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Macroeconomic Theory Chapter 4 Monetary Policy Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University ‘reaction function’ ‘reaction function’ is what the CB uses to respond to shocks to the economy and steer it toward an explicit or implicit inflation target. Tasks of the reaction function are: 1. To provide a ‘nominal anchor’ for the medium run, which is defined in terms of an inflation target. 2. To provide guidance as to how the CB’s policy instrument, the interest rate, should be adjusted in response to different shocks so that the medium-run objective of stable inflation is met while minimizing output fluctuations CBs in the last two decades in OECD economies and in many transition and developing countries have shifted toward inflation-targeting regimes of this broad type. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University why low inflation-targets have been adopted. We begin by asking two questions: 1. What is wrong with inflation? 2. What is the ‘ideal’ rate of inflation? is it zero, positive or negative? we shall see the role played by the following six key variables in CB policy making: 1. the CB’s inflation target 2. the CB’s preferences 3. the slope of the Phillips curve 4. the interest sensitivity of aggregate demand 5. the equilibrium level of output 6. the stabilizing interest rate. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Inflation, disinflation, and deflation In the medium-run equilibrium, inflation is equal to the CB’s inflation-target, if the CB seeks to stabilize unemployment around the ERU. In the IS/LM version of the model, in the medium-run equilibrium, inflation is equal to the growth rate of the money supply set by the CB The Phillips curves are therefore indexed by lagged inflation (πI = π−1) and shift whenever π−1 changes: π = πI + α(y − ye) = π−1 + α(y − ye). Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University With linear Phillips curves, the sacrifice ratio is constant and independent of the CB’s preferences. Although the time path of unemployment is affected by the choice between a policy for (cold turkey) and a more gradualist policy, the cumulative amount of unemployment required to achieve a given reduction in inflation does not depend on the degree of inflation aversion of the CB. However, with non-linear Phillips curves, this is no longer the case: when the Phillips curves become flatter as unemployment rises, a ‘cold turkey’ policy of disinflation favored by a more inflation-averse CB entails a higher sacrifice ratio than does a ‘gradualist’ policy favored by a less inflation-averse CB. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Rising inflation rising inflation reflects a situation in which workers’ real wage aspirations are systematically frustrated: the real wage is typically on the PS curve, not on the WS curve. If there are lags in price setting as well as in wage setting, then the aspirations of neither workers nor firms are fully satisfied (the real wage lies between the PS and WS curves). This reflects distributional conflict as different social groups (wage setters/employees and price setters/employers) seek to protect their interests. for disinflation to be costless, expectations of inflation must be formed using the Rational Expectations Hypothesis, the commitment of the government and CB to a policy of low inflation at equilibrium unemployment has to be believed by the private sector and there must be no lags in the adjustment of wages and prices. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University For countries experiencing episodes of moderate inflation up to double digit rates per annum, these conditions do not appear to have been met Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Very high inflation and hyperinflation Hyperinflation has traditionally been defined as referring to a situation in which inflation rates rise above 50% per month Situations of very high and hyperinflation are normally the result of governments being unable to finance their expenditure through normal means (borrowing or taxation) and they therefore resort to monetary financing. This is known as seignorage. There is some evidence that the deterioration in the economic environment is associated with very high inflation. Very high inflation is typically associated with very poor performance: investment, consumption, and output are all depressed. The length of wage contracts becomes very short and there is increasing recourse to the use of foreign currency for transactions. It requires that the causes of the unsustainable fiscal stance be addressed and that the CB be prevented from financing the deficit through the creation of money but as is often the case in macroeconomics, this is easier said than done. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Volatile inflation When inflation is high it also seems to be more volatile. Volatile inflation is costly because it creates uncertainty and undermines the informational content of prices. Unexpected changes in inflation imply changes in real variables in the economy: if money wages and pensions are indexed by past inflation and there is an unanticipated jump in inflation, real wages and pensions will drop. Equally, the real return on savings will fall because the nominal interest rate only incorporates expected inflation. Volatile inflation masks the economically relevant changes in relative prices and therefore distorts resource allocation. In short, volatile inflation has real effects on the economy that are hard to avoid. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Constant inflation—what level is optimal? Imagine that we move from a situation in which prices are rising at 3% per year to a rate of 10% per year. We assume that this change is announced well in advance and that the tax system is indexed to inflation so that all the tax thresholds are raised by 10% p.a. The same is assumed to be true of pensions and other benefits. The consequence of this change will be that all wages, benefits, and prices will now rise at 10% p.a. and the nominal interest rate will be 7% points higher. All real magnitudes in the economy remain unchanged. The economy moves from a constant inflation equilibrium with π = 3%p.a. to a constant inflation equilibrium with π = 10% p.a. The real interest rate and the levels of output and employment remain unchanged. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University At high inflation, people wish to hold lower money balances they wish to economize on their holdings of money so for equilibrium in the money market, the real money supply must be lower than in the initial low inflation equilibrium. Since MS/P = L(i, y) = L(r + πE, y), at equilibrium output with low inflation, πL, we have: MS/Phigh = L((re + πL), ye) and at equilibrium output with high inflation, πH, we have: MSlPlow = L((re + πH), ye). This highlights the fact that even in our simple example the shift from inflation of 3% to 10% p.a. is not quite as straightforward as it seems at first. After the move to 10% inflation, money wages, prices, the nominal money supply, and nominal output will rise by 10% each year. But at the time of the shift, there has to be an additional upward jump in the price level to bring down the real money supply (MS/P) to its new lower equilibrium level ((MS/P)low) consistent with the demand for lower real money balances when inflation is higher. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University What are the real costs of people economizing on money balances when inflation is high? These costs are sometimes referred to as ‘shoe-leather’ costs. Other costs (so-called menu costs) arise because of the time and effort involved in changing price lists frequently in an inflationary environment. These costs are estimated to be quite low We note here an apparent paradox: if the rate of inflation does not matter much, why should governments incur the costs of getting inflation down from a high and stable level to a Low and stable one? One response is that it seems empirically to be the case that inflation is more volatile when it is higher and as noted above, volatile inflation brings additional costs. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Another reason is that the initiation of disinflation policies frequently begins with high and rising inflation. In this case, since costs will be incurred in stabilizing inflation, it may be sensible for the government to go for low inflation as part of a package that seeks to establish its stability-oriented credentials. Once we relax our assumption that indexation to inflation is widespread in the economy and that adjustment to higher inflation is instantaneous because all parties are fully informed and can change their prices and wages at low cost, it is clear that the costs of switching to a high inflation economy are likely to be more substantial. The continuous reduction in individuals’ living standards between wage adjustments gives rise to anxiety. Distributional effects are also likely to occur: unanticipated inflation shifts wealth from creditors to debtors. It is also likely to make the elderly poorer since they rely on imperfectly indexed pensions and on the interest income from savings. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Can we infer from this analysis that the optimal rate of inflation is zero or even negative? In thinking about the optimal inflation rate, we are led first of all to consider the following: The return on holding high-powered money (notes and coins) is zero so with any positive inflation rate, the real return turns negative. The negative real return leads people to waste effort economizing on their money holdings. If we follow the logic of this argument then with a positive real rate of interest, for the nominal interest rate to be zero, inflation would have to be negative. This was Milton Friedman’s view of the optimal rate of inflation: the rate of deflation should equal the real rate of interest, leaving the nominal interest rate equal to zero. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Deflation Is deflation optimal? If inflation is negative (e.g. −2% p.a.), prices and wages will be 2% lower in a year’s time than they are now. In a world of perfect information, there would only be benefits from this as we have already seen-shoe leather would be saved. In spite of these arguments, there are two main reasons why deflation is not viewed as a good target by CBs. The first reason relates to the apparent difficulty in cutting nominal wages. If workers are particularly resistant to money wage cuts, then a positive rate of inflation creates the flexibility needed to achieve changes in relative wages. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University The second reason stems from the need for the CB to maintain a defense against a deflation trap. A deflation trap can emerge when weak aggregate demand leads inflation to fall and eventually become negative. For this to happen, two things are necessary: (i) the automatic self-stabilizers that operate to boost aggregate demand when inflation is falling fail to operate sufficiently strongly and (ii) policy makers fail to stop prices falling. Attempts to use monetary policy to stimulate the economy result in the nominal interest rate falling. A nominal interest rate close to zero (as low as it can go) combined with deflation implies a positive real interest rate. This may be too high to stimulate private sector demand. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Continued weak demand will fuel deflation and push the real interest rate up, which is exactly the wrong policy impulse. This will tend to weaken demand further and sustain the upward pressure on the real interest rate. Once deflation takes hold, it can feed on itself and unlike a process of rising inflation, it does not require the active cooperation of the CB for the process to continue Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Monetary policy paradigms The first paradigm, the money supply model or LM paradigm, characterized by the following propositions: (1) The ultimate determinant of the P and π is MS; (2) the instrument of monetary policy is MS; (3) the mechanism through which the economy adjusts to a new equilibrium with constant inflation following a shock is that embodied in the IS/LM model plus the inertia-augmented (or expectations-augmented) Phillips curve. The second paradigm, the interest rate reaction function or MR paradigm, characterized as follows: (1) the ultimate determinant of the P and π is policy; (2) the instrument of policy is the short-term nominal interest rate; (3) the mechanism through which the economy adjusts to a new equilibrium with constant inflation following a shock is encapsulated in an interest rate rule. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University The monetary policy rule in the 3-equation model We pin down the role played by the following six key variables in CB policy making: (1) the CB’s inflation target, πT (2) the CB’s preferences, β (3) the slope of the Phillips curve, α (4) the interest sensitivity of aggregate demand (i.e. the slope of the IS curve), a (5) the equilibrium level of output, ye (6) the stabilizing interest rate, rS. In order to make the discussion of monetary policy rules concrete, we shall: (1) Define the CB’s utility function in terms of both output and inflation. This produces the policy maker’s indifference curves in output-inflation space. