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Economics 514 Macroeconomic Analysis Final Exam December 16, 2004 1. (15 points) There is an economy with a short-run aggregate supply curve, so that when output is above long run potential, inflation will be above the inflation plans of the corporate sector, πt – π = θ∙ (qt – q) A government likes price stability and wants output to be near some target level, q* which is greater than the potential output, q. The government chooses an inflation policy function πP to minimize min a×( πP)2 + b×(qt –q* )2 subject to πP – π = θ∙ (qt – q). a. Draw the first order condition that describes inflation policy, πP as a linear function of the inflation plans of the corporate sector π . Explain the intercept and the slope of the line. πP π b. Discuss the equilibrium inflation rate when the corporate sector has rational expectations. Why is it difficult to build credibility for a zero inflation policy? [Note: I do not expect an explicit solution. I am interested in the intuitive explanation using graphs and words] Under rational expectations, planned inflation must equal inflation policy. This means that the corporate sector must expect that the government will raise inflation policy to that point where inflation becomes so costly that they wouldn’t want to push it up farther even to achieve their high output goal. Any lower expectations would lead to systematic errors in under-predicting inflation by the corporate sector. Since inflation policy has little effect on output, the government might like lower inflation. But the government has a difficult time achieving credibility for a low inflation policy, because if the public ever did expect low inflation, the government would no longer want to follow a low inflation policy. 2. (25 points) In an economy, the demand for goods is set by Mt ∙Vt = Pt ∙Qt . In this economy, money supply is always constant, so we can write the demand curve with actual inflation as a function of velocity growth, νt, and the output gap, 1) πt = νt – (qt – q). On the supply side, firms set an inflation plan, π, actual inflation increases as a function of costs which are determined by the output gap and an oil price shock, φt. 2) πt = π + θ∙ (qt – q) + φt Finally inflation plans are set according to rational expectations. 3) π = REt-1[πt ] Solve for the effect of demand shocks (νt) and supply shocks (φt). Hint: Follow the following steps. i. Solve for π remembering that REt-1[ t ] = 0; ii. Plug this formula into equation 2) and solve for qt and πt. a. Assume that there are no oil shocks, φt = 0. Velocity shocks are a random walk, i.e. they are a sum of yesterdays output and an unpredictable shock. 4) νt = νt-1 + tV . REt-1[νt] = νt-1 Solve for qt as a function of q, νt-1 , and tV . First equation 2 along with model consistent expectations suggests REt-1[πt ]= REt-1[π ] + θ∙ (REt-1[qt] – q) implies REt-1[πt ]= REt-1[πt ]+ θ∙ (REt-1[qt] – q) implies REt-1[qt] = q. Equation 1 and model consistent expectations implies REt-1[πt ]= REt-1[νt ]-( REt-1[qt] -q) = REt-1[νt] = νt-1 from equation 4. Substitute into equation 2) to get πt = = νt-1 + θ∙ (qt – q) . Combine with equation 1 to get νt-1 + θ∙ (qt – q) = νt – (qt – q) which implies (1+ θ)∙ (qt – q)= νt - νt-1 which implies tV qt q 1 b. Assume that there are no demand shocks, νt = 0. Oil shocks follow a random walk. 5) φt = φt-1 + tO . REt-1[φt] = φt-1 Solve for qt as a function of q, φt-1, and tO . From equation 1, we write πt = – (qt – q). So, REt-1[πt ]= -RE[(qt – q)] From equation2), we get πt = π + θ∙ (qt – q) + φt, so REt-1[πt ]= REt-1[πt ]+ θ∙RE[(qt – q)]+ RE[ φt] implying RE[(qt – q)] = - φt-1/ θ = REt-1[πt ] We then write πt = – (qt – q) = - φt-1/ θ + θ∙ (qt – q) + φt-1 + tO . This means that qt q tO 1 t 1 . 1 1 c. Do supply or demand shocks have persistent effects on output under rational expectations? Only supply shocks have persistent effects. 3. (25 points) Construct a small model of the economy that includes A) a Planned expenditure curve; B) a Taylor rule; and C) an expectations augmented Philips curve A. The planned expenditure curve has two parts. The first part sets domestic demand as a negative function of the real interest rate and an exogenous demand shock, αt. The second part includes net exports. yt t b rt nxt Net exports are a function of the real exchange rate, st. The real exchange rate adjusts so that net exports are equal to outward, capital flows that are themselves a negative function of the gap between foreign and domestic interest rates. nxt f (st 1) cft g (r f rt ) So the expenditure curve can be written as yt t b rt g (r f rt ) t g r f (b g )rt B. The Taylor rule sets the real interest rate as an increasing function of the inflation rate rt r d t C. The SRAS curve sets inflation acceleration as a function of the output gap. t t 1 yt y Calibrate the parameters of the model equal to r = .1, rf = .1, q = 1, b = 2 , d = .5, f = 2, g = 2 and θ = .5. t 1 = 1.2 and that at time t-1, the economy is operating at potential output, i.e. t 1 t 2 and yt 1 y 1 . Solve for the inflation a. Assume that rate, real interest rate, and real exchange rate in this economy. Combine the expenditure curve and the Taylor rule, yt t g r f (b g )rt t g r f (b g )r d (b g ) t to get the aggregate demand curve. Fill in the numbers to get yt t .2 2 t We can write If t 1 = 1.2 and yt 1 y 1 then 1 1.2 .2 2 t 1 t 1 0 . Then rt 1 r d 0 r .1 . Then nxt 1 g (.1 .1) 0 and 0 f ( st 1 1) st 1 1 b. Now, assume that at time t, an increase in fiscal spending leads to a shift out in domestic demand and t = 1.5. Solve for the new level of output, real interest rate, inflation, and real exchange rate. Does the increase in government spending cause an increase or decrease in net exports? If t 1 0 then the supply curve is t .5 yt 1 .5 yt .5 , so solving for yt we write Consider an alternate, extremely globalized economy with very strong yt t .2 2 t 1.5 .2 2 .5 yt 1 2.3 yt yt 1.15 t .075 rt .1375 nxt g (r f rt ) .075 st .9625 c. capital movements, g = 10. Now, solve for the increase in output, qt that occurs when demand shifts from t 1 = 1.2 to t = 1.5. Is a globalized economy, more or less stable in terms of output? Not covered this year. Midterm 2 Economics 514 Macroeconomic Analysis 2. Rational Expectations and Persistent Shocks Demand follows the quantity theory of money t vt yt yt t where mt is (logged) money supply, vt is velocity, pt is the price level, and qt is the output level. We will assume a fixed money supply , so mt = m. The supply curve is given by t te yt y t e t where p is the expectation of the price level formed with the information available at time t-1 and yt is the potential output level. Potential output is given by a constant, q, plus random technology, xt. yt q xt Both supply shocks (xt) and demand shocks (vt) follow persistent movements. vt vt 1 tD where the innovation terms, tD xt xt 1 tS and tS are unpredictable, so vte vt 1 xte xt 1 where vte is the expectation of the time t velocity formed at time t-1 and xte is the expectation of the time t technology formed at time t-1. Assume that expectations are formed rationally. Normalize t =q = 0. Solve for the expectation of the output level and the price level as a function of vt-1, xt-1, Solve for the actual output and price level as a function of vt-1 and xt-1 tD and tS . Output can deviate from potential output only if actual prices to differ from expected prices. As we cannot rationally expect the actual prices will be at some different level from expectation, expected output must be equal to potential output. After the e normalization, the expectation of potential output is equal to y t xte xt 1 . Given a constant money stock, the price level is equal to velocity minus output. Thus, the expecte price level equals expected velocity minus expected output. te vt 1 xt 1 . Substituting this into the supply curve we get t vt 1 xt 1 qt xt 1 tS The equation for the price level is t vt yt vt 1 tD yt Substitute this into the vt 1 tD yt vt 1 xt 1 yt xt 1 tS tD yt xt 1 yt xt 1 tS supply equation to solve for qt. yt xt 1 t vt 1 1 1 tS 1 D t 1 tD xt 1 1 tS 3. Baumol-Tobin Model An econometrician estimates a version of the Baumol Tobin Model of money demand. Her results suggest that velocity can be