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EC201: Worksheet 4 with Romer model Boromeus Wanengkirtyo Department of Economics University of Warwick 1 Introduction to IS-MP The Romer model consists of three equations: MP : IS : rt = a1 + φπ (πt − π ? ) + φy (yt − y N ) rt = c − d · yt I A : π t = π t −1 + f ( y t −1 − y N ) + ε t MP is the Taylor rule, which describes the interest-rate policy rule that the central bank (CB) follows. The parameter φπ describes the response of the CB to deviations of inflation from the target inflation rate. Likewise, φy is the response of the CB to output gap fluctuations. a1 contains the natural rate of interest (the interest rate the CB sets when there is no inflation or output gap deviations), as well as a discretionary term. The CB can change a1 if it wants to deviate away from the Taylor rule prescribed interest rate (‘discretionary policy’). Note that as we plot the MP curve on r-Y space, the curve is for a given level of inflation. If inflation changes, the MP curve will shift. The IS curve describes the demand-side (consumption, investment etc). 1/b is the interest elasticity of output. Therefore, there is a negative relationship between consumption and investment demand, with interest rates. We use the MP and IS curve to derive the AD curve. Finally, the supply side of the economy is modelled with the IA (inflation adjustment) curve. In the Romer model, inflation is pre-determined (i.e. it is only affected by last period’s output gap and inflation, and not current output gap). Therefore, it is a flat curve since we are plotting the IA curve on πt − yt space. ε t is shocks to price setting (‘inflation shocks’). I explain this in more detail later. See Figure 1. Diagrammatically, to plot the AD curve, simply take some inflation π1 and find the equilibrium output in the IS-MP graph (since the MP curve is for a given rate of inflation). Then take a higher inflation point π2 . This shifts up the MP curve as the central bank raises rates endogenously (i.e. autonomously). This gets a lower equilibrium output, so we end up with a downward sloping AD curve. The emphasis on endogenous movements of interest rates is crucial. 1 A shift of the MP curve due to inflation changes causes a movement along the AD curve. There could be other things that shift the MP curve (for example, a1 , π ? , y N etc.) These things will not only shift the MP curve, but also shift the AD curve too. This is because for a given inflation, there is a different equilibrium output. Therefore, a change in the CB’s discretionary policy (a1 ) will cause a shift in the AD curve. To get the AD curve mathematically, eliminate the variable rt from the IS and MP equations (i.e. equate them), to get: πt = π ? + i 1 h c + φy y N − a1 − (b + φy )yt φπ The slope of the AD curve is therefore −(b + φy )/φπ . 2 Q5: Adverse Demand Shocks See Figure 2. An adverse demand shock could be the example of a fall in consumer confidence, like in Romer’s paper. The effect on impact is a shift of IS to the left (a fall in the parameter c of the IS curve). This will also shift the AD curve to the left, since for a given inflation rate, there is lower equilibrium output as determined by the IS-MP graph. Therefore, we jump from point (1) to point (2). What happens afterwards depends on the behaviour of the CB. 2.1 Rule-based policy One possibility is that the CB adheres to the Taylor rule. At point (2) it will set a lower interest rate endogenously, due to the lower output. Note that this is a movement along the MP curve, because we are changing output. At this point, there is a recession (negative output gap, y < y N ), so the IA curve starts shifting down and inflation falls. This will continue to happen until the output reaches y N . At the same time, because inflation falls, the MP curve also shifts down. Therefore, the long-run equilibrium under this monetary policy decision is a return to y N , but at a lower level of inflation. 2.2 Discretionary policy The above policy is suboptimal. The economy undergoes a recession, and in the long-run, inflation is below target (exposing the CB to possible zero lower bound issues). The CB can instead intervene directly if it recognises the IS shock quickly. 2 This is an exogenous change in the Taylor rule (a1 ). This means that the MP curve shifts down straight to (r3 , y N ). The AD curve shifts back to the original position. Therefore, the economy does not suffer from a recession while inflation is on target. 2.3 Divine Coincidence This is the ‘divine coincidence’ result — that is, in the face of demand shocks, when the central bank stabilises inflation, it stabilises output too. A way to see the divine coincidence result mathematically is to have a strict inflation targeting central bank (i.e. φπ → ∞). In other words, the central bank responds very aggressively to any inflation deviations, to keep to its inflation target. This implies that the slope of the AD curve is zero, meaning that demand shocks (changes in c) has no effect on output and inflation (a leftward shift of a horizontal AD curve is still the same AD curve. 