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University (2) Define the constraints faced by the policy maker: these are the Phillips curves. (3) Derive the optimal monetary rule in output-inflation space: this is the MR line. Hidden in this relationship is the policy instrument, r, that the CB will use to secure the appropriate level of aggregate demand and output. (4) We can also derive the interest rate rule, which tells the CB how to adjust the r in response to current economic conditions. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University The CB’s utility function We assume that CB has two concerns: the rate of inflation, π, and the level of output, y. 1. The rate of inflation 2. We assume that CB has a πT and that it wants to minimize fluctuations around πT, i.e., it wants to minimize the loss function: (π − πT)2 This particular loss function has two implications: First, the CB is as concerned to avoid inflation below πT as it is above πT. If πT = 2% the loss from π = 4% is the same as the loss from π = 0%. In both cases (π − πT )2 = 4. Second, it attaches increased importance to bringing π back to πT the further it is away from πT ; the loss from π = 6% is 16, compared to the loss of 4 from π = 4%. The CB’s marginal disutility is increasing as π − πT grows. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University 2.Output and employment. Assume the CB’s target is ye and it seeks to minimize the gap between y and ye. The CB’s loss as a result of y being different from its ye is: (y − ye)2. the CB understands the model and realizes that inflation is only constant at y = ye. If y < ye then this represents unnecessary unemployment that should be eliminated. If y > ye , this is unsustainable and will require costly increases in unemployment to bring the associated inflation back down. Adding the two loss functions together, we have the CB’s objective function: L = (y − ye)2 + β(π − πT )2, (CB loss function) where β is the relative weight attached to the loss from inflation. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University β is a critical parameter: a β > 1 means the CB places less weight on deviations in employment from its target than on deviations in inflation, and vice versa. The loss function is simple to draw: with β = 1, each indifference curve is a circle with (ye, πT) at its centre (see Fig. 5.1(a)). The loss declines as the circle gets smaller. When π = πT and y = ye , the circle shrinks to a single point (called the ‘bliss point’) and the loss is at a minimum, which is zero. With β = 1, the CB is indifferent between inflation 1% above (or below) πT and output 1% below (or above) ye. They are on the same loss circle. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University If β > 1 (inflation avert), the CB is indifferent between (say) inflation 1% above (or below) πT and output 2% above (or below) ye. They are on the same loss curve. This makes the indifference curves ellipsoid with a horizontal orientation, Fig. 5.1(b). A CB with less inflation aversion (β < 1) will have ellipsoid indifference curves with a vertical orientation (Fig. 5.1(c)). The indifference curves are steep reflecting that the CB is only willing to trade off a given fall in inflation for a smaller fall in output than in the other two cases. If the CB cares only about inflation then β = ∞ and the loss ellipses become one dimensional along the line at π = πT. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Figure 5.1 β=1 β>1 β<1 Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University The Phillips curve constraint Assume that the CB can control y by using monetary policy to control aggregate demand, yD. However, it cannot control inflation directly -only indirectly via y. As discussed before, output affects inflation via the Phillips curve: π = π−1 + α.(y − ye). This is shown in Fig. 5.2. For simplicity assume that α = 1, so that each Phillips curve has a slope of 45◦. Assume that π−1 = πT = 2%. The CB is in the happy position of being able to choose the bull’s eye point B or (πT , ye) at which its loss is zero. Suppose, that inflation is 4%. The bull’s eye is no longer obtainable. The CB faces a trade-off: if it wants a level of output of y = ye next period, then it has to accept an inflation rate above target, i.e. π = 4 = πT (point A). Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University If it wishes to hit πT it must accept a lower level of output (point C). Point A corresponds to a fully accommodating monetary policy in which the objective to hit the ye (β = 0), and point C corresponds to a non-accommodating policy, in which the objective is to hit the inflation target (β = ∞). The CB can do better by minimizing its loss function by choosing point D, where the PC (πI = 4) line is tangential to the indifference curve of the loss function closest to the bull’s eye, output = y1 which will in turn imply an inflation rate of 3%. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Figure 5.2 Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Deriving the monetary rule, MR For simplicity, we use the form of the loss function in which β = 1 so that we have loss circles as in Fig. 5.2 above. This implies: L = (y − ye)2 + (π − πT )2. Using the simplest version of the Phillips curve in which α = 1 so that each PC has a 45◦ slope as in Fig. 5.2: π = π−1 + y − ye . In Fig. 5.3, the points of tangency between successive Phillips curves and the loss circles show the level of output that the CB needs to choose to minimize its loss at any given level of π−1. Thus when π−1 = 3, its loss is minimized at C; or when π−1 = 4 at D. Joining these points (D,C, B) produces the MR line that we used in Chapter 3. We can see from Fig. 5.3 that a one unit rise in π−1 implies a half unit fall in y, for example an increase in π−1 from 3% to 4% implies a fall in y from y2 to y1. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Figure 5.3 Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University We can derive the monetary rule explicitly as follows. By choosing y to minimize L we can derive the optimal value of y for each value of π−1. Substituting the Phillips curve into L and minimizing with respect to y, we have: ∂L/∂y= 2(y − ye) + 2(π−1 + (y − ye) − πT) = 0 = (y − ye) + (π−1 + (y − ye) − πT) = 0. Since π = π−1 + y − ye , ∂L/∂y= (y − ye) + (π − πT) = 0 =⇒ (y − ye) = −(π − πT ). (MR equation) The monetary rule in the Phillips diagram shows the equilibrium for the CB: it shows the equilibrium relationship between the π chosen indirectly and y chosen directly by the CB to maximize its utility (minimize its loss) given its preferences and the constraints it faces. