Chapter 10 Appendices
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Chapter 10 Appendices. Outline Finding equilibrium GDP algebraically. Finding the effects of a change in autonomous spending. The tax multiplier. Finding Equilibrium GDP Alegebraically. We start with the equation for the consumption function:

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Chapter 10 Appendices

  • Outline

  • Finding equilibrium GDP algebraically.

  • Finding the effects of a change in autonomous spending.

  • The tax multiplier.


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]


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 to real GDP. That is:(1 –b):

We use this equation to solve for equilibrium GDP (Y)


AE = C + I to real GDP. That is:P + G + NX

C = 2,000 + 0.6YDIP = 700G = 500NX = 400T = 2,000

Example

To solve for equilibrium GDP (Y), use the following formula:


AE to real GDP. That is:

AE = 2,400 + 0.6Y

2,400

450

0

6,000

Y


Effect of changes in autonomous expenditure to real GDP. That is:

How do I compute the change in equilibrium GDP resulting from a change in a, IP, G, or NX?


Let to real GDP. That is:denote achangein autonomous expenditure. To compute the change in equilibrium GDP:

For example, let  = G = $40. Compute the change in equilibrium GDP:


The graph to real GDP. That is:

2

AE2 = 2,440 + 0.6Y

AE

AE1 = 2,400 + 0.6Y

1

2,440

2,400

450

0

6,000

6,100

Y


The Tax Multiplier to real GDP. That is:

  • A change in autonomous spending (a; IP; G; or NX) impinges on aggregate expenditure (AE) directly.

  • A change in net taxes (T) impinges on AEindirectly, by its affect on disposable income (YD).

YD

T

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 (I to a change in net taxes (T)P), the initial change in AE is given by:

AE = IP = $1,000

But, for the decrease in net taxes, the initialchange 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:


To compute the effect of a change in net taxes ( and negative in signT) on equilibrium GDP (Y).

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


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