3 3.1 Q6: Transitory adverse supply shocks Definition of supply shocks Firstly, we have to clarify that there are to types of supply-side shocks that the economy faces: • Inflation shocks ε t — shocks to price-setting • Supply shocks — shocks to y N The first causes shifts in the IA curve on impact. Oil price shocks would be an example of an inflation shock. The second causes changes in y N . In turn, this may shift the IA curve slowly if there are output deviations from the natural rate. 3.2 Inflation shocks See Figure 3. I take at this point the interpretation a temporary inflation shock. Therefore, the IA curve jumps up, and we go from point (1) to point (2). There is higher inflation, π2 , so the MP curve shifts up. Note that this is the endogenous response of the Taylor rule to higher inflation, so this is merely a movement along the AD curve. There is a recession, so this puts downward pressure on the IA curve, so in the long-run, we return to point (1) at (y N , π1 ). 3 3.3 Inflation-output tradeoff What if the CB intervenes and changes a1 ? Now there is no divine coincidence, the CB chooses between stabilising output or stabilising inflation (or somewhere in the middle). If it chooses to stablise output, it can do so by shifting the MP curve at the same time as the inflation shock, boosting output. This shifts the AD curve to the right, so we end up at point (3). There is no recession, as y = y N , but the cost is high inflation. Another possibility is inflation stabilisation. The CB can raise interest rates instead, creating a large recession. But in return, the IA curve shifts down rapidly so it returns to the low-inflation outcome more quickly at point (1). 4 Q8: Permanent adverse supply shock (doves vs hawks) Firstly, we need to clarify the properties of a dovish and hawkish policymaker. A dove cares about stabilising output (large φy , small φπ ), while a hawk cares more about stabilising inflation (small φy , large φπ ). There is no perfect mapping between the CB’s preferences over inflation and output and the parameters as the Taylor rule is only an ad-hoc, reduced-form, way of modelling CB behaviour. Later, I show mathematically that the source of dovishness (φy or φπ ) matters for the equilibrium outcome. However, the optimal (discretionary) policy for both dovish and hawkish policymakers is the same. In addition to the endogenous rise in interest rates from the fall in y N , the CB should contract monetary policy even more to ensure that y falls directly to y N . Therefore, there will be no inflationary pressure on the IA curve, and output is at its new lower natural level. I show below that the endogenous response of interest rates is never enough to fully reduce output to y N , no matter the parameterisation of the Taylor rule. 4.1 Dynamics What I analyse now is the dynamics of a rules-based policy, comparing a hawkish and dovish stance. The fall in potential output creates more complex dynamics as now there are two effects, and requires mathematics to quantify the two effects. We want to see what parameters determine the long-run rise in inflation from this shock. See Figure 4. 4 4.1.1 Shift of AD curve The first effect is a shift of the MP curve, and therefore, a leftward shift of the AD curve. Rearrange the AD curve to make y the subject: yt = i 1 h φπ c + φy y N − a1 − (πt − π ? ) b + φy b + φy Given that πt is a given (i.e. flat IA line), we can take it as a constant for the moment. Thus, if y N falls by 1 unit, yt falls by φy /(b + φy ). This is the horizontal movement along the IA curve. But given that φy ≥ 0 and b > 0, this number is always less than 1. Therefore, it must be the case that after the shock, y > y N . Obviously, the number is equal to zero if φy = 0 (i.e. there is no shift in the AD curve). 4.1.2 Slope of AD curve After the shock, there are no more shifts of the AD curve as all the other parameters remain constant. The second effect is the adjustment occurs upwards along the AD curve, due to inflationary pressure on the IA curve from y > y N . This adjustment stops when output equals its natural rate. What determines the inflation rise from this adjustment is the slope of the AD curve: −(b + φy )/φπ . 4.2 Overall effects Quantifying the inflation increase is easy, because it is just the height of the triangle. The distance (y − y N ) that the economy has to adjust until it reaches the higher inflation equilibrium is the difference between the fall in y N (1 unit) and the jump: 1− φy b = b + φy b + φy The height is just the distance covered times the slope of the AD curve: ∆π = b + φy b b · = b + φy φπ φπ Thus, we arrive at a result: the long-run inflation increase does not depend on the amount of output targeting — rather, only on the response to inflation deviations. 