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University This shows the monetary rule as an inverse relation between π and y with a negative 45◦ slope (Fig. 5.3). Specifically, it shows that the CB must reduce yD and y, below ye so as to reduce π below πT by the same percentage. Thus this could be thought of as monetary policy halfway between: (i) completely non-accommodating when the CB cuts output sufficiently to bring π straight back to πT at the cost of a sharp rise in unemployment; (ii) a completely accommodating one, which leaves π (and y) unchanged. If the monetary rule was flat at πT we would have a completely non-accommodating monetary policy; if it was vertical at ye , we would have a completely accommodating monetary policy. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University The monetary rule ends up exactly halfway between an accommodating and a non accommodating policy because of the two simplifying assumptions. By relaxing these assumptions, we learn what it is that determines the slope of the monetary rule. The first factor that determines the slope of the monetary rule is the degree of inflation aversion of the CB is captured by β in the CB loss function: L = (y − ye)2 + β(π − πT )2. If β > 1, the CB attaches more importance to the inflation target than to the output target. This results in a flatter monetary rule as shown in Fig. 5.4. Given these preferences, any inflation shock that shifts the PC upward implies that the optimal position for the CB will involve a more significant output reduction and hence a sharper cut in inflation along that PC than in the neutral case. Using the same reasoning, β < 1 implies that the monetary rule is steeper. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Figure 5.4 Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University The second factor that determines the slope of the monetary rule is the responsiveness of inflation to output (i.e. the slope of the PC): π −π−1 = α(y −ye). If α > 1 so the PCs are steeper, any given cut in y has a greater effect in reducing inflation than when α = 1. As we can see from Fig. 5.5, this makes the MR line flatter than in the case in which α = 1: MR0 is the old and MR1 the new monetary rule line obtained by joining up the points D, C, and B. Steeper PCs make the MR line flatter. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Figure 5.5 Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Let us now compare the response of a CB to a given rise in inflation in the case where the PCs are steep with the case where they have a slope of one. Our intuition tells us that steeper PCs make things easier for the CB since a smaller rise in unemployment (fall in output) is required to achieve any desired fall in inflation. In the left hand panel of Fig. 5.6 we compare two economies, one with flatter PCs (dashed) and one with steeper ones. The MR line is flatter for the economy with steeper PCs: this is MR1. Suppose there is a rise in inflation in each economy that shifts the PCs up: each economy is at point B. We can see that a smaller cut in aggregate demand is optimal in the economy with the steeper PCs (point D). Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Fig. 5.6 Identical PC two different preferences Inflation ave Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University In the right hand panel, we compare two economies with identical supply sides (same PC) but in which one has an inflation-averse CB (the oval-shaped indifference ellipse) and show the CB’s reaction to inflation at point B. The more inflation-averse CB always responds to this shock by cutting aggregate demand (and output) more (point D). Derivation of the general form of the CB’s monetary rule. By choosing the interest rate in period zero, the CB affects y and π in period 1. We assume it is only concerned with what happens in period 1. This is the reason that its loss function is defined in terms of y1 and π1. If we let β and α take any positive values, the CB chooses y to minimize: L = (y1 − ye)2 + β(π1 − πT)2 (5.2) subject to: π1 = π0 + α(y1 − ye) (5.3) Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University By substituting (5.3) into (5.2) and differentiating with respect to y1 (since this is the variable the CB can control via its choice of the interest rate), we have: ∂L/∂y1 = (y1 − ye) + αβ(π0 + α(y1 − ye) − πT) = 0. (5.4) Substituting equation (5.3) back into equation (5.4) gives: (y1 − ye) = −αβ(π1 − πT ). (monetary rule, MR) Now it can be seen directly that the larger is α or the larger is β the flatter will be the slope of the monetary rule. In the first case (larger α) this is because any reduction in aggregate demand achieves a bigger cut in inflation. In the second case (lager β), this is because, whatever the labor market it faces, a more inflation- averse CB will wish to reduce inflation by more than a less ‘extreme’ one. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Using the IS-PC-MR graphical mode Given the determinants of the slope MR slope, the role of each of the six key inputs to the deliberations of the CB is now clear. (1) the CB’s inflation target, πT: this affects the position of the MR; (2) the CB’s preferences, β: this determines the shape of the loss ellipses and affects the slope of the MR; (3) the slope of the PC, α: this also affects the slope of the MR line; (4) the interest sensitivity of yD, a: this determines the slope of the IS; (5) the equilibrium level of output, ye : this determines the position of the vertical PC and affects the position of the MR line; (6) the stabilizing interest rate, rS: the CB adjusts r relative to rS so it must always analyze whether this has shifted, e.g. as a result of a shift in the IS or due to a change in the equilibrium level of output, ye . Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University On the basis of the previous discussion, the IS-PC-MR model can be used to analyze a variety of problems. An example to clarify the CB’s decision and to highlight the role played by the lag in the effect of monetary policy on yD and y. The example shows that the CB is engaged in a forecasting exercise: it must forecast next period’s PC and IS curve. We assume that the economy starts off with ye and πT of 2% as shown in Fig. 5.7. We take a permanent positive aggregate demand shock, the IS moves to IS’. As y is above ye, π will rise above πT; in this case to 4%. This defines next period’s PC (PC(πI = 4)) along which the CB must choose its preferred point: point C. The CB forecasts that the IS curve is IS’, i.e. it judges that this is a permanent shock and by going vertically up to point C’ in the IS diagram, it can work out that the appropriate interest rate to set is r’. As the PC shifts down with falling inflation, the CB reduces the interest rate and the economy moves down the MR line to point Z and down the IS’ curve to Z’. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Fig. 5.7 Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University This example highlights the role of the stabilizing real interest rate, rS: following the shift in the IS curve, there is a new stabilizing interest rate and, in order to reduce inflation, the interest rate must be raised above the new rS, i.e. to r’. The CB is forward looking and takes all available information into account: its ability to control the economy is limited by the presence of inflation inertia. In the IS equation it is the interest rate at time zero that affects output at time one: y1 − ye = −a(r0 − rS). This is because it takes time for a change in the interest rate to feed through to consumption and investment decisions. In Fig. 5.7 in order to choose its optimal point C on the PC (πI = 4), the CB must set the interest rate now at r’. However, it is interesting to see what happens if the CB could affect y immediately, i.e. if y0 − ye = −a(r0 − rS). Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University In this case, as soon as the IS shock is diagnosed, the CB would raise the interest rate to rS’. The economy then goes directly from A’ to Z’ in the IS diagram and it remains at A in the Phillips diagram, i.e. points A and Z coincide. Since the aggregate demand shock is fully and immediately offset by the change in the interest rate, there is no chance for inflation to rise. This underlines the crucial role of lags and hence of forecasting for the CB: the more timely and accurate are forecasts of shifts in aggregate demand, the greater is the chance that the CB can offset them and limit their impact on inflation. Once inflation has been affected, the presence of inflation inertia means that the CB must change the interest rate and get the economy onto the MR line in order to steer it back to the inflation target. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University A Taylor Rule in the IS-PC-MR model Interest rate rules: We now show how to derive an interest rate rule, which directly expresses the change in the interest rate in terms of the current state of the economy. We then show how it relates to the famous Taylor Rule. We bring together the three equations: π1 = π0 + α(y1 − ye) (Phillips curve) y1 − ye = −a(r0 − rS) (IS) π1 − πT = −1/αβ (y1 − ye). (MR) From these equations, we want to derive a formula for the interest rate, r0 in terms of period zero observations of inflation and output in the economy. If we substitute for π1. Using the Phillips curve in the MR, we get: π0 + α(y1 − ye) − πT = −1/αβ (y1 − ye) π0 − πT = −α + 1/αβ(y1 − ye) Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University and if we now substitute for (y1 − ye) using the IS, we get the interest-rate rule: r0 − rS = 1/(a(α + 1/αβ)) (π0 − πT). (Interest rate rule) We can see that r0 − rS = 0.5 π0 − πT if a = α = β = 1 Two things are immediately apparent: First, only inflation and not output deviation is present in the rule Second, all the parameters of the 3-equation model matter for the CB’s response to a rise in inflation. If inflation is 1% point above the target, then the interest rate rule says that the real interest rate needs to be 0.5% higher. Since inflation is higher by 1%, the nominal interest rate must be raised by 1 + 0.5, i.e. by 1.5% in order to secure a rise in the real interest rate of 0.5 percentage points. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University For a given deviation of inflation from target, and in each case, comparing the situation with that in which a = α = β = 1, we can see that a more inflation-averse CB (β > 1) will raise the interest rate by more; • when the IS is flatter (a > 1), the CB will raise the interest rate by less; • when the Phillips curve is steeper (α > 1), the CB will raise the interest rate by less. Let us compare the interest rate rule that we have derived from the 3-equation model with the famous Taylor Rule, r0 − rS = 0.5.(π0 − πT) + 0.5.(y0 − ye), (Taylor Rule) The Taylor Rule states that if output is 1% above equilibrium and inflation is at the target, the CB should raise the interest rate by 0.5 percentage points relative to stabilizing interest rate. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Interest rate rules and lags The interest rate rule derived from the 3-equation model is similar to Taylor’s rule. However, it only requires the CB to respond to inflation. This seems paradoxical, given that the CB cares about both inflation and output (equation 5.2). It turns out that to get an interest rate rule that is like the Taylor rule in which both the inflation and output deviations are present, we need to modify the 3-equation model to bring the lag structure closer to that of a real economy. As before we assume that there is no observational time lag for the monetary authorities, i.e. the CB can set the interest rate (r0) as soon as it observes current data (π0 and y0). We continue to assume that the interest rate only has an effect on output next period, i.e. r0 affects y1. The new assumption about timing that is required is that it takes a year for output to affect inflation, i.e. the output level y1 affects inflation a period later, π2. This means that it is y0 and not y1 that is in the Phillips curve for π1. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University The empirical evidence is that on average it takes up to about one year for the response to a monetary policy change to have its peak effect on demand and production, and that it takes up to a further year for these activity changes to have their fullest impact on the inflation rate. The “double lag” structure is shown in Fig. 5.8 and emphasizes that a decision taken today by the CB to react to a shock will only affect the inflation rate two periods later, i.e. π2. When the economy is disturbed in the current period (period zero), the CB looks ahead to the implications for inflation and sets the interest rate so as to determine y1, which in turn determines the desired value of π2. Since the CB can only choose y1 and π2 by its interest rate decision, its loss function is: L = (y1 − ye)2 + β(π2 − πT )2. Given the double lag, the three equations are: π1 = π0 + α(y0 − ye) (Phillips curve) y1 − ye = −a(r0 − rS) (IS) π2 − πT = − 1/αβ (y1 − ye). (MR) Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University FIGURE 5.8 Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University By repeating the same steps as above, we can derive the interest rate rule, which takes the form of a Taylor rule: r0 − rS = 1/(a(α + 1/αβ) ((π0 − πT) + α(y0 − ye)). And r0 − rS = 0.5(π0 − πT) + 0.5(y0 − ye) (Taylor rule in 3-eq. (double lag) model) if a = α = β = 1. In Fig. 5.9, the initial observation of output and inflation in period zero is shown by the large cross, ×. To work out what interest rate to set, the CB notes that in the following period, inflation will rise to π1 and output will still be at y0 since a change in the interest rate can only affect y1. The CB therefore knows that the constraint it faces is the PC(π1) and it chooses its best position on it to deliver π2. The best position on PC(π1) is shown by where the MR line crosses it. This means that output must be y1 and therefore that the CB sets r0 in response to the initial information shown by point ×. This emphasizes that the CB must forecast a further period ahead in the double lag model in order to locate the appropriate PC, and hence to determine its optimal r choice for today: it chooses r0 → y1 → π2. Once the economy is on the MR line, the CB continues to adjust the interest rate to guide the economy along the MR back to equilibrium. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University FIGURE 5.9 Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University The remaining task is to give a geometric presentation of the double lag model and the associated Taylor Rule: rt −rS = 0.5 . (πt −πT)+0.5 .(yt −ye). Fig. 5.10 shows the example in Fig. 5.9 again. As shown in the left hand panel of Fig. 5.10, the two components of the Taylor Rule are shown by the vertical distances equal to α(y0 −ye) and π0 −πT , where α is the slope of the Phillips curve. If these are added together, we have the forecast of π1−πT . Just one more step is needed to express this forecast in terms of (r0 − rS) and therefore to deliver a Taylor Rule. As shown in the right hand panel of Fig. 5.10, the vertical distance π1 − πT can also be expressed as (α + γ) . a(r0 − rS), where α and γ = 1/αβ reflect the slopes of the Phillips curve and the monetary rule curve, respectively and a reflects the slope of the IS curve. Thus, we have: (α + γ) . a(r0 − rS) = (π0 − πT) + α(y0 − ye) Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University FIGURE 5.10 Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University by rearranging to write this in terms of the interest rate, we have a Taylor Rule: r0 − rS = 1/(α + γ) a π0 − πT + α(y0 − ye) = 0.5 . (π0 − πT) + 0.5 . (y0 − ye) if α = γ = a = 1 Once we modify the model to reflect the fact that a change in output takes a year to affect inflation (the double lag model), then both the inflation and output deviations appear in the interest rate rule and it resembles Taylor’s Rule. The reason is that the current period output deviation serves as a means of forecasting future inflation to which the CB will want to react now. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Problems with using an interest rate rule The CB may sometimes be dissatisfied in its attempt to use an interest rate rule to stabilize the economy: One reason would be if aggregate demand fail to respond or to respond enough to the change in the interest rate. Empirical evidence for the impact of changes in the cost of capital relative to the expected rate of return is rather weak. Another reason arises from the fact that the interest rate that is relevant to investment decisions is the long term real interest rate. The CB can affect the short-term nominal interest rate. The relationship is referred to as the term structure of interest rates. The long-term interest rate refers to the interest rate now (i.e. at time t) on an n-year bond. We can express the long-term interest rate as follows: Int = 1/n . [i1t + i1t +1|t + i1t +2|t + … + i1t +n−1|t] + φnt . (5.5) In words, this means the long-term interest rate is equal to the average of the expected interest rate on one-year bonds for the next twenty years plus the term (phi) φnt , which is called the ‘uncertainty premium’. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University In calm times, we would expect the long-term interest rate to exceed the short-term rate by the uncertainty premium and we would expect short- and long-term interest rates to move in the same direction. Monetary policy will then have the desired effect. As a counter-example, suppose the CB cuts the short-term interest rate to stimulate the economy because it fears a recession is imminent. If the financial markets believe that higher inflation will prevail in the long run, markets will believe a higher long-run real interest rate will be necessary. Higher long-term interest rates are likely to dampen interest-sensitive spending at a time when the authorities are trying to stimulate the economy. A third example comes from the fact that the nominal interest rate cannot be negative. In a very low inflation economy, there is therefore limited scope to use monetary policy to stimulate aggregate demand if the required real interest rate is negative, e.g. with an inflation target of 2%, the zero floor to the nominal interest rate means that real interest cannot be reduced below −2%. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University To summarize, the reasons that monetary policy can fail to have its desired effect on output include the following: 1. investment is insensitive to the real interest rate; 2. the long-run real interest rate does not move in line with changes in the short-term nominal interest rate; 3. the CB wishes to stimulate demand but the nominal interest rate is close to zero. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University The deflation trap The simplest way to see how a deflation trap may operate is to combine the fact that i cannot be negative with the fact that r is approximately: r = i − πE. Since i ≥ 0, the minimum r is r = -π. When inflation is positive, this does not matter very much in general since the minimum r is negative. But when π < 0 the minimum r is positive. The problem that can arise is that the real rate needed to stabilize demand at ye is less than the minimum feasible real rate, i.e. rs < min r(π) = -π. This condition is shown in Fig. 5.11 where rs is below the minimum feasible rate of 1%. Given the depressed state of aggregate demand depicted by the position of the IS curve, if inflation has fallen to −1%, then it will be impossible to achieve the equilibrium level of output. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Figure 5.11 Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University The monetary policy approach of using i to set r associated with aggregate demand at equilibrium output ceases to work. Assume the CB sets the lowest r possible, r = −π, so that y = y0 and the economy is at point A. Since y0 < ye , the consequence is that inflation falls. That implies that the minimum r rises, further reducing output and hence increasing the speed at which inflation falls. The economy is thus caught in a vicious circle or a deflation trap. It is clear from Fig. 5.11 that getting out of the deflation trap requires either (1) a successful fiscal expansion or recovery of autonomous investment or consumption that shifts the IS curve to the right or (2) the creation of more positive inflation expectations. But the only way to create expectations of inflation in the future is to create expectations of future higher aggregate demand: if the authorities do not take measures to create the demand, it is no good hoping that people will expect higher inflation. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University There is an additional channel through which a deflation trap can be sustained. Just as unanticipated inflation shifts wealth from creditors to debtors in the economy as the real value of debts is eroded, unanticipated deflation has the opposite effect. If asset prices in the economy are falling as well as goods prices, then debtors in the economy will not only find that the real burden of their debt is rising but also that the assets that they have used as security or collateral for the debt are shrinking in value. This so-called balance sheet channel may make investment less sensitive to changes in the real interest rate thereby steepening the IS curve and weakening the investment response even if positive inflation expectations could be generated. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Credibility, time inconsistency, and rules versus discretion Backward-looking Phillips curves and credibility In the IS-PC-MR model, the PC is backward looking: π = π−1 + α.(y − ye), This is consistent with the evidence that disinflation is costly, i.e. in order to reduce inflation, output must be reduced. The debate about how best to model the inflation process is a very lively one in macroeconomic research at present. The key point to highlight here is that although the inertial or backward-looking PC matches the empirical evidence concerning inflation persistence, it has a major shortcoming, it does not allow a role for ‘credibility’ in the way monetary policy affects outcomes. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University We can demonstrate the point using an example. In Fig. 5.12, we assume that the CB’s inflation target is 4% and the economy is initially at point A with high but stable inflation of 4% (on PC(πI = 4)). The CB now decides to reduce its inflation target to 2%, i.e. πT1 = 2%.With backward-looking PC, it is clear from that disinflation will be costly and following the announced change in inflation target, unemployment first goes up (shown by point B). The economy then shifts only gradually to the new equilibrium at Z as the CB implements the monetary rule. Whether or not the CB’s decision is announced and if so whether it is believed by the private sector makes no difference at all to the path of inflation. The inflation that is built into the system takes time to work its way out. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Figure 5.12 Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University 6.2 Introducing inflation bias In the IS-PC-MR model, medium-run equilibrium is characterized by inflation equal to the CB’s inflation target and output at equilibrium. However, since we have seen that imperfect competition in product and labor markets implies that ye is less than the competitive full-employment level, the government may have a higher target. We assume that the government can impose this target on the CB. How do things change if the CB’s target is full-employment output, or more generally a level of output above ye? A starting point is to look at the CB’s new objective function. It now wants to minimize: L = (y − yT )2 + β(π − πT )2, (5.6) where yT > ye . This is subject as before to the Phillips curve, π = π−1 + α(y − ye) (5.7) Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University In Fig. 5.13 the CB ideal point is now point A (where y = yT and π = πT) rather than where y = ye and π = πT (i.e. point C). If we assume that α = β = 1 then each indifference circle has its centre at A. To work out the CB’s monetary rule, consider the level of output it chooses if πI = 2% Fig. 5.13 shows the PC corresponding to πI = 2%. The tangency of PC(2) with the indifference circle shows where the CB’s loss is minimized (point D). Since the CB’s monetary rule must also pass through A, it is the downward- sloping line MR in Fig. 5.13. The government’s target, point A, does not lie on the Phillips curve for πT = 2%: the economy will only be in equilibrium with constant inflation at point B. This is where the monetary rule (MR) intersects the vertical Phillips curve at y = ye . At point B, inflation is above the target: the target rate is 2%but inflation is 4%: this gap between the πT and π inflation in the equilibrium is called the inflation bias. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University the CB chooses its preferred point on the πI = 2% PC and the economy is at D. But with y above ye, inflation goes up to 3% and the PC shifts up. The process of adjustment continues until point B: output is at the equilibrium and inflation does not change so the PC remains fixed. But neither inflation nor output are at the CB’s target. We can derive the same result mathematically. Minimizing the CB’s loss function - equation (5.6) - subject to the PC curve - equation (5.7) implies y − yT + αβ(π−1 + α(y − ye) − πT) = y − yT + αβ(π − πT) = 0. So the new monetary rule is: y − yT = −αβ(π − πT) (5.8) This equation indeed goes through (πT, yT). Since equilibrium requires that π−1 = π when y = ye , we have ye = yT − αβ(π−1 − πT) ⇒ π = π−1 = πT + ((yT − ye)/αβ). (inflation bias) inflation bias Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University figure 5.13 Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University In equilibrium, inflation will exceed the target by (yT−ye) αβ . This is called the inflation bias. The significance of this result is that π > πT whenever yT > ye . The steeper is the CB’s monetary rule, the greater will be the inflation bias. A lower α also raises the inflation bias. A lower α implies that inflation is less responsive to changes in output. Therefore, any given reduction in inflation is more expensive in lost output; so, in cost-benefit terms for the CB, it pays to allow a little more inflation and a little less output loss. