1. Chapter 10 Appendices
Outline
Finding equilibrium GDP algebraically.
Finding the effects of a change in autonomous spending.
The tax multiplier.

2. Finding Equilibrium GDP Alegebraically
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]

3. AE = a - bT + bY + IP + G + NX [6]
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]

4. Y(1 – b) = a - bT + IP + G + NX [10]
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]

5. We use this equation to solve for equilibrium GDP (Y)
Now divide both sides of the equation by (1 – b):
We use this equation to solve for equilibrium GDP (Y)

6. AE = C + IP + G + NX
C = 2, YD IP = 700 G = 500 NX = 400 T = 2,000
Example
To solve for equilibrium GDP (Y), use the following formula:

7. AE
AE = 2, Y 
2,400
450
6,000 Y

8. Effect of changes in autonomous expenditure
How do I compute the change in equilibrium GDP resulting from a change in a, IP, G, or NX?

9. Let  denote a change in autonomous expenditure
Let  denote a change in autonomous expenditure. To compute the change in equilibrium GDP:
For example, let  = G = $40. Compute the change in equilibrium GDP:

10. 2 1 The graph AE2 = 2,440 + 0.6Y AE AE1 = 2,400 + 0.6Y 2,440 2,400
450
6,000
6,100 Y

11. YD T C AE The Tax Multiplier
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).
YD T C
AE

12. 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?

13. AE =b YD = b T = (0.6)($1,000) = $600
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

14. 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:

15. To compute the effect of a change in net taxes (T) on equilibrium GDP (Y).
Thus we compute the effect of a $1,000 decrease in net taxes on equilibrium GDP (Y) as follows: