Worked Problems 1. Consider an economy described by the following equations: Y = C + I + G + X (Income identity) C = 100 + .9 Yd (Consumption) with investment I = $200 billion, government spending G = $200 billion, net eScports X = $100 billion, and the tax rate 1' = .2. a. What is the level of income when spending balance occurs? What is the multiplier? b. Suppose government spending increases to $300 billion. What is the new level of income? a . From the income identity, Y = C + I + G + X. Using the consumption function, C = 100 + .9 Yd, and the denition of disposable income, Yd : Y .ZY, Y =100+.9(Y.2Y)+200+200+100 =600+.72Y. Subtract .72Y from both sides of the equation, Y .72Y = 600, factor out Y, Y(1.72) = 600, 281' =600, and multiply both sides by 1/28 = 3.571 to obtain the solution, Y = 3.571(600) = $2,143 billion. By solving for income, we automatically derive the multiplier. It is the number, 3.571, that multiplies the exogenous compo- nents of consumption, investment, government spending, and net exports, 600, to get income. Alternatively, we can use the formula 1 1 1 1 1 - b(1 - t) 1-.9(1 -.2) 1-.72 .28 = 3.571 to find the multiplier. It is important that you learn how to derive the multiplier rather than just memorize the formula, which is correct only if investment, government spending, and net exports all do not depend on income. b. This can be solved in two ways. One is to recalculate Y using the new level of G, 300. Y = 100 + .9(Y -.2Y) + 200 + 300 + 100 = 700 + .72Y = 3.571(700) = $2,500 billion. A second is to use the multiplier to calculate the change in Y, AY = 3.571 AG = 3.571(100) = 357, and then add the change in Y, 357, to the original value of Y, 2,143, to get the new value, $2,500 billion