CH-15 A Dynamic Model of Economic Fluctuations – Macroeconomics : Book Guide

Answers to Textbook Questions and Problems CHAPTER 15 A Dynamic Model of Economic Fluctuations Questions for Review 1. The equation for the dynamic aggregate supply curve is pt = pt-1+f(Yt -Yt )+ut. Recall that  is a positive parameter that measures how rapidly firms adjust their prices in response to output fluctuations. When output in the economy rises above its natural level, firms experience rising marginal costs and will increase prices. There is therefore a positive relationship between the level of output and inflation in the economy. The dynamic aggregate supply curve is upward sloping. The steepness of the dynamic aggregate supply curve depends on how quickly marginal costs rise when output is above its natural level and on how quickly firms respond to the rising marginal cost with an increase in prices. The dynamic aggregate supply curve will be steeper if marginal costs rise more quickly and if firms respond by increasing prices more quickly. The dynamic aggregate supply curve is illustrated in Figure 15-1. 2. The equation for the dynamic aggregate demand curve is Y =Y -ØŒ aqn øœ p -p* +ØŒ 1 øœe . t Œº1+aqY œß ( t t ) Œº (1+aqY )œß t The dynamic aggregate demand curve is defined by a given monetary policy rule and illustrates a negative relationship between the quantity of output demanded and inflation. When inflation changes, the central bank follows its monetary policy rule and changes the nominal interest rate. The monetary policy rule specifies that the nominal interest rate will change by more than the inflation rate so that there is a change in the real interest rate, and hence the demand for goods and services. Thus, if inflation rises, the central bank will raise the nominal interest rate, the real interest rate will rise, the amount of goods and services demanded will fall, and the level of output will fall. The dynamic aggregate demand curve is steeper if the central bank is more tolerant of high inflation (θπ is smaller), if the central bank is less tolerant of deviations in output away from the natural level (θY is larger), and if the public’s spending is less responsive to changes in the real interest rate (α is smaller). The dynamic aggregate demand curve is illustrated in Figure 15-2. 3. The dynamic aggregate demand curve is drawn for a given monetary policy rule. If the central bank changes the rule by increasing the target inflation rate, then the dynamic aggregate demand curve will shift to the right. Looking at the equation for the dynamic aggregate demand curve, an increase in the target inflation rate will increase output for any given level of the inflation rate. When the central bank increases the target inflation rate, the current inflation rate will be below the target. As a result, the central bank will have to lower both nominal and real interest rates. The lower real interest rate will increase the demand for goods and services at the current inflation rate and output will rise. The shift in the aggregate demand curve is illustrated in Figure 15-3. Since output is above its natural level, marginal costs will rise and firms will increase prices. The economy moves from its original equilibrium at point A to its new short-run equilibrium at point B. As the level of inflation rises, so will the expected inflation rate, and the dynamic aggregate supply curve will shift up and to the left, as illustrated in Figure 15-3. As inflation rises, the central bank will follow its new policy rule and increase the nominal interest rate. Eventually, the economy reaches its new long-run equilibrium, identified by point Z. Notice that inflation has risen from 2 percent to 3 percent. The nominal interest rate will be higher in the long run because there is no change in the long run real interest rate, and the nominal interest rate in the long run is equal to the real interest rate plus the target inflation rate. 4. If the central bank decides to increase the response of interest rates to changes in inflation (the parameter θπ), then the central bank has become less tolerant of inflation. In this case, any increase in inflation will elicit a larger increase in nominal and real interest rates in an attempt to reduce the demand for goods and services and prevent further increases in inflation, such that the dynamic aggregate demand curve is flatter. Mathematically, the slope of the dynamic aggregate demand curve is given by 1+aqY . aqp When the parameter θπ increases in value, the slope becomes smaller in absolute terms and the dynamic aggregate demand curve becomes flatter. Intuitively, when the central bank is less tolerant of inflation, they are willing to put up with larger deviations of output from the natural rate, making the dynamic aggregate demand