Intermediate Macroeconomic Theory II, Fall 2006 Solutions to Problem Set 4 (35 points) 1. (16 points) For all of the questions below, draw the relevant curves. (a) (2 points) Suppose that the government wants to raise investment but keep output constant. Using the IS LM model, what mix of monetary and fiscal policy would achieve this goal? (b) (2 points) Two classical economies differ only in one respect: in economy A money growth and inflation have been low and stable for many years, but in economy B, money growth and inflation have fluctuated erratically between very low and very high levels. When producers in economy B observe changes in the prices of the goods they produce, from past experience they usually attribute these changes to fluctuations in the overall price level, rather than to changes in the relative prices of their goods. Will the slope of the SRAS for economy B be flatter or steeper than the slope of the curve for economy A? What about the slope of the Phillips curve? (Hint: draw supply curves and convince yourself that your answer is right.) Use the IS LM model to determine the short run and long run effects of each of the following on the output (real income), the real interest rate, consumption, investment, the price level, and the real money balances. Draw the relevant diagrams to show how you arrived at your answer. Assume that consumption is not responsive to changes in the real interest rate; and that the economy is initially at the natural level of output. Track the effects for normal cases, i.e., do not bother about vertical/horizontal IS/LM curves. (c) (4 points) The expected rate of inflation rises. (d) (4 points) The introduction of automatic teller machines reduces the demand for money. (e) (4 points) An increase in consumer confidence, as consumers expect their incomes to be higher in the future. (a) To raise investment, we need to reduce the real interest rate. Thus, we need to adopt a policy that shifts the LM curve to the right, i.e., to pursue an expansionary, loose, monetary policy. For a stable IS curve this will translate into a lower real interest rate, higher investment, and higher output. To maintain output at its previous level, we need to contract spending in the economy and move the IS curve to the left this can be achieved, say, by a decrease in government expenditures. Thus, a loose monetary policy and a tight fiscal policy would produce the desired outcome. 1
(b) The slope of the SRAS curve will be much steeper in the economy B, because producers increase their output only by a small amount in response to an increase in price. The economy A s SRAS curve will be flatter, as people are likely to perceive own price changes as changes in the relative prices rather than in the aggregate price level, and thus will respond more strongly to changes in own prices. The short-run Phillips curve will also be steeper in economy B, since unemployment, like output, will not respond much to a change in inflation. In economy A, where (un-)employment and output respond more strongly to the price changes, the short-run Phillips curve will be flatter. (c) (d) Reduction in the money demand shifts the LM curve to the right for a given output, price and money supply, we will have a lower real interest rate. Since the IS is stable, the real interest rate falls, the investment increases, and the national income increases. Compared to the initial equilibrium, in the SR: Y, I, r, C (since the real income increases), P, M/P. In the long run, the price level increases in response to an increase in expenditures, M/P falls, and the LM shifts to the left, back to where it was initially. The real interest rate increases 2
to its original level, investment falls and output reverts to its natural level. Compared to the initial equilibrium, in the LR: Y, I, r, C, P, M/P. (e) An increase in consumer confidence raises planned expenditures and therefore shifts the IS curve to the right, leading to an increase in the real interest rate, and the fall in investment. Compared to the initial equilibrium, in the SR: Y, I, r, C, P, M/P. In the long run, the price level increases, M/P falls and the LM shifts to the left, increasing the real interest rate and depressing investment even further. Compared to the initial equilibrium, in the LR: Y, I, r, C (since autonomous consumption increases for any level of income), P, M/P. 2. (14 points) Consider the following economy: Planned consumption: C = 1, 275 + 0.5 (Y T ) 200 r. Planned investment: I = 900 200 r. Real money demand: (M/P ) d = 0.5 Y 200 i. 3