written as a function of the nominal interest rate Vt 14 i . Assume the real interest rate is equal to the real output growth rate. What is the seignorage maximizing growth rate of the money supply? What percentage of output can possibly be collected as seignorage? In the long run, if the real interest rate is equal to the real growth rate, the nominal interest rate is equal to the money growth rate. Seignorage is equal to S ( gM ) gM gM M gM Q gM Q 4 4 Q 1 g M P 1 g M V 1 g M i 1 g M 1 S '( g M ) 0 2 gM 1 g M 4 gM 1 g M 2 1 gM g M 1 S (1) 2Q 2 1 g M Practice Problems 4. Rational Expectations and Monetary Policy Shocks Demand follows the quantity theory of money t t yt y t where mt is (logged) money supply, vt is velocity, pt is the price level, and qt is the output level. The supply curve is given by t REt 1 t yt y Assume a constant potential output. To simplify algebra, normalize potential output, q = 0. Monetary policy follows a random walk. a. Monetary policy follows a random walk so that the growth rate of money is unpredictable white noise: t t . This implies that te REt 1 ( t ) 0 . Assume that velocity is constant, t =0. Calculate the expected price level. Calculate actual levels of output as a function of εt. Velocity is constant vt = v. So expected prices are t t v yt tE Et 1 t Et 1 t v yt Et 1 t Et 1 t v Et 1 yt t 1 v y t Actual output is yt y t 1 v y v y ( { y y }) 1 t t t t t t 1 1 yt y t t t 1 1 1 b. Assume that velocity follows the process, t t 1 tv where ρ < 1. This implies that te REt 1 ( t ) t 1 . Assume that monetary policy systematically targets velocity, t t 1 te t 1 . Calculate output and prices as functions of t 1 and tv . What effect does the choice of λhave on output? It is still the case that expected output is potential output. Thus, expected prices are Et 1 t Et 1 t Et 1 vt Et 1 yt (1 )vt 1 y t We can also calculate that actual prices are t t vt yt (1 )vt 1 t yt We can then calculate output yt y t as (1 )v 1 t 1 t yt (1 )vt 1 t y t 1 ( t {qt y t }) 1 What effect 1 yt y t t t (1 )vt 1 t yt 1 1 1 1 does the choice of λhave on output?The parameter of systematic monetary policy has no effect on output because it is automatically incorporated into prices. 5. Okun’s Law and Time Consistency Each % point increase in output above potential output results in θ% extra inflation. t REt 1 t yt y Each % point increase in output above potential output reduces unemployment below the natural rate by ψ%. urt ur NR qt q t REt 1 t urt ur NR The government is interested in low inflation and pushing unemployment toward some target level ur*. The government sets inflation policy to minimize 2 2 a t b urt ur * subject to t REt 1 t ur ur NR taking expectations as given. t a. Calculate the optimal inflation policy, πP, as a function of ur*, urNR, and REt 1 t . The optimal policy design is to minimize 2 2 min a t b urt ur * REt 1 t urt ur NR t FOC a P urt ur * b P REt 1 t urt ur NR urt ur * ur * ur NR 2b urt ur * 2a P , P REt 1 t P 1 a b2 2 ur NR a P ur * ur NR b ur * 1 REt 1 t a 2 1 b2 b. What will equilibrium inflation and unemployment be when REt 1 t P Rational expectations suggests that when inflation is set according to a predictable policy, that policy will be incorporated into expectations. Thus, inflation will not push output above potential output. So unemployment will equal the natural rate. Then equilibrium inflation is P 1 a b2 2 ur NR ur NR ur * 1 P P . ur * a a 2 1 b b2 c. If the government can choose a target unemployment rate, ur*, in order to have zero inflation, what would it be? Whenever the government has a bias toward unemployment below the natural rate, inflation policy will require positive inflation. Therefore if ur* = urNR, then equilibrium inflation is zero. 6. Adaptive Expectations and Dynamics Construct a small model of the economy that includes A) a Planned expenditure curve; B) a Taylor rule; and C) an expectations augmented Philips curve A. The planned expenditure curve sets demand as a negative function of the real interest rate and an exogenous demand shock, αt. yt t b rt B. The Taylor rule sets the real interest rate as an increasing function of the