curve flatter. In this case, a supply shock that shifts the dynamic aggregate supply curve up and to the left will cause a larger reduction in the level of output and a smaller increase in the inflation rate, as illustrated in Figure 15-4. Under the new policy, the economy moves from point A to point C in response to the supply shock, as opposed to moving from point A to point B under the old policy. Note that, if the economy is in long-run equilibrium at the time of the central bank policy change, the economy will still remain in long-run equilibrium, but with a flatter dynamic aggregate demand curve. Problems and Applications 1. The five equations that make up the dynamic aggregate demand–aggregate supply model can be manipulated to derive long-run values for the variables. In this problem, it is assumed that there are no shocks to demand or supply and inflation has stabilized. Since inflation has stabilized, inflation in time t is equal to inflation in time t – 1(πt = πt–1). We also know that expected inflation is equal to last period’s inflation, or Et–1πt = πt–1. We start with the Phillips curve equation on line 1 below and use these two facts to find the following: pt =Et-1pt +f(Yt -Yt )+ut pt =pt-1+f(Yt -Yt )+ut pt =pt +f(Yt -Yt )+ut From here, it follows that output must equal natural output since the supply shock parameter υt equals zero. Moving to the demand for goods and services equation next, it now follows that the real interest rate equals the natural rate of interest since the demand shock parameter εt equals zero and Yt = Yt : Yt =Yt -a(rt -r)+et . Turning to the Fisher equation on line 1 below, we can show the nominal interest rate is equal to the natural interest rate plus the current inflation rate. Since inflation has stabilized, expected inflation equals current inflation (Etπt+1 = πt), and we have just demonstrated that the real interest rate is equal to the natural rate of interest (rt = ρ): rt = it – Ett+1 rt = it – t it = rt – t it = ρ – t. Moving now to the monetary policy rule equation on line 1 below, given current inflation equals the target inflation rate, the third term on the right zeros out. Likewise, the fourth term on the right side will zero out since output is at the natural level: it =pt +r+qp(pt -pt*)+qY (Yt -Yt ) it =pt +r The final values are as follows: Yt =Yt rt =r pt =pt* Etpt+1 =pt* ii =r+pt* 2. If the central bank has the wrong natural rate of interest, then it is using a value ρ that is different from the real value ρ. Suppose that the wrong natural rate of interest is defined as follows: r¢= r+Dr In this case, if Δρ equals zero, then the central bank has the correct natural rate of interest. If the natural rate of interest is wrong, then the long-run equilibrium values will change. The five equations that make up the dynamic aggregate demand–aggregate supply model can be manipulated to derive longrun values for the variables. In this problem, it is assumed that there are no shocks to demand or supply and inflation has stabilized. Since inflation has stabilized, it must be true that inflation in time t is equal to inflation in time t – 1(πt = πt – 1). We also know that expected inflation is equal to last period’s inflation, or Et–1πt = πt–1. We start with the Phillips curve on line 1 below and use these two facts to find the following: pt =Et-1pt +f(Yt -Yt )+ut pt =pt-1+f(Yt -Yt )+ut pt =pt +f(Yt -Yt )+ut From here it follows that output must equal natural output since the supply shock parameter υt equals zero. Moving to the demand for goods and services equation below it now follows that the real interest rate equals the natural rate of interest since the demand shock parameter εt equals zero and Yt = Yt : Yt =Yt -a(rt -r)+et Turning to the Fisher equation on line 1 below, we can show the natural interest rate is equal to the nominal interest rate minus the current inflation rate. Since inflation has stabilized, expected inflation equals current inflation (Etπt+1 = πt, and we have just demonstrated that the real interest rate is equal to the natural rate of interest (rt = ρ): rt = it –Ett+1 rt = it – t ρ = it – t. The monetary policy rule equation on line 1 below has the wrong natural rate of interest ρ. Substitute in the relationship between the correct and incorrect rates of natural interest and rearrange terms: it =pt +r¢+qp(pt -pt*)+qY (Yt -Y it =pt +(r+Dr)+qp(pt -pt*)+q Y -Y ) it =pt = (r+Dr)+qp(pt -pt*)+qY (Yt -Yt ) The third term on the right side will zero out since output is at the natural level. Now, combine the rewritten Fisher equation with the rewritten monetary policy rule equation above: it =pt +(r+Dr)+qp(pt -pt*)+qY (Yt -Yt ) r=(r+Dr)+qp(pt -pt*) 0=Dr+qppt -qppt* pt =pt* - Dr qp The final values are as follows: Yt =Yt rt =r pt =pt* - Dr qp Etpt+1 =pt* -Dr qp it =r+pt* - Dr qp Intuitively, if