Stabilization policy with rational expectations. IAM ch 21.

Transcription

1 Stabilization policy with rational expectations. IAM ch 21. Ragnar Nymoen Department of Economics, UiO Revised 20 October 2009

2 Backward-looking expectations (IAM 21.1) I From the notes to IAM Ch 20, we have the final equation for inflation π t = or: 1 (1 + γα) π t 1 + (π t π ) = where γ (1 + γα) z t + 1 (1 + γα) (π t 1 π ) + α = 1 (1 + γα) s t + γα (1 + γα) π γ (1 + γα) z t + α 2 h (1 + α 2 b), β = γα, and z t = α 1 (g t ḡ) + v t (1 + α 2 b) (1) 1 (1 + γα) s t (2)

3 Backward-looking expectations (IAM 21.1) II We now investigate inflation expectations errors, assuming that t = 0 is the initial situation, and that expectation for period t = 1, 2,.. are made at the start of period 1, using information available at the end of period 0. Expectations are made for the (infinite) horizon t = 1, 2, 3,... Although we have so far referred to backward-looking expectations formation as static, we are in fact "using the model" to generate the agents expectations for period t = 2, 3,... In more detail, the expectations formation is π e t = π 0, for t = 1 but (3) π e t = π emod t = π t 1, for t = 2, 3 > 1 (4) where π emod t denotes the model based forecast and π t 1 is the period t 1 solution obtained by the use of the final equation.

4 Backward-looking expectations (IAM 21.1) III Note that this do not entail that the agents has to know the model (i.e. to sit in this auditorium). Hence expectations are in fact not backward-looking in the strict sense of π e t = π 0 for all t. To analyze the expectations errors, we assume that g t = ḡ and v t = s t = 0 both in the economy and in the expectations formation process. This is the same simplification as on p 630 in IAM. The period 1 error (remember that 0 is the initial period): e1 π = π 1 π e 1 = (π 1 π ) (π e 1 π ) = 1 (1 + γα) (π 0 π ) (π 0 π ) = γα (1 + γα) (π π 0 ) (5)

5 Backward-looking expectations (IAM 21.1) IV Period 2 error: e2 π = π 2 π e 2 = (π 2 π ) (π e 2 π ) ( ) 1 2 = (π 0 π 1 ) 1 + γα (1 + γα) (π 0 π ) And generally: = (β 2 β)(π 0 π ) = β(β 1)(π 0 π ) e π t = β t 1 (β 1)(π 0 π ) for t = 2, 3,.. (6) which is equation (10) in IAM p 630. Assume that π 0 = π. Then (5) and (6) shows that the expectations generated by (3) and (4) are always accurate.

6 Backward-looking expectations (IAM 21.1) V If π 0 π, the forecasts are biased but 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% and π = 0. The critique against backward-looking expectations is that the forecast bias (systematic error) only goes away gradually, cf also the Lucas critique at the end of this set.

7 AD-AS model with rational expectations (IAM 21.2) I The rational expectations hypothesis (REH) equates agents subjective expectations about a variable, for example π e t+1, with the mathematical expectation conditional on an information set I t. Rational expectations are without systematic errors, which, in macroeconomics entails that the true model of the economy is part of the information set. With this extra assumption, RE amounts to model consistent expectations. The agents must know the model before they can form rational expectations. Let π e t t 1 and y t t 1 e denote the rational expectations for period t conditional on information available at the end of period t 1.

8 AD-AS model with rational expectations (IAM 21.2) II The short-run AD-AS model with RE is given by y t ȳ = α 1 (g t ḡ) α 2 (r t r) + v t, (7) r t = i t π e t+1 t 1, (8) i t = r + π e t+1 t 1 (π + h e t1 t 1 π ) (9) ) +b (yt t 1 e ȳ, π t = π e t t 1 + γ(y t ȳ) + s t. (10) v t and s t are assumed to have zero expectations. They have variances σ 2 v and σ 2 s, and they are uncorrelated. For simplicity we set g e t t 1 = ḡ

9 Solving the RE model I Step 1 Express y t and π t in terms of expectations and exogenous variables y t = ȳ + α 1 (g t ḡ) (11) ] +α 2 [h (π e t t 1 π ) + b(yt t 1 e ȳ) + v t π t = π e t t 1 + γα 1 (g t ḡ) (12) ] +γα 2 [h (π e t t 1 π ) + b(yt t 1 e ȳ) + γv t + s t.

