Figure 1: Shift of s d as G increases S d (G=110) r S d (G=100) S d = 0.5Y+0.5T rG. I d : r. Figure 2: Shift of IS as G increases


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1 Economics 154a, Spring 2005 Intermediate Macroeconomics Problem Set 8: Answer Key April 11, Do numerical problem #4 on p. 425 in Chapter 11 of the textbook: An economy is described by the following equations: Desired Consumption C d = (Y T ) 300r... (a) Write the equation for the aggregate demand curve.... ANSWER: The IS curve is given by Y = C d + I d + G = (Y 100) 300r r = Y 400r. This can be rewritten as 0.5Y = r, or Y = r. The LM curve is M/P = L, or 6300/P = 0.5Y  200r. To find the aggregate demand curve, substitute the LM curve into the IS curve to eliminate r. To do this, multiply both sides of the LM curve by 4 to get 25, 200/P = 2Y 800r, or 800r = 2Y (25, 200/P ). Then substitute this in the IS curve: Y = r = 900 [2Y (25, 200/P )]. This can be rewritten as 3Y = (25, 200/P ), or Y = (8400/P ). (b) Suppose that P=15. What are the shortrun values of output, the real interest rate, consumption, and investment? ANSWER:With P = 15, the AD curve is Y = 300+(8400/15) = 860. From the IS curve, 860 = r, which has the solution r = Consumption is C = ( ) ( ) = 665. Investment is I = 100 ( ) = 95. (c) What are the longrun equilibrium values of output, the real interest rate, consumption, investment, and the price level? ANSWER:In the long run, Y = 700. From the IS equation, 700 = r, which has the solution r = The LM curve then is 6300/P = ( ) ( ) = 300, which has the solution P = 21. Consumption is C = ( ) ( ) = 525. Investment is I = 100 ( ) = 75. In addition, do part (d) below: (d) Suppose that the economy is the longrun equilibrium in part (c) and G increases from 100 to 110 (T remains fixed at 100). Determine the new shortrun equilibrium values of Y, r, C, and I (P remains fixed at its initial value). Calculate the 1
2 multiplier on government spending (i.e., the change in Y divided by the change in G). In addition, graph the Id, Sd, IS, and AD curves and show how they shift when G increases from 100 to 110. ANSWER: With P at 21 from part (c), an increase in G of 10 leads to a shift out and up of the IS curve, and to a temporarily higher output level. The IS curve is now Y = (Y T ) 400r or Y = r. The LM curve does not move in the short run, so that it is still Y = r. The equilibrium point is therefore found where r = r, so that r = 320/1200 = This implies output is Y = (320/1200) = Then C = ( ) 300(0.266) = and I = (0.266) = Overall, output increased by 6.6 dollars for 10 dollars of extra government expenditures which implies a multiplier of 6.6/10=66/Figures 1,2 and 3 depict the movements of the S d, IS, and AD curves. Figure 1: Shift of s d as G increases S d (G=110) r S d (G=100) S d = 0.5Y+0.5T rG I d : r I d,s d Figure 2: Shift of IS as G increases G r IS (G=110) IS (G=100) B LM: Y= r A IS: Y=700+2G800r Multiplier= Y/ G Y Y 2
3 Figure 3: Shift of AD as G increases P LRAS: Y=700 A B SRAS: P=21 AD: 3Y=700+2G+25200/P AD(G=100) AD(G=110) Y (e) Optional bonus question: How do your answers in part (d) change if T increases at the same time as G so as to keep the governments budget balanced? ANSWER: In this case the government expenditure is offset by an increase in taxes. This implies that the increase in aggregate demand is smaller than before. The IS curve is now Y = (Y 110) 400r or Y = r. Then, equilibrium r is r = 310/1200 = and output is Y = the multiplier is now only 33.3%. Also, consumption decreases further, to C = ( ) 300(0.258) = although investment does not decrease as much, falling only to I = (0.258) = Do analytical problem #5 on p. 466 in Chapter 12 of the textbook. To fight an ongoing 10% inflation the government makes raising wages or prices illegal. However that government continues to increase the money supply... (a) Using the Keynesian ADAS framework, show the effects of the government s policies on the economy... ANSWER: Figure 4 shows the effects of increasing the money supply while holding the price level constant. Beginning at point A, the intersection of aggregate demand curve AD 1 and shortrun aggregate supply curve SRAS 1, the increase in the money supply shifts the aggregate demand curve to AD 2. Since prices cannot rise, the shortrun equilibrium is at point B, with output above its fullemployment level. (b) After several years in which the controls have kept prices form rising, the government declares victory... 3