the central bank thinks that the natural rate of interest is higher than it really is, then it will be setting interest rates higher than they should be set. The higher interest rates will result in lower demand for goods and services, and in the long run, this will result in an inflation rate that is lower than the target inflation rate. In the short run, higher interest rates will temporarily cause real interest rates to be higher than normal, causing the dynamic aggregate demand curve to shift down and to the left. In being wrong about the natural rate, the central bank has effectively forced the economy through a recessionary cycle, which has resulted in the inflation rate coming in below the target rate. As the lower inflation rate persists, the expected inflation rate will decrease and the dynamic aggregate supply curve will shift down and to the right until a new long run equilibrium is reached. This is illustrated in Figure 15-5. From the derived long-run values above, the inflation rate is below the target rate, expected inflation equals actual inflation and is also below the target rate, and the nominal interest rate is lower than it would otherwise be. 3. “If a central bank wants to achieve lower nominal interest rates, it has to raise the nominal interest rate.” In long-run equilibrium, the nominal rate of interest is equal to the natural rate of interest plus the target inflation rate. To lower the long-run nominal interest rate, the central bank must lower the target inflation rate and, ultimately, the actual inflation rate. In the short run, the central bank must increase the nominal interest rate in order to reduce spending and output in the economy. This will reduce inflation and, ultimately, expected inflation. The economy will adjust to a new long-run equilibrium in which the nominal interest rate, the target inflation rate, and the actual inflation rate are all lower. Graphically, lowering the target inflation rate will shift the dynamic aggregate demand curve down and to the left, forcing the economy through a recessionary cycle, and in the short run, output and inflation will be lower. As expected inflation adjusts over the long run, the dynamic aggregate supply curve will shift down and to the right. In the long run, output is equal to the natural level and inflation is lower. 4. The sacrifice ratio measures the accumulated loss in output associated with a one-percentage-point reduction in the target inflation rate. Graphically, the reduction in the target inflation rate will shift the dynamic aggregate demand curve down and to the left, resulting in a short-run equilibrium with a lower level of output and a lower inflation rate. Over time, expected inflation will adjust and the dynamic aggregate supply curve will shift down and to the right until output again equals potential output. For each year that output remains below potential, the percentage deviation of actual output from potential output can be calculated, and these results can be summed to find the accumulated lost output in percentage terms. For the twelve years included in the text simulation, the accumulated lost output is 2.59 percent. During this same period, the inflation rate fell from 2 percent to 1.35 percent, which is a decrease of 0.65 percent. The implied sacrifice ratio is therefore 2.59/0.65 = 3.98. We can derive this same result directly from the dynamic aggregate demand–aggregate supply model. Start with the Phillips curve equation on line 1 below and use the adaptive expectations assumption to rewrite as follows: pt = Et-1pt +f(Yt -Yt )+ut pt =pt-1pt +f(Yt -Yt )+ut From this equation, we see that, in the absence of supply shocks (υt = 0), a one-percentage-point decrease in output below its natural level causes inflation to decrease by θ percentage points. (Recall that the natural level of output is 100 so that a one-unit deviation of output from its natural level is equivalent to a one-percentage-point deviation.) Turning this result around, we find that, in order to reduce the inflation rate by one percentage point, output must decline by 1/θ percentage points. From the simulation, the value of θ is 0.25 so that 1/θ is equal to 4. Note that this is very close to the value of 3.98 that was obtained directly from the simulation results. 