Lecture 9: Stabilization policy with rational expecations; Limits to stabilization policy; closed economy case. Ragnar Nymoen Department of Economics, University of Oslo October 17, 2008 1
Ch21andch22inIAM Comprehensive coverage of issues and premises that are central in modern discussion of stabilization policies Limitation: Closed economy. Focus on the issues that are most analytically tractable. 2
Backward-looking vs forward-looking expectations (IAM 21.1 and 21.2) Lecture 8 established the reference case with backward-looking expectations. IAM refers to that model as static expectations, but as we show see below inflation expectations are really dynamic since they follow the same dynamic process as π t itself,butwithoneperiodlag. Start with the subsection in IAM called case against backward-looking expectations which points out as a weakness that after a policy change expectations adjust gradually, even when the policy change is perfectly credible. This motivates the rational expectations hypothesis, REH. 3
The case against backward-looking expectations The short-run AD-AS model of lecture 8 consists of and: π t = π 1 α (y t z t ), (1) with α = α 2h 1+α 2 b and z t = α 0 + v t + α 1 g t α 2 r + α 2 bȳ. 1+α 2 b π t = π t 1 + γ(y t ȳ)+s t. (2) The short-run aggregate demand (SRAD) function, and the short-run aggregate supply (SRAS) function. They define the equilibrium solutions for y t and π t as functions of the exogenous and predetermined variables:g t, v t, s t and π t 1. We established last week that the SRAD was invariant to how inflation expectations are formed. 4
But SRAS clearly depends on expectations formation, here we have used so called static expectations π e t = π t 1. The case against backward-looking expectations rests on the behaviour of inflation expectations in the face of a credible change (reduction) in the inflation target. To analyze this, we need the dynamic equation for inflation. To obtain that, write the SRAD schedule with y t on the left-hand side, and insert into (2). This gives: π t = 1 1+γα π t 1 + for t =1, 2, 3, γα 1+γα π + γ 1+γα (z t ȳ)+ 1 1+γα s t, (3) which is the final equation for π t since it has only predetermined and exogenous variables on the right hand side. The autoregressive coefficient is the same as in the final equation that we derived for y t last week. Is that a coincidence? 5
It gives insight to re-write (3) in terms of deviations from target, as in IAM eq (9): subtract π on both sides of the equation to give π t π = 1 1+γα (π t 1 π )+ IAM uses the notation γ 1+γα (z t ȳ)+ 1 1+γα s t, (4) β = 1 1+γα. which is positive but less than one under the assumptions we made in lecture 8. Thestablesteady-statesolutionforπ t,denoted π, is: π = π + 1 ( z ȳ) α where ȳ is full capacity output, and z is the steady-state value of z t. As we discussed in the last lecture, z is an endogenous variable in the steady-state with π = π. Since z = α 0 + α 1 ḡ α 2 r + α 2 bȳ 1+α 2 b 6
the natural choice of endogenous variable is the equilibrium real-interest rate r. 7
Expectation errors We now investigate expectations errors, assuming that t = 0 is the initial situation, and that expectation for period t =1, 2,.. is made at the start of period 1, using information that is available at the end of period 0. Forecast aremadeforthe(infinite) horizon t =1, 2, 3,... Although we have so far referred to the backward-looking expectations formation as static, we are in fact using the model to generate the agents forecast for period t =2, 3,... Hence, for the purpose of this lecture where π e,mod t πt e = π 0,for t = 1 but (5) πt e = πt e,mod, for t>1 (6) denotes the model based forecast. π e,mod t is defined as: 8
πt e,mod = 1 1+γα π t 2 + γα 1+γα π + γ 1+γα (ze t 1 ȳ), for t>1 (7) For period t =2, 3,..., economic agents use solution for π t 1 (form (3)) to generate their forecasts, but with s 2 = s 3... =0,andv 2 = v 3... =0since there is no way that agents (at the start of period 1) can anticipate the sequence of future supply and demand shocks. zt e denotes z twith v t =0(forallt) However, in order to simplify, we assume that g t =ḡ and v t = s t =0 both in the economy and in the expectations. This is the same simplification as on p 630 in IAM. π e,mod t canbewrittenas: πt e,mod = 1 1+γα πe,mod t 1 + γα 1+γα π + γ ( z ȳ), for t>1 1+γα where z is the constant obtained by imposing g t =ḡ and v t = s t =0inthe expression for z t. Hence expectations follow the same dynamic equations as inflation itself: 9