10 Solving the RE model II Step 2 Obtain the functional relationships for the mathematical expectations π e t t 1 and y t t 1 e from the expressions in Step 1: ) ] yt t 1 e = ȳ+α 1 (gt t 1 e ḡ +α 2 [h (π e t t 1 π ) + b(yt t 1 e ȳ) }{{} =0 (13) and ) π e t t 1 = π e t t 1 + γα 1 (gt t 1 e ḡ + [ ]} γ {α 2 h (π e t t 1 π ) + b(yt t 1 e ȳ) 0 = h ( π e t t 1 π ) + b(yt t 1 e ȳ) (14)

11 Solving the RE model III Together with (13) this is seen to imply: y e t t 1 = ȳ, (14) π e t t 1 = π We now have found the rational expectations for GDP and inflation. Since these are variables in the AD and AS equations, the last step of the solution is to insert these REs into the expressions in Step 1.

12 Solving the RE model IV Step 3 Insertion back into (11) and (12) gives the rational expectations solution: y t = ȳ + α 1 (g t ḡ) + v t (15) π t = π + γα 1 (g t ḡ) + γv t + s t (16) Which is a static model. There are no persistence or business cycles in this solution

13 Policy ineffectiveness I (g t ḡ) is a fiscal policy surprise, which does not appear in IAM since they set g t = ḡ from the outset. We have used the weaker assumption that g e t t 1 = ḡ. A simple model for g t which is consistent with this is g t = ḡ + ε gt where ε gt has conditional expectation zero. Using, g t ḡ = ε gt, the RE solution can be expressed as: y t = ȳ + α 1 ε gt + v t, (17) π t = π + γv t + s t. (18) Since none of the policy variables or parameters (π, h or b) enter into (17), this is called the policy ineffectiveness proposition.

14 Policy ineffectiveness II Systematic monetary policy does not affect GDP. This is because the rule based monetary policy cannot generate inflation surprises which drives y t via the AS function (y t ȳ) = 1 γ (π t π e t t 1 ) + 1 γ s t which in this form is known as Lucas supply function. To avoid the conclusion about policy ineffectiveness the model needs to be modified: Assume that the central bank acts on the basis of actual inflation and output. Instead of the Taylor rule (9), we use eq (28) in IAM r t = r + h (π t π ) + b (y t ȳ) to represent the situation that central bank can react to π t and y t after inflation expectations have been formed in the private sector.

15 Policy effectiveness under RE With the modified Taylor rule, IAM p 636 and 637, shows that the RE solution is (use Step 1 - Step 3), and set g t = ḡ as in the book): π t = π + (1 + α 2b)s t + γv t 1 + α 2 (b + γh), (19) y t = ȳ + v t α 2 hs t 1 + α 2 (b + γh) (20) GPD is now influenced by systematic monetary policy. For example: Increased weight (b) on GDP in Taylor rule reduces the impact of supply and demand shocks. The reason is that the central bank can affect the real interest rate by changing the nominal rate after inflation expectations have been set.

16 Optimal stabilization policy under RE I The analysis is in terms of the same social loss function as in ch 20. What matters is the variability of inflation and GDP. Taking variances on both sides of (19) and (20): σ 2 y = σ 2 v + (α 2 h) 2 σ 2 s [1 + α 2 (b + γh)] 2, (21) σ 2 π = γ2 σ 2 v + (1 + α 2 b) 2 σ 2 s [1 + α 2 (b + γh)] 2 (22) We also follow the method of first analyzing the case of pure demand shocks, and then the case of isolated supply shocks.

17 Optimal stabilization policy under RE II Demand shocks only (σ 2 s = 0): σ 2 y = σ 2 π = σ 2 v 1 + α 2 (b + γh) γ 2 σ 2 v [1 + α 2 (b + γh)] 2 Both variances are reduced by choosing high positive values for b and h. Optimal to choose highest possible values of both b and h, subject to i t 0. No conflict between GDP and inflation stabilization. This is the same qualitative conclusion as with static expectations.