5. Follow the hint given in the problem and solve for the long-run equilibrium with the new assumption that the demand shock parameter εt is not zero. Since inflation has stabilized, it must be true that inflation in time t is equal to inflation in time t – 1(πt = πt–1). We also know that expected inflation is equal to last period’s inflation, or Et–1πt = πt–1. Start with the Phillips curve on line 1 below and use these two facts to find the following: pt =Et-1pt +f(Yt -Yt )+ut pt =pt-1pt +f(Yt -Yt )+ut pt =pt +f(Yt -Yt )+ut From here, it follows that output must equal natural output since the supply shock parameter υt equals zero. From the demand for goods and services equation on line 1 below, it now follows that the real interest rate equals the natural rate of interest plus a new term: Yt =Yt -a(rt -r)+et 0=-art +ar+et art =ar+et et rt =r+ a Turning to the Fisher equation on line 1 below, we can show the nominal interest rate is equal to the natural interest rate plus the current inflation rate plus a new term. Since inflation has stabilized, expected inflation equals current inflation (Etπt+1 = π)t, and we have just demonstrated that the real interest rate is equal to the natural rate of interest plus a new term: et : rt =r+ Ł ał rt = it - Etpt+1 rt = it -pt it = rt +pt et it =r+pt + a Moving now to the monetary policy rule equation on line 1 below, substitute in for the nominal rate of interest from the rewritten Fisher equation above, and then note that the fourth term on the right side will zero out since output is at the natural level: it =pt +r+qp(pt -pt*)+qY (Yt -Yt ) et +pt =pt +r+qp(pt -pt*) r+ a et =qppt* a pt =pt*+ et qpa The final values are as follows: Yt =Y rt =r+ pt =pt + qpa Etpt+1 =pt*+ et qpa it =r+pt*+et a If the demand shock parameter εt were to increase permanently, such that it remained a constant positive number, the dynamic aggregate demand curve would shift to the right permanently. This would cause a short-run increase in output and inflation and a long-run increase in the inflation rate as the economy adjusted to its new long run equilibrium. This is consistent with the newly derived longrun values above, where the inflation rate is higher than the target inflation rate. Note that expected inflation is also higher, as is the nominal and the real interest rate. To deal with this issue, the central bank could decrease its target inflation rate. This would effectively offset the permanent increase in the demand shock parameter t and shift the dynamic aggregate demand curve back to its original position. 6. The equation for the dynamic aggregate demand curve is given below: Ø aq ø Ø Yt =Yt -ŒŒº (1+aqp Y )ßœœ (pt -pt*)+ŒºŒ (1+aq1 Y )œøœßet . The parameter θπ measures the central bank’s responsiveness to changes in the inflation rate. When θπ is large, the central bank aggressively responds to changes in the inflation rate. When θπ is small but still positive, the central bank has a weak response to changes in the inflation rate, and the dynamic aggregate demand curve becomes very steep. If θπ becomes negative, the dynamic aggregate demand curve actually has a positive slope, as can be seen in the equation above. In this case, a supply shock that shifts the dynamic aggregate supply curve up and to the left will lead to ever-increasing inflation, even if the shock is temporary. This is due to the fact that output remains above its natural level since the central bank’s increase in nominal interest rates is not enough to increase real interest rates. The supply shock will shift the dynamic aggregate supply curve up and to the right as rising production costs increase the inflation rate. Since nominal interest rates rise by less than the inflation rate, real interest rates will fall and therefore output will rise. In Figure 15-6, this is shown as a movement from point A to point B. Since output is above the natural rate, inflation will continue to rise, and the dynamic aggregate supply curve will continue to shift up and to the left as people adjust their expectations about inflation. This analysis reinforces the Taylor principle as a guideline for the design of monetary policy in that the central bank wants to maintain low and stable inflation. 7. Suppose that the natural rate of interest is not a constant parameter but varies over time so that it is now written as ρt. a. The equation for dynamic aggregate supply is not affected by this change because its derivation does not involve the natural rate of interest. The equation for dynamic aggregate demand is not affected by this change either because, although the variable ρt is involved in the derivation of the dynamic aggregate demand curve, it cancels out and does not end up as part of the final equation. b. A shock to ρt would not cause a shift to either dynamic aggregate demand or dynamic aggregate supply because the variable does not appear in either equation. Output and inflation would not be affected. However, the real and nominal interest rates would both change by the amount of the change in ρt. c. If the natural rate of interest varied over time, it would make the setting of monetary policy more difficult. If the central bank knows that the natural rate of interest is 4 percent, for example, and it is aiming for target inflation rate of 2 percent, then a nominal interest rate of 6 percent will be the long-run target. If, on the other hand, the natural rate of interest varies over time, then the target long-run interest rate will also vary over time. It is more difficult to hit a moving target than a target that is standing still. In particular, in contrast to what is implicitly assumed above, if the natural rate of interest is always moving, the central bank might have trouble knowing the natural rate of interest at every point in time. Moreover, if the variation in the natural rate of interest is in any way linked to the rate of inflation, the central bank will face major challenges in its inflation targeting exercises. 