The expectations error in period t is: e π t = π t πt e,mod =(π t π ) (π t 1 π ) = β(π t 1 π )+ γ 1+γα ( z ȳ) (π t 1 π ) =(β 1)(π t 1 π )+ γ ( z ȳ) 1+γα =(β 1)β(π t 2 π γ )+(β 1) ( z ȳ)+ γ 1+γα =(β 1)β(π t 2 π γ )+β ( z ȳ) 1+γα Continue with repeated backward substitution to obtain: ( z ȳ) 1+γα e π t = π t πt e,mod =(β 1)β t 1 (π 0 π )+β t 1 γ ( z ȳ) (8) 1+γα for t =2, 3,... which is the counterpart to equation (10) in IAM. Note that this result is based on agents knowing how the economy operates. Hence expectations are not backward-looking in the strict sense of setting π e t = π 0 for all t. 10
The case against (partly) backward-looking expectations Assume that π 0 = π so that z = ȳ for consistency. Then (8) shows that expectations generated by (5) and (6) are always accurate. If π 0 6= π and z 6= ȳ, the forecasts become gradually closer to the inflation target. The inflation forecast during the phase of adjustment may be as depicted in Figure 21.1 in IAM, for the case of π 0 =3%andπ =0. The stronger case against backward-looking expectations is the following construction: Assume that π 0 = π at the time that the agents form their forecast. Then, immediately after, the central bank changes the target to π 0 =0. 11
From period 1 and onwards the economy and expectations evolve according to π t = 1 α (y t z), t =1, 2,... π e,mod t = ( π t = π e,mod t 1 1+γαπe mod t 1 + γα 1+γα π + + γ(y t ȳ)+s t π 0,fort =1 ( z ȳ) fort =2, 3 γ 1+γα The first equation is the SRAD curve after the policy change at the start of period 1. The second equation has the expectations evolving according to the old version of the economy. The third line is the Phillips curve. Now there is a logical inconsistency in the model: Thedemandsideofthemodelisconsistentwithπ t 0ast. The supply side, because expectations are linked to the old regime where π e,mod t π and π t π. 12
Hence it is difficult to claim that this system has a meaningful solution,apart from the short run model, for t =1. This seems to be a much stronger case against (what IAM call) backwardlooking expectations. However, a minutes reflection reveals that things may not be so dramatic. First, note that the short-run model is logically consistent, since in period 1, i.e. the period of the policy change, we have π 1 = 1 α (y 1 z), π e,mod 1 = π 0 π 1 = π 0 + γ(y 1 ȳ) 13
In period 2, the first period after the shock, there is no reason to expect that agents stick to their old forecasting rule. Instead they will use πt e,mod = βπt 1 e mod + γ ( z ȳ), for t>1, 1+γα since, once agents are able to condition their expectations on the policy change, they will replace their expectations by the updated forecasts.hence we may re-specify the model with backward-looking expectation and a policy-change from π > 0toπ = 0 at the start of period 1 as: π t = 1 α (y t z), t =1, 2,... π e,mod t = ( π t = π e,mod t βπ e mod t 1 + γ 1+γα π 0,fort =1 ( z ȳ) fort =2, 3 + γ(y t ȳ)+s t and πt e,mod new target and π t new target, just as in the analysis with no policy change. There are no logical inconsistencies. The feature with gradual adjustment toward the target remains, though. Can there be a faster adjustment? 14
Rational expectations The rational expectations hypothesis (REH) equates agents subjective expectations about a variable, for example πt+1 e,withthemathematical expectation conditional on an information set I t. In macroeconomics it is assumed that the model coincides with the true relationships of the economy, apart from the error terms that are completely random. With this extra assumption, REH amount to the same thing as model consistent expectations. We also use this connotation of rational expectations in the following. Let π t t 1 e denote the rational expectations for period t inflation conditional on information available at the end of period t 1. The short-run model with rational expectations (RE): 15
