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):

b

NXGIbTaY

P

1

We use this equation to solve for equilibrium

GDP (Y)

AE = C + IP + G + NX

C = 2,000 + 0.6YD

IP = 700G = 500NX = 400T = 2,000

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

b

NXGIbTaY

P

1

000,6$4.0

400,2

6.01

400500700)]000,2)(6.0[(000,2

Y

AE

Y

AE = 2,400 + 0.6Y

2,400

0 6,000

450

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:

bY

1

1

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

100$)5.2)(40(6.01

140

1

1

bGY

AE

Y

AE1 = 2,400 + 0.6Y

2,400

0 6,000

450

1

2

2,440

AE2 = 2,440 + 0.6Y

6,100

•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:

b

b

b

b

b

11

)1(11

1

1

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

b

bTTY

1

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

000,3$)5.1)(000,2(06.1

6.0000,2

Y