8. Suppose people’s expectations of inflation are subject to random shocks so that Et–1πt = πt–1 + ηt–1. a. The dynamic aggregate supply curve equation is derived from the Phillips curve and the expectations equation. In this case, start with the Phillips curve equation on line 1 below and substitute in for the expected inflation term using the expression above: pt = Et-1pt +f(Yt -Yt )+ut pt =pt-1+ht-1+f(Yt -Yt )+ut The dynamic aggregate demand curve is derived from the demand for goods and services equation, the Fisher equation, and the monetary policy rule equation. In this problem, the Fisher equation will be modified to include the new expected inflation equation. Start with the demand for goods and services equation on line 1 below, then use the Fisher equation and monetary policy rule equation to make the necessary substitutions: Yt =Yt -a(rt -r)+et Yt =Yt -a(it - Etpt+1-r)+et Yt =Yt -a(it -(pt +ht )-r)+et Yt =Yt -a p(( t +r+qp(pt -pt*)+qY (Yt -Y))-(pt +ht )-r)+et Yt =Yt -aq( p(pt -pt*)+qY (Yt -Y)-ht )+et With a few more algebraic manipulations, you end up with the following equation for the dynamic aggregate demand curve: Yt =Yt -غaqp / 1( +aqY )øß (pt -pt*)+ºØa/ 1( +aqY )ßøht +غ1/(1+aqY )øßet . b. If ηt is greater than zero for one period only, then the dynamic aggregate demand curve will shift to the right and the dynamic aggregate supply curve will not shift. Note that the dynamic aggregate supply curve depends on the lagged value of this shock parameter so that it will be affected in period t + 1. As the dynamic aggregate demand curve shifts to the right, output and inflation will both rise. Based on the central bank’s monetary policy rule, nominal and real interest rates will both be increased. Intuitively, if people expect inflation to be higher next year, then they will increase purchases today to take advantage of the still-lower prices. c. In period t + 1, the dynamic aggregate demand curve will shift back to its original position (because ηt+1 is zero), and the dynamic aggregate supply curve will shift to the left (because ηt is positive and also because lagged inflation has increased). In comparison to long-run equilibrium, output will be lower and inflation will be higher. The economy is experiencing stagflation. Inflation is higher because of higher expectations of inflation, and output is lower because of the higher real interest rates that resulted from higher inflation. d. In subsequent time periods, the dynamic aggregate supply curve will slowly shift back to its original position as the lower level of output reduces inflation, and hence expectations of future inflation. Although the parameter ηt+1 was positive for only one time period, the dynamic aggregate supply curve does not immediately return to its original position because the short-run increase in inflation has caused expected inflation to rise above its long-run value. e. This problem shows that inflation scares are often self-fulfilling. When people believe inflation will rise, they act in such a way that inflation does actually rise, and the economy goes through a period of higher inflation. 9. Use the dynamic AD–AS model to solve for inflation as a function of only lagged inflation and the two shocks. Start with the dynamic aggregate supply curve and substitute in for Yt using the dynamic aggregate demand curve equation as is done on line 1 below. Now, solve for inflation through a few algebraic manipulations: pt =pt-1+fØŒº Yt -غaqn / 1( +aqY )øß (pt -pt*)+غ1/(1+aqY )øßet -Yt øßœ +ut pt ØŒº1+(faqp / 1( +aqY ))œßø =pt-1+(faqY )pt*+ºØf/ 1( +aqY )øßet +ut (1+aqY ) (faqp) * f (1+aqY ) pt = pt-1+ pt + et + ut (1+aqY +faqp) (1+aqY +faqp) (1+aqY +faqp) (1+aqY +faqp) a. A supply or demand shock will lead to an increase in current inflation. As the economy adjusts and returns to long-run equilibrium, the inflation rate will return to its target level. Note that the coefficient on the lagged inflation variable in the equation above is positive but less than 1. This means that inflation in time t + 1 will be less than inflation in time t, and that inflation will eventually return to its target rate. b. If the central bank does not