y t ȳ = α 1 (g t ḡ) α 2 (r t r)+v t, α 1 > 0,α 2 > 0 (9) r t = i t πt+1 t 1 e (10) i t = r + π e t+1 t 1 + h(π t π )+b(y t ȳ),h>0,b>0 (11) m t π t p t 1 = m 0 m 1 i t + m 2 y t,m i > 0, i =1, 2 (12) π t = πt t 1 e + γ(y t ȳ)+s t (13) wherewehaveadoptedthesamenotionasiniamalsofortheproductmarket, hence ȳ = α 0 + α 1 ḡ α 2 r. In addition, it is assumed that g t =ḡ, asiniam. 16
To solve (9)-(13) under the assumption of RE we first write y t and π t in terms of their respective expectations, and exogenous variables: Insertinginto(13)gives y t ȳ = α 2 h(π t π ) α 2 b(y t ȳ)+v t, (14) π t = π e t t 1 + γ [ α 2h(π t π ) α 2 b(y t ȳ)+v t ]+s t. (15) We have now two equations in the two endogenous variables y t ȳ and π t : (1 + α 2 b)(y t ȳ)+α 2 h(π t π )=v t (16) γα 2 b(y t ȳ)+γα 2 h(π t π )=πt t 1 e + γv t + s t (17) The solution is: (y t ȳ) = v t α 2 hs t α 2 h(π e t t 1 π ) 1+α 2 (b + γh) π t π = (1 + α 2h)(π e t t 1 π )+(1+α 2 b)s t + γv t 1+α 2 (b + γh) (18) (19) 17
As a second step we solve for the inflation expectations under the assumption that agents do not know the realization of v t and s t, they rationally set v t = s t = 0 in (19). Therefore: π e t t 1 = π (20) Finally, we solve for the two endogenous variables y t and π t,using(20)and (18) and (19). y t =ȳ + v t α 2 hs t 1+α 2 (b + γh) π t = π + (1 + α 2b)s t + γv t 1+α 2 (b + γh) which is (35) and (34) in IAM. (21) (22) 18
There are three important properties to note about this solution 1. The RE solution consists of static models for y t and π t. There is no effects from initial conditions. There is no propagation of shock. The only nonzero interim multipliers from supply and demand shocks are the impact multipliers. 2. Inflation expectations are equal to the target π,alsowhenπ t 6= π. 3. The RE solution is affected by (systematic) monetary policy, since the Taylor coefficients enter into the solutions (21) and (22). Regarding 1. Expectations in this models are static in the sense that π t t 1 e = π, and since the only source of dynamics is expectations, the solution is also static. Therefore the solution does not contains any persistence or propagation of shocks. 19
Regarding 2 Expectations are always equal to the target, so there is no transition period during which expectations stay below or above target for a long time. This does not mean that expectations are always correct. For example RE gives π e t t 1 = π π e t+1 t 1 = π also in the case when the target is changes to 0 at the start of period t. But once the expectations are updated, by conditioning on period t, they are fully adjusted: Regarding 3 π e t+1 t =0 We will not discuss optimal monetary policy, but refer to IAM pp 637-640. The conclusion is that the qualitative conclusions of the analysis with backward looking expectations still apply: If demand shocks dominate, there is no conflict between price and output stabilization, but there is with supply-shocks. 20
Policy ineffectiveness proposition RE is often association with the policy ineffectiveness proposition, which states that systematic monetary (or fiscal) policy, in the form of a Taylor-rule for example, does not affect the solution. To invoke this result within our framework we need to replace the Taylor-rule with this alternative i t = r + πt+1 t e + h(πe t t 1 π )+b(yt t 1 e r t = h(πt t 1 e π )+b(yt t 1 e ȳ) ȳ) which is (14) in IAM. In this case the central bank is forecasting period t inflation and output using thesameinformationsetfromperiodt 1 as the wage-and price setters, who form expectation about π t t 1 e in the Phillips-curve. In the Taylor-rule above we assume that the central bank observes y t and π t, or that the central bank 21