5 r r2 MP(π 2 ) r1 MP(π 1 ) IS Y π π2 π1 IA AD Y Figure 1: Derivation of AD Curve 6 r r1 2 1 MP(π 1 ) 3 r2 IS 2 MP(π 2 ) IS 1 Y π π1 2 1 IA 1 3 π2 IA LR AD2 Y2 YN Figure 2: Adverse Demand Shock 7 AD1 Y r r1 2 MP(π 2 ) 1 r2 MP(π 1 ) IS 1 Y π π1 2 IA 2 1 π2 IA1 AD1 Y2 YN Figure 3: Transitory Inflation Shock 8 Y 4.1.1 Given that now pt isthere a given IA line), we can mathematics take it as a constant for the the two effects. are (i.e. two flat effects, and requires to quantify Shift of AD curve 4.1.1 Shift of AD curve N moment. 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Thisshock, the AD curve, due to inflationary pressure on the IA curve from y > y adGiven that p is a given (i.e. flat IA line), we can take it as a constant for the Obviously, the number is equalthe tot number zero if fis 0 (i.e. is fno = shift in the ADis no shift in the y = Obviously, equal to there zero if 0 (i.e. there AD y N when Thus, outputifequals itsby natural What inflation moment. y falls 1 unit,rate. yt falls bydetermines fy /(b + fy )the . This is the horizontal curve). justment stops curve). rise from this adjustment is the of the But AD given curve:that(bf+y fy )0/fand p . b > 0, this number is movement along theslope IA curve. always less than 1. Therefore, it must be the case that after the shock, y > y N . Slope of AD 4.1.2 curve Slope of AD curve Obviously, 4.2 Overall effects the number is equal to zero if fy = 0 (i.e. there is no shift in the AD After the shock, there are the no more thenoAD curve as all the AD other paramcurve). After shock,shifts thereofare more shifts of the curve as all the other paramπ Quantifying the inflation increase is easy, because it is just the height of the trieters remain constant. The second effect is the adjustment occurs upwards along eters remain constant. The second effect is the adjustment occurs upwards along N angle.due Thetodistance (y pressure y ) that on thethe economy has to adjust reaches theN Nuntil the AD curve, inflationary IA curve from > yIA . curve Thisitad4.1.2 ADtocurve the ADSlope curve,of due inflationary pressure ony the from y > y . This adN inflation equilibrium isnatural the difference between the fall ininflation y (1 unit) and justment higher stops when output equals its rate. What determines the justment stops when output equals its natural rate. What determines the inflation After isthe shock, there no more shifts the AD as all the other paramrise fromthe thisjump: adjustment the slope of theare AD curve: b +the fofy )AD /fpcurve: . curve rise from this adjustment isy the slopeb(of (b + fy )/fp . f eters remain constant. 1 The second = effect is the adjustment occurs upwards along b + fy b + fy the AD curve, due to 3inflationary pressure on the IA curve from y > y N . This ad4.2 Overall effects effects IA The height 4.2 is justOverall theπstops distance covered theitsslope of the ADWhat curve: justment when outputtimes equals natural rate. LR determines the inflation 3 4.1.2 rise from thisthe adjustment is the itslope of thebecause AD curve: (btri+the fy )height /fp . of the triQuantifying the inflation increase is inflation easy, because height ofisthe Quantifying easy, it just b +isfisyjust bincrease b N Dp = · = angle. The distanceangle. (y y The ) that the economy adjust reaches the until it reaches the ftopthe fpuntil ithas distance (yb +yfNy)has that economy to adjust N 2 higher inflation equilibrium is the effects difference between fall1 in ybetween (1 unit) 4.2 Overall π higher inflation equilibrium is the the difference theand fall in y N (1 unit) and IA 1 1 at a result: the long-run inflation increase does not depend on the the jump:Thus, we arrive the jump: finflation b y rather, increase Quantifying the — isfeasy, because it is justdeviations. the height of the triamount of output targeting response bto inflation y 1 = only1on the = AD N b + f b + f y y angle. The distance (y y ) that bthe 1 to adjust until it reaches the + feconomy b + fhas y AD 2 y 5 higher inflation equilibrium is theofdifference between theYfall in y N (1 unit) and The height is just theThe distance times the slope the AD curve: N the N heightcovered is just the distance covered times slope of the AD curve: Y2 Y1 the jump: b + fy b b bfy b + f b Dp = · =1 = y = b b + fy fp Dp =fpb b++f fy· fb + f y fp y p The height is just theinflation distanceincrease covereddoes times the slope of curve: Thus, we arrive at a Thus, result: thearrive long-run not depend onthe theAD we at a result: the long-run inflation increase does not depend on the Figure 4: Permanent Supply Shock amount of output targeting on the to inflation deviations. bon + the fy response amount — of rather, output only targeting —response rather,b only to inflation deviations. b Dp = · = b + fy fp fp 5 5 Thus, we arrive at a result: the long-run inflation increase does not depend on the amount of output targeting — rather, only on the response to inflation deviations. 5 9