding the effects of a change in
autonomous spending.
•
The tax multiplier.
We start with the equation for the consumption
function:
C = a + bYD
[1]
Remember that disposable income (YD) is the
difference between real GDP (Y) and net taxes (T):
YD = Y – T
[2]
Now substitute [2] into [1]:
C = a + b(Y – T)
[3]
Now rearrange [3]:
C = (a - bT) + bY
[4]
[4] is the equation for the consumption-income line. Notice
that the intercept of the line is given by (a - bT) and the
slope of the consumption-income line is given by b.
The equation for aggregate expenditure (AE) is given by:
AE = C + IP + G + NX
[5]
Now substitute [4] into [5]:
AE = a - bT + bY + IP + G + NX
[6]
We know that, in equilibrium, aggregate
expenditure is equal to real GDP. That is:
Y = AE
[7]
Now substitute [6] into [7]
Y = a - bT + bY + IP + G + NX
[8]
Now, rearrange [8] to obtain:
Y – bY = a - bT + IP + G + NX
[9]
Now, rearrange [9] to obtain:
Y(1 – b) = a - bT + IP + G + NX
[10]
Now divide both sides of the equation by (1 – b):
a  bT  I P  G  NX
Y
1 b
We use this
equation to solve
for equilibrium
GDP (Y)
AE = C + IP + G + NX
C = 2,000 + 0.6YD
IP = 700
G = 500
NX = 400
T = 2,000
To solve for equilibrium GDP (Y), use the
following formula:
a  bT  I P  G  NX
Y
1 b
2,000  [(0.6)( 2,000)]  700  500  400 2,400
Y

 $6,000
1  0.6
0.4
AE

AE = 2,400 + 0.6Y
2,400
450
0
6,000
Y
How do I compute the
change in equilibrium
GDP resulting from a
change in a, IP, G, or
NX?
Let  denote a change in autonomous expenditure. To
compute the change in equilibrium GDP:
1
Y   
1 b
For example, let  = G = $40. Compute the
change in equilibrium GDP:
1
1
Y  G 
 40 
 ( 40)( 2.5)  $100
1 b
1  0.6
2
AE
AE2 = 2,440 + 0.6Y
AE1 = 2,400 + 0.6Y
1
2,440
2,400
450
0
6,000 6,100
Y
•A change in autonomous spending (a; IP; G; or
NX) impinges on aggregate expenditure (AE)
directly.
•A change in net taxes (T) impinges on AE indirectly,
by its affect on disposable income (YD).
T
YD
C
AE
Initial impact of a change in autonomous spending
compared to a change in net taxes (T)
Will a $1,000
decrease in T have
the same initial
effect as a $1,000
increase in IP?
For the increase in the planned investment (IP), the initial
change in AE is given by:
AE = IP = $1,000
But, for the decrease in net taxes, the initial change in AE
is given by:
AE =b YD = b T = (0.6)($1,000) = $600
Hence, the impact of a
change in net taxes is
not as great as a change
in a, IP, G, or NX
The tax multiplier is 1.0 less than the spending
multiplier, and negative in sign
Let  denote the tax multiplier. Thus we can say:
 = - (spending multiplier – 1).
Because the multiplier is equal to 1/(1 – b ), we can
substitute to get:
1  (1  b)
b
 1

 
 1  

1 b
1 b
1  b

To compute the effect of a change in net taxes (T)
on equilibrium GDP (Y).
b
Y  T    T 
1 b
Thus we compute the effect of a $1,000 decrease in
net taxes on equilibrium GDP (Y) as follows:
 0.6
Y  2,000 
 (2,000)( 1.5)  $3,000
1  .06