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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 G NX Y 1 b P 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