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]

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]

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]

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)

AE = C + IP + G + NX C = 2,000 + 0.6YD IP = 700 G = 500 NX = 400 T = 2,000 Example To solve for equilibrium GDP (Y), use the following formula:

AE AE = 2,400 + 0.6Y  2,400 450 6,000 Y

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?

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:

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

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

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?

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

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