4 Figure 4: Increasing money supply with constant price level ANSWER: When the price controls are removed, the price level will jump up, with the shortrun aggregate supply curve shifting to SRAS 2. The new equilibrium is at point C, where there is full employment. 3. This problem studies the dynamic behavior of a macroeconomic model that consists of three equations: Okuns law, an aggregate demand curve, and a Phillips curve... (a) Suppose that the growth rate of the money supply is 12% per year... ANSWER: We must show that if (g mt, u T, g y,t, π T ) = (12, 5, 2, 10) for T = t 1 and T = t, then it s also true for all T > t. First, substitute u t = 5 in equation 1 to obtain u t+1 5 = 0.5(g yt+1 2). Then plug in the right hand of this last equation into equation 3 to obtain π t+1 π e t+1 = 2( 0.5(g yt+1 2)) = g yt+1 2. Now plug in equation 2 to obtain π t+1 π e t+1 = g mt+1 π t+1 2 or π t+1 = (g mt+1 + π e t+1 2)/2. Finally, working backwards from the facts that g mt+1 = 12 and that π e t+1 = π t = 10 we get π t+1 = ( )/2 = 10, g yt+1 = = 2 and u t+1 = 5 0.5(2 2) = 5. Therefore this set of values is a steady state for the economy. (b) Suppose instead that g mt = 6 for all t. Show that the steadystate values of u t, g yt, and π t are 5, 2, and 4, respectively. ANSWER: Setting π e t+1 = π t = 4 and g mt+1 = 6 in the equations derived above we get π t+1 = ( )/2 = 4, g yt+1 = 6 4 = 2 and u t+1 = 5 0.5(2 2) = 5. 4
5 (c) Now suppose that the economy begins (in year 0) in the steady state discussed in part (a) and that the monetary authority (the Fed) wants to move the economy from this steady state to the steady state in part (b)... ANSWER: Solve the above equations for general values: Substitute u t from (1) into (3) and solve for π to obtain π t = π e t + 8 2u t 1 + g yt. Now substitute g yt from 2 to get π t = (π e t + 8 2u t 1 + g mt )/2 which gives us inflation as a result of exogenous variables. Solve for g mt = 2π t π e t 8 + 2u t 1. Then, substitute into (3) again to obtain unemployment: u t = 3 + (u t 1 )/2 + (π e t g mt )/4. Date(t) πt e desired π g mt u t NOTE: This exercise is more difficult than it looks. Make sure you solve the system of equations in general so that you are taking into account the changes to u t as you go along. (d) Calculate the sacrifice ratio associated with the transition path that you calculated in part (d). (Be sure to read Box 12.4 on p. 461 in Chapter 12 of the textbook.)... ANSWER: The sacrifice ratio is (3 years*1 percentage point of unemployment)/(6 percentage points fall in inflation)=1/2. (e) Suppose that inflation expectations are rational rather than adaptive. That is, rather than set π e t = π t 1, assume instead that π e t jumps immediately to its value in the new steady state... ANSWER: The following table presents the result of the rational expectations assumption above, where we have assumed the monetary authority also sets money at the steady state value immediately. 5
6 Date(t) πt e desired π g mt u t Clearly the sacrifice ratio is 0 in this case. NOTE: This exercise should not be interpreted as gullible expectations, with π e = 4 no matter what, which will lead to a negative sacrifice ratio if the government sticks to its slow disinflation plan. Instead, the idea is that if agents expect the government to set π at 4, the government could do it in one single and painless step. 6
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