can nowcast these two variables very accurately. The RE solution obtained on p634-5iniamis π t = π + γv t y t =ȳ + v t where the policy coefficients b and h do not enter. The intuition for policy ineffectiveness is that from the Phillips curve part of themodel,wehave: (y t ȳ) =γ 1 ³ π t π e t t 1 + γ 1 s t showing that the only way that policy can affect output in period t is by generating an inflation surprise: π t π t t 1 e 6= 0. The possibility of this is assumed away when the Taylor-rule is in terms of (π t t 1 e π )and(y t t 1 e ȳ). Today, most economy take the view that there is some room for affecting the π t and y t after the private sector has committed itself to a inflation forecast for period t. Nominal rigidities is a common term that captures this stance. 22
The Lucas critique One aspect of the Lucas critique is that models with backward looking expectations give wrong predictions about policy effects. Theexamplewithareductioninπ in period t serves as an illustration: The AD-AS model with backward-looking expectations then predicts a (long) period with falling inflation and unemployment (y t+j < ȳ). But the model with RE shows that expectations adjust fully in period t +1, or already in period t if the change was announced in period t 1. Moreover, output is unaffected meaning that there is less welfare loss under RE. Today, the validity Lucas critique is simply taken for granted in large parts of the profession. But it hinges on the REH being a sufficiently good approximation to real world expectations. 23
The limits to stabilization policy (ch 22) Fine tuning the economy to avoid large and persistent deviation from full employment is a difficult task, also in the case where full employment is the only steady-state of the economy (as the PCM implies). Ch 22 in IAM identifies three types of assumptions that have been implicit so far: 1. No information lags. The policy maker knows the state of the economy (the initial situation). There are no serious measurement problems. 2. There are no adjustments lags in policy decisions and policy instruments canbeadjusted bothupanddown. 3. The announced policy is credible. IAM discusses all three. Here we concentrate on 3. 24
Rules vs discretion and time consistency Use again the AD-AS model, but simplify even more than before by assuming v t = s t =0,g t =ḡ, π =0andγ =1 We assume RE. From (21) and (22) the equilibrium under a conventional Taylor-rule (with π =0)is and from (20) y t =ȳ, (23) π t =0 (24) π e t t 1 =0. (25) 25
Assume that the central bank would like to minimize the loss-function where SL =(y t y ) 2 + κπ 2 t (26) y =ȳ + ω, ω>0. ω is a parameter that reflects labour market distortions/inefficiencies. ȳ is not really the full employment output. The value of SL under the Taylor-rule: SL R = ω 2 > 0 Is this the minimum SL? Note that SL is given by Since γ =1, we have the PCM SL = {(y t ȳ)+(ȳ y )} 2 + κπ 2 t π t π e t t 1 = y t ȳ 26
and therefore SL can be written: SL = n (π t πt t 1 e )+(ȳ y ) o 2 + κπ 2 t (27) Assume that the central bank has led the public to believe that it will ensure price stability, so π t t 1 e =0asintherulebasedREsolution. Whatisthe optimal π t if the central can deviate from its rule? The cheating inflation rate is: πt c =min{(π t ω) 2 + κπt 2 } or Insertion of π e t t 1 πt c ω + κπt c =0 πt c = ω 1+κ > 0 = 0 into the PCM equation π t π e t t 1 =(y t ȳ) 27
gives yt c =ȳ + ω 1+κ. These cheatingvalues oftherateofinflation and output give zero SL, which is optimal: The effect from distortions ω on output are reduced. However, the cheating policy is self-defeating since agents will learn to form expectations from (π t π e t t 1 ω)+κπ t =0 which is the 1oc for minimization of (27) with respect to π t. Implying that insertion into the PCM π e t t 1 = ω κ = π t (28) (28) and (29) define the time-consistent RE equilibrium. Inflation is higher than in the Taylor-rule equilibrium. y t =ȳ (29) 28
Therefore loss of credibility in monetary policy can be said to lead to an inflation bias. This is a result which has been very influential in practice. For example: One of the main rationale for credibility building, which often involves delegation of monetary policy and central bank independence. One interesting point made in IAM is that the case for central bank independence with a strict mandate for inflation stabilization is strongest when demand shocks dominate. With dominating supply shocks, the results may be excess variability in output. 29