respond to changes in output so that θY is zero, then the economy will still return to its target inflation rate after a supply or demand shock because the coefficient on the lagged inflation variable in the equation above is still positive but less than 1. In this case, inflation should return more quickly to its target rate. This is because the coefficient on lagged inflation has become smaller (the change in the numerator is larger in comparison to the change in the denominator). The dynamic aggregate demand curve is relatively flat when the central bank only cares about inflation. c. If the central bank does not respond to changes in inflation so that θπis zero, then the coefficient on lagged inflation in the above inflation equation equals 1. In this case, the economy will not return to its target inflation rate after a demand or supply shock. The demand or supply shock will increase inflation in time t. When θπ is zero, inflation in time t + 1 is equal to inflation in time t. d. The Taylor rule says that a one-percentage-point increase in inflation will increase the nominal interest rate by 1 + θπ percentage points. If the central bank increases the nominal interest rate by only 0.8 percentage points for each one-percentage-point increase in the nominal interest rate, then this means θπ is equal to –0.2. When θπ is negative, the dynamic aggregate demand curve is upward sloping. A shock to demand or supply will set the economy on a path of ever-increasing inflation. This path of ever-increasing inflation will occur because real interest rates will continue to fall and output will remain above the natural level. You can see this phenomenon in the above equation for inflation: If θπ is negative, the coefficient on lagged inflation is greater than 1. That larger-than-one coefficient is the mathematical manifestation of explosive inflation. IN THIS CHAPTER, YOU WILL LEARN: ▪ how to incorporate dynamics into the AD-AS model we previously studied ▪ how to use the dynamic AD-AS model to illustrate longrun economic growth ▪ how to use the dynamic AD-AS model to trace out the effects over time of various shocks and policy changes on output, inflation, and other endogenous variables Introduction ▪ The dynamic model of aggregate demand and aggregate supply gives us more insight into how the economy works in the short run. ▪ It is a simplified version of a DSGE model, used in cutting-edge macroeconomic research. (DSGE = Dynamic, Stochastic, General Equilibrium) Introduction ▪ The dynamic model of aggregate demand and aggregate supply is built from familiar concepts, such as: ▪ the IS curve, which negatively relates the real interest rate and demand for goods & services ▪ the Phillips curve, which relates inflation to the gap between output and its natural level, expected inflation, and supply shocks ▪ adaptive expectations, a simple model of inflation expectations How the dynamic AD-AS model is different from the standard model ▪ Instead of fixing the money supply, the central bank follows a monetary policy rule that adjusts interest rates when output or inflation change. ▪ The vertical axis of the DAD-DAS diagram measures the inflation rate, not the price level. ▪ Subsequent time periods are linked together: Changes in inflation in one period alter expectations of future inflation, which changes aggregate supply in future periods, which further alters inflation and inflation expectations. Keeping track of time ▪ The subscript “t ” denotes the time period, e.g. ▪Yt = real GDP in period t ▪Yt -1 = real GDP in period t – 1 ▪Yt +1 = real GDP in period t + 1 ▪ We can think of time periods as years. E.g., if t = 2010, then ▪Yt = Y2010 = real GDP in 2010 ▪Yt -1 = Y2009 = real GDP in 2009 ▪Yt +1 = Y2011 = real GDP in 2011 The model’s elements ▪ The model has five equations and five endogenous variables: output, inflation, the real interest rate, the nominal interest rate, and expected inflation. ▪ The equations may use different notation, but they are conceptually similar to things you’ve already learned. ▪ The first equation is for output… Output: The Demand for Goods and Services Yt = Yt −  (rt − +) t output natural real   0, 0 level of interest output rate Negative relation between output and interest rate, same intuition as IS curve. Output: The Demand for Goods and Services Yt = Yt −  (rt − +) t demand measures the shock, interest-rate sensitivity of demand “Natural rate of interest.” In absence of demand shocks, random and zero on average Y Yt = t when rt = The Real Interest Rate: The Fisher Equation ex ante (i.e. expected) real interest nominal expected rate interest inflation rate rate t+1 = increase in price level from period t to t +1, not known in period t Et t+1 = expectation, formed in period t, of inflation from t to t +1 Inflation: The Phillips Curve 