inflation rate rt r d t C. The SRAS curve sets inflation acceleration as a function of the output gap. t t 1 yt y Calibrate the parameters of the model equal to r = .1, q = 1, b = .5 , d = .5 and θ = .5. We might first solve these three equations for qt , rt , and πt as a function of exogenous variables. First combine the planned expenditure curve and the Taylor rule to develop the AD curve. yt t b r d t t b r b d t . Combine the SRAS t t 1 yt y with the AD curve to solve for output. yt t b r b d t t b r b d t 1 yt y 1 b d yt t b d y b r b d t 1 t b d y b r b d t 1 1 b d Then plugging into an inverted expenditure curve t d y r d d t d y r d t 1 t 1 t t 1 Then rt yt b b b b 1 b d 1 b d t y b r t 1 using the inverted Taylor rule t r 1 rt . d d 1 b d For part a)-e), we will keep to r = .1, y = 1, b = .5, and θ = .5 constant. For a)-d) we ..075 .25 t 1 keep d = .5 . For simplicity, fill in these numbers. yt t amd 1.125 .25 t .15 .5 t 1 .5 t .525 t 1 and t rt 1.125 1.125 yt a. Assume that at time t-1, the economy was in a steady state where t 1 t 2 and t 1 =1.10. Solve for output, yt-1, the interest rate, rt-1, and inflation πt-1. .5 t 1 .525 t 2 Start with t 1 . If t 1 t 2 and t 1 =1.10. then 1.125 .025 t 1 .025 1.175 .25 .2 t 1 t 1 .2 t 2 . Then qt 1 1 q . Then .125 1.125 1.125 .225 rt 1 .2 1.125 b. Assume that unexpectedly t =1.10 drops to 1.05. Using the answer from 3a for πt-1, solve for the new real interest rate output, qt, the interest rate, rt, and inflation πt. c. Using πt from 3b, solve for yt+1 and πt+1. Using πt+1 solve for yt+2 and πt+2. Using πt+2 solve for qt+3 and πt+3. Graph the dynamic response of inflation and output to this demand shock. d. In the far future, inflation will again reach a constant level. Solve for the constant level of inflation when t =1.05. Using t = 1.05, we can write the inflation equation function as a dynamic equation. 8 t t 1 . The output and real interest rate functions are written as equations in 9 1.125 .25 t 1 .1125 .5 t 1 2 4 t 1 . as qt 1 t 1 and rt .1 t 1 . 9 9 1.125 1.125 I use Excel to solve the dynamic behavior of these equations over time. π t-1 t t+1 t+2 t+3 t+4 t+5 t+6 t+7 t+8 t+9 t+10 t+∞ 0.2 t-1 0.177778 t 0.158025 t+1 0.140466 t+2 0.124859 t+3 0.110986 t+4 0.098654 t+5 0.087692 t+6 0.077949 t+7 0.069288 t+8 0.061589 t+9 0.054746 t+10 0 t+∞ q r 1 t-1 0.95555556 t 0.96049383 t+1 0.9648834 t+2 0.96878525 t+3 0.97225355 t+4 0.97533649 t+5 0.97807688 t+6 0.98051278 t+7 0.98267803 t+8 0.98460269 t+9 0.9863135 t+10 1 t+∞ 0.2 0.188889 0.179012 0.170233 0.16243 0.155493 0.149327 0.143846 0.138974 0.134644 0.130795 0.127373 0.1 Those equations will reach their steady state at infinity with π = 0, q = q = 1, r = .1. e. Re solve 3a and 3b for qt-1 and qt under the assumption that the central bank sets an aggressive response to inflation, d = 2. How does an aggressive response to inflation affect the response of output to a demand shock? Explain Fill in the new numbers. qt t .45 t 1 1.5 , rt t .9 2 t 1 1.5 and .5 t .525 t 1 . When t 1 = 1.10, and t 1 t 2 then 1.5 .5 1.1 .525 t 1 t 1 .05 and qt-1 = q =1 and 1.5 .45 t 1 1.05 .45 .05 1.45 qt t 0.966666667 . Output doesn’t declince 1.5 1.5 1.5 as much and the output gap is not as large. When demand shifts down, inflation goes down. When the inflation rate goes down, under the second monetary policy the t central bank reduces interest rates sharply which offsets some of the drop in demand. Weak Stance d=2 π SRAS Strong Stance d=2 Q q 4. (20 points) A government starts time 0 with outstanding financial wealth equal to (1+r) B-1. In the long run, government debt (i.e. negative financial wealth) must equal the present value of primary surpluses, as the government must collect more in future taxes to repay the debt. (1 r ) B1 TAX 0 G0 (TAX1 G1 ) (TAX 2 G2 ) (TAX 3 G3 ) .... 2 3 1 r 1 r 1 r The ratio of outstanding debt to GDP is 90%. If the real interest rate is 21% (r = .21) and the GDP growth rate is 10%, go = .1, solve for a permanent level of primary surpluses to GDP, psy, which will keep the debt to GDP ratio constant at 90%. Hint: (1 r ) B1 1 g .9 psy [1 Q0 1 r Q 1 g 1 g 2 Q 1 r 2 3 Q 1 r 3 ....