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Time inconsistency and inflation bias We can link the problem of inflation bias to problems of credibility and time inconsistency by adopting a forward-looking Phillips curve. The simplest assumption to make is that inflation expectations are formed rationally and that there is no inflation inertia: i.e. πE = π + εt. The intuition is that wage setters know that whatever their expected rate of inflation, the condition for πE = π is that y = ye. This is the so-called Lucas surprise supply equation: yt − ye = 1/α(πt − πEt) yt = ye + 1/α(πt − πEt) (Lucas surprise supply equation) = ye + ξt We continue to assume that the CB chooses y (and hence π) after wage setters have chosen πE. This defines the CB as acting with discretion. Now, in order for wage setters to have correct inflation expectations, they must choose πE such that it pays the CB to choose y = ye. That must be where the CB’s monetary rule cuts the y = ye vertical line, i.e. at point B in Fig. 5.13. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Solutions to the time-inconsistency problem The inflation bias presents a problem. the loss to the CB at B is greater than the loss to the CB at C since output is the same but inflation is higher at B. So the CB would clearly be better off at C. Moreover, wage setters would be just as happy at C as at B, since employment and the real wage are the same in each case. What is to stop the CB being at C? When wage and price setters are forward looking, the problem is called that of time inconsistency. Although the CB claims to have an inflation target of πT , if wage setters act on the basis of this target (2%), when it comes to act, the CB does not choose the output level consistent with its target. In short, at point B there is no incentive for the CB to cheat; whereas at point C, there is an incentive. We have seen that the time-inconsistency problem arises under the following circumstances: 1. the CB has an over-ambitious output target (i.e. yT > ye) 2. wage and price setters form expectations using rational expectations 3. the CB uses a rule-based reaction function but operates with discretion, i.e. chooses its desired level of aggregate demand after inflation expectations have been formed in the private sector. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University There are three broad approaches to solving time-inconsistency problem. 1. Replacing discretion by a rule: If the timing of the game between the CB and private sector is changed so that the CB cannot choose the rate of inflation after wage and price setters have formed their expectations, then the inflation bias disappears. This entails a structure through which the CB is prevented from optimizing after the private sector has set wages and prices and is referred to as a policy of commitment rather than discretion. 2. Delegation The inflation bias is equal to (yT−ye).αβ , and this may reflect a situation in which the government rather than the CB controls monetary policy. The government could reduce the inflation bias by transferring control of monetary policy to an independent CB. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University Fig. 5.14 illustrates the reduction in inflation bias through delegation of monetary policy to the CB. The flatter sloped monetary rule is that of the CB, MRCB, and the more steeply sloped that of the government, MRG. MRG evidently implies a higher inflation bias with the equilibrium at point B. MRCB on the other hand implies that equilibrium is at point A, with π = 3%. Wage and price setters rationally expect a smaller inflation surprise when faced with an independent CB than when faced by the government. For delegation to produce a costless move from high to low inflation, there must be no inflation inertia and expectations must be formed rationally. In this case, if wage setters believe that the policy maker’s preferences have changed in the appropriate way, the economy will shift directly down the vertical Phillips curve at ye from point B to the new equilibrium with π = 3% at point A. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University 5.14 Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University One problem with this proposed solution is that if the government can delegate powers to the CB, why can’t it take them back when it wants to? It would pay the government to take back those powers at the moment that wage setters chose a low πE corresponding to the loss function parameters of the CB. For then the government would be tempted to opt for a level of output greater than ye. 3. Reputation A third solution to the problem of inflation bias lies with the government or CB building a reputation for being tough on inflation. Suppose that the government has delegated monetary policy to the CB but wage setters remain unsure of just how independent the CB is. They only know that there is a probability p that the CB is independent and a probability (1 − p) that it is a puppet of the government. The only way that they can find out is by observing the decisions taken by the CB. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University The situation is one in which the CB interacts with wage setters more than once. we can say that it is possible for the CB to build a reputation for toughness as a method of solving the inflation bias problem. Let us begin with the case in which the interaction between the CB and wage setters occurs twice: in period 1, wage setters choose πE1 with no knowledge of whether the CB is weak or tough; the CB then chooses output in period 1, y1 knowing πE1. In period 2, the wage setters choose πE2 knowing y1; the CB then chooses y2 knowing πE2. The result is that a weak CB will choose to act like a tough one in the 1st period, which will establish a low expected inflation rate in the 2nd period, thereby providing bigger gains from boosting output in the 2nd period. The CB gains because in the 1st period, the outcome is inflation at its target (no inflation bias) and output at the equilibrium, whilst in the 2nd period, it can gain by setting output above the equilibrium. When the game is extended from two to many periods, the benefits to the CB from behaving as if it were tough increase. This is because the situation in period one is repeated again and again until the last period. Macroeconomic Theory Prof. M. El-Sakka CBA. Kuwait University

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