0 indicates how much average inflation responds when output fluctuates around its natural level Expected Inflation: Adaptive Expectations Et  t+1 = t For simplicity, we assume people expect prices to continue rising at the current inflation rate. The Nominal Interest Rate: The Monetary-Policy Rule it = + +    t ( t − t*) + Y (Yt −Yt ) nominal central interest rate, bank’s set each period inflation by the central target bank natural rate of interest  0, Y 0 The Nominal Interest Rate: The Monetary-Policy Rule it = + +    t ( t − t*) + Y (Yt −Yt ) measures how much the central bank adjusts the interest rate when inflation deviates from its target measures how much the central bank adjusts the interest rate when output deviates from its natural rate CASE STUDY The Taylor rule ▪ Economist John Taylor proposed a monetary policy rule very similar to ours: iff =  + 2 + 0.5 ( – 2) – 0.5 (GDP gap) where ▪iff = nominal federal funds rate target Y Y− ▪ GDP gap = 100 x Y = percent by which real GDP is below its natural rate ▪ The Taylor rule matches Fed policy fairly well.… ▪ Endogenous variables: Yt = t = rt = it = Et t+1 = Output Inflation Real interest rate Nominal interest rate Expected inflation Yt = t* = t = t = Natural level of output Central bank’s target inflation rate Demand shock Supply shock ▪ Exogenous variables: ▪ Predetermined variable: t−1 = Previous period’s inflation ▪ Parameters: = = =  = Y = Responsiveness of demand to the real interest rate Natural rate of interest Responsiveness of inflation to output in the Phillips Curve Responsiveness of i to inflation in the monetary-policy rule Responsiveness of i to output in the monetary-policy rule The model’s long-run equilibrium ▪ The normal state around which the economy fluctuates. ▪ Two conditions required for long-run equilibrium: ▪ There are no shocks:  t = t = 0 ▪ Inflation is constant:  t−1 = t The model’s long-run equilibrium ▪Plugging the preceding conditions into the model’s five equations and using algebra yields these long-run values: Yt = Yt rt =   t = t* Et  t+1 = t* it =  + t* The Dynamic Aggregate Supply Curve ▪The DAS curve shows a relation between output and inflation that comes from the Phillips Curve and Adaptive Expectations:   t = t−1 + (Yt −Yt ) +t (DAS) The Dynamic Aggregate Supply Curve   t = t−1 + (Yt −Yt ) +t πDAS slopes upward: high levels of output are associated with high inflation. DAS shifts in response to changes in the natural level of output, previous inflation, and supply shocks. Y ▪To derive the DAD curve, we will combine four equations and then eliminate all the endogenous variables other than output and inflation. Start with the demand for goods and services: Yt = Yt −  (rt − +) t Yt = Yt −(it − Et   t+1 − +) t result from previous slide Yt = Yt −(it − Et   t+1 − +) t using the expectations eq’n Yt = Yt −   (it − − +t ) t using monetary policy rule Y Yt = −t    [ t + + ( t − +t*) Y (Y Yt − − − +t )   t ] t Yt = Yt −  [ ( t − t*) + Y (Yt −Yt )] +t result from previous slide Yt = Yt −  [ ( t − t*) + Y (Yt −Yt )] +t combine like terms, solve for Y Y Yt = t − A( t − t*) + Bt , (DAD) where A =   0, B = 1  0 1+Y 1+Y The Dynamic Aggregate Demand Curve Y Yt = t − A( t − t*) + Bt π DADt Y DAD shifts in response to changes in the natural level of output, the inflation target, and demand shocks. The short-run equilibrium π In each period, the intersection of DAD and DAS determines the short-run eq’m values of inflation and output. In the eq’m shown here at A, output is below Y its natural level. Long-run growth Period DAS shifts t: initial because eq’m at A economy can produce more DASt +1 g&s. Period t + 1: ππt 1New DAD shifts Longgrowth eq’m-run at B; t + =ibrncome grows because higher income emains stable.ut inflation increases thenatural rate raises of output. demand Y for g&s. A shock to aggregate supply Period Period Period Period tt t+ +:– 1 21:: Supply shock As inflation falls, initiaSupply shock eq’m at A inflation s over ((ν > 0) shifts ν = 0) but DAS does notexpectations fall, DAS upward; πtreturn tDAS moves inflation rises, its initial πt + 2pdownward, centralsition due to bank t -1 higheroutput rises. responds byinflation expectations.This process raising real πt – 1continues until interest rate, output returns to output falls. its natural rate. Yt Yt + 2Yt –1 LR eq’m at A. Parameter values for simulations Yt = 100 t* = 2.0 = 1.0 = 2.0 = 0.25  = 0.5 Y = 0.5 Thus, we can interpret Yt – Yt as the percentage deviation of Central bank’s inflation The following graphs artarget is 2 percent.A 1-percentage-point ine called crease vel. output from its natural le impulse response functionsin the real interest rate reduces . They show the output demand by 1 percent of its natural level.The natural rate of interest is response of the endogenous variables to the 2When