Expected inflation: π e = 0. (a) (2 points) Suppose T = G = 450 and M = 9, 000. Find an equation for the IS curve. (Hint: Set the national savings to be equal to the national investment, and solve for the relationship between r and Y, given P. Express the IS curve as Y being a function of r.) (b) (2 points) Find an equation for the LM curve. (Hint: Set real money supply and real money demand equal, and solve for the relationship between r and Y, for a given P. Express the LM curve as Y being a function of r.) (c) (2 points) Find an equation for the aggregate demand curve. (Hint: Use the IS and LM equations to find a relationship between Y and P. Express the AD curve as Y being a function of P.) (d) (2 points) If the full employment output Y = 4, 600, what are the equilibrium values for the level of aggregate prices, output, consumption, investment, and the real interest rate? (e) (2 points) Suppose that the money supply falls to M = 4, 500. What is the equation for the AD now? (Express the AD curve as Y being a function of P.) (f) (2 points) Repeat (2d) given than money supply falls to 4,500. (g) (2 points) Explain what you find using the IS LM diagram, and the LRAS curves drawn in the r Y space. Argue whether LM, IS shifted or not and why. (a) r = i π e = i. S = Y C G = Y 1, 275 0.5 (Y T )+200 r G = 0.5 Y +200 r 1, 275+0.5 T G. Equating S to I, gives the following equation for the IS curve: 0.5 Y + 200 r 1, 275 + 0.5 T G = 900 200 r, or 0.5 Y = 400 r + 1, 275 0.5 T + G + 900. I.e., the IS curve is: Y = 800 r T + 2 G + 4, 350. Plugging G, and T, the IS curve is: Y = 800 r + 4, 800. (b) Equating the demand for real money balances to supply, the LM curve is: 9, 000/P = 0.5 Y 200 i = 0.5 Y 200 r, or Y = 18, 000/P + 400 r. (c) From the IS curve, 400 r = 1/2 Y + 2, 400. Plugging 400 r into the LM curve gives the AD: Y = 18, 000/P 1/2 Y + 2, 400, or 18, 000/P + 2, 400 = 3/2 Y, or Y = 12, 000/P + 1, 600. (d) Since our supply curve is Y = 4, 600, equilibrium output is 4,600. Equilibrium level of P is obtained by setting the aggregate quantity of output demanded equal to the quantity supplied: 12, 000/P = 3, 000, and so P = 4. The real interest rate can be obtained from the IS curve: r = 1/4. C = 1, 275 + 0.5 (4, 600 450) 200 (1/4) = 3, 300. I = 900 200 (1/4) = 850. (e) When the money supply falls to 4,500, the LM curve will be Y = 9, 000/P + 400 r, and the new AD curve is: Y = 6, 000/P + 1, 600. (f) Output stays at Y = 4, 600. From the AD curve, P = 6, 000/3, 000 = 2. From the IS curve, r = 1/4. C=3,300. I=850. 4
Money is neutral here, i.e. it does not affect the real variables. The fall in the money supply just causes the fall in the aggregate price level. (g) Your IS LM curve, and the LRAS curve should intersect at the point where r = 1/4, and Y = Y = 4, 600. For a given price level and the real interest rate, there is a shift in the LM curve to the right. But, since the price level drops to P = 2, the real money balances fall to their previous level, and the LM curve shifts back to its original position. Thus, in the end, changes in the money supply lead only to changes in the aggregate price level the monetary neutrality result of classical economics. 3. (5 points) Consider an economy in long-run equilibrium with an inflation rate π of 12% (=0.12) per year and a natural unemployment rate u n of 6% (=0.06). The Phillips curve is π = π e 2(u u n ). Assume that Okun s law holds so that a 1% increase in the unemployment rate maintained for one year reduces GDP by 2% of full-employment output. Consider a four-year disinflation according to the following table: Year 1 2 3 4 π 0.08 0.04 0.04 0.04 π e 0.10 0.08 0.06 0.04 (a) (2 points) What is the cyclical unemployment rate in each of the four years? (b) (2 points) By what percentage does output fall short of full-employment output each year? (c) (1 point) What is the sacrifice ratio for this disinflation? Use equations: u = 0.06 0.5 (π π e ). Output shortfall=2.0 (u 0.06). (a) (b) (c) Year π π e u u u n Output shortfall 1 0.08 0.10 0.07 0.01 0.02 2 0.04 0.08 0.08 0.02 0.04 3 0.04 0.06 0.07 0.01 0.02 4 0.04 0.04 0.06 0.0 0.0 5