output is 1 percent percent. (the shock). impulse above its natural level, inflation rises by 0.25 percentage point. These values are from the Taylor rule, which approximates the actual behavior of the Federal Reserve. A shock to aggregate demand t +4 Yt + 5 Yt –1 Yt Periods Periods Period Period e iod tt t t+t:+– ++ 1 51 62:: Higher inflation iand higher:toDAS is higher iniPositive t ial + 4eq’m : at An tDAS gradually Higher inflation due to higher demand shock rais d inflation expectatshifts down as inflation in (ε previ > 0) shifts us o s for inflation and preceding periodDAD eriod raises t + to1, the , shifting DAS up.inflation but demand righ ; output Inflation rises expectations fall,shock ends and and inflation, more, output falleconomy shifts DAS up.DAD returns to rise. s. gradually Inflatits initial positionon rises, . recovers until output falls.Eq’m at G. reaching LR eq’m at A. CHAPTER 15 Dynamic Model of Economic Fluctuations 35 A shift in monetary policy Period Period Subsequent Period t t– t+ :1 1 : : targThe fall in periods:Central bank t inf ation πt rate reduced This process lowers target π* = 2%, πt – 1 = 2%initial inflation continues untilto π*eq’m = 1%, at A πtexpectationsoutput returnsraises real for to its natural interest rate, t + 1, final shifting DAS rate and shifts DAD downward. inflation leftward. πfinal = 1%Output rises, reaches its neOutput and w inflation falls.target. inflation fall. CHAPTER 15 Dynamic Model of Economic Fluctuations 40 ▪ A supply shock reduces output (bad) and raises inflation (also bad). ▪ The central bank faces a tradeoff between these “bads” – it can reduce the effect on output, but only by tolerating an increase in the effect on inflation…. CASE 1: θπ is large, θY is small In this case, a small change in inflation has a large effect on The shock has t – 1, t a large effect Y on output but a Yt Yt –1 small effect on inflation. CASE 2: θπ is small, θY is large π In this case, a large change in inflation has only a small effect on πtt – 1 output, so DAD is relatively steep. πt –1 Now, the shock has only a small Y effect on output, Yt Yt –1 but a big effect on inflation. ▪ The Taylor principle (named after John Taylor): The proposition that a central bank should respond to an increase in inflation with an even greater increase in the nominal interest rate (so that the real interest rate rises). I.e., central bank should set θπ > 0. ▪ Otherwise, DAD will slope upward, economy may be unstable, and inflation may spiral out of control. Yt = Yt −  ( t − t*) + 1 t (DAD) 1+Y 1+Y it = + +    t ( t − t*) + Y (Yt −Yt ) (MP rule) If θπ > 0: ▪ When inflation rises, the central bank increases the nominal interest rate even more, which increases the real interest rate and reduces the demand for goods & services. ▪ DAD has a negative slope.  *) + 1 t (DAD) Yt = Yt − ( t − t 1+Y 1+Y it = + +    t ( t − t*) + Y (Yt −Yt ) (MP rule) If θπ < 0: ▪ When inflation rises, the central bank increases the nominal interest rate by a smaller amount. The real interest rate falls, which increases the demand for goods & services. ▪ DAD has a positive slope. ▪ If DAD is upward-sloping and steeper than DAS, then the economy is unstable: output will not return to its natural level, and inflation will spiral upward (for positive demand shocks) or downward (for negative ones). ▪ Estimates of θπ from published research: ▪ θπ = –0.14 from 1960–78, before Paul Volcker became Fed chairman. Inflation was high during this time, especially during the 1970s. ▪ θπ = 0.72 during the Volcker and Greenspan years. Inflation was much lower during these years. ▪ The DAD-DAS model combines five relationships: an IS-curve-like equation of the goods market, the Fisher equation, a Phillips curve equation, an equation for expected inflation, and a monetary policy rule. ▪ The long-run equilibrium of the model is classical. Output and the real interest rate are at their natural levels, independent of monetary policy. The central bank’s inflation target determines inflation, expected inflation, and the nominal interest rate. ▪ The DAD-DAS model can be used to determine the immediate impact of any shock on the economy and can be used to trace out the effects of the shock over time. ▪ The parameters of the monetary policy rule influence the slope of the DAS curve, so they determine whether a supply shock has a greater effect on output or inflation. Thus, the central bank faces a tradeoff between output variability and inflation variability. ▪ The DAD-DAS model assumes that the Taylor principle holds, i.e. that the central bank responds to an increase in inflation by raising the real interest rate. Otherwise, the economy may become unstable and inflation may spiral out of control. . Solution Manual for Macroeconomics Gregory N. Mankiw 9781464182891, 9781319106058