18 Optimal stabilization policy under RE III Supply shocks only (σ 2 v = 0): From (21) and (22) we have: σ 2 y = σ 2 π = (α 2 h) 2 σ 2 s [1 + α 2 (b + γh)] 2 (1 + α 2 b) 2 σ 2 s [1 + α 2 (b + γh)] 2 It is not possible to see, by direct inspection, how choices of h and b influence these variances. Therfore we need the derivatives with respects to b and h.

19 Optimal stabilization policy under RE IV To calculate, it is practical to first take logs and both sides ln σ 2 y = ln[(α 2 h) 2 σ 2 s ] 2 ln [1 + α 2 (b + γh)] ln σ 2 π = ln[(1 + α 2 b) 2 σ 2 s ] 2 ln [1 + α 2 (b + γh)] and then find the derivatives. σ 2 y b σ 2 y h σ 2 π b σ 2 π h = = = = 2α 2 [1 + α 2 (b + γh)] σ2 y < 0 if b > 0 2(1 + α 2 b) h [1 + α 2 (b + γh)] σ2 y 2α 2 2 γh (1 + α 2 b) [1 + α 2 (b + γh)] σ2 π 2α 2 [1 + α 2 (b + γh)] σ2 π < 0 if b > 0

20 Optimal stabilization policy under RE V We see that the direct derivatives σ2 y b and σ2 π h are negative subject to b > 0 (countercyclical interest rate setting). Remember that h > 0 from the Taylor principle But σ2 y h and σ2 π b are both positive (given that b > 0), meaning that when supply shocks are dominant, there is a conflict of priorities between GDP stabilization and inflation stabilization. So in general, there is a trade-off, reflected by the choice of parameter κ in the social loss function. Again, these conclusions are qualitatively the same as in the case with static inflation expectations.

21 A modified Taylor-rule (gradualism) I In the RE solution, all endogenous variables adjust instantaneously to a shock. This includes the nominal interest rate. IAM contains an important paragraph ( The optimality of a modified Taylor rule under forward-looking expectations ) explaining why a Taylor rule that comes closer to empirical Taylor-rules may actually be optimal. In empirical studies we often estimate: i t = r + h(π t π ) + h(y t ȳ) + ci t 1, c > 0 (23) Which corroborates that many central banks normally prefer to adjust the interest rate in small but frequent steps. ie, gradualism in interest rate setting.

22 A modified Taylor-rule (gradualism) II The theoretical point is that, with c rather large, an adjustment of i t will signal a higher interest rate in the future. The result is that a policy interest rate change is likely to affect mortgage rates and other long term rates more effectively than would be the case if c = 0. The policy instrument gets a stronger effect in the monetary transmission mechanism.

23 The Lucas critique I The Lucas critique states that models with backward-looking expectations give wrong predictions about policy effects. The example with a reduction in π in period t serves as an illustration. Remember that the AD and AS equations with static expectations are: (y t ȳ) = α(π t π ) z t, π t = π t 1 + γ(y t ȳ) + s t showing that if π is reduced π in period t, GDP is reduced. This due to increased real interest rate. Already, this is different from the RE solution, compare (17) and (20), which show no reduction in GDP when π is reduced.

24 The Lucas critique II The dynamic effects on GDP in the static model are intuitively clear: There must be a gradual increase in GDP, back to ȳ. This is achieved by a gradual reduction in inflation, and a reduction of the real interest rate. It may be confusing that the final equation that we (and IAM) have derived for (y t ȳ) does not contain π as a parameter, so the algebraic solution of the model does not seem to match the economic interpretation. The explanation is that in the derivation, we have assumed that π is a constant parameter, and it therefore drops out from the solution.

25 The Lucas critique III To allow for a time varying inflation target, we write the model as (y t ȳ) = α(π t π t ) z t π t = π t 1 + γ(y t ȳ) + s t and re-do the derivation of the final equation, which becomes (y t ȳ) = β(y t 1 ȳ) + β(z t z t 1 ) αβs t + αβ(π t π t 1) for GDP, and for inflation. π t = βπ t 1 + γαβπ t + γβz t + βs t β = αγ, and (1 β) = αγ 1 + αγ as before.

26 The Lucas critique IV We now see that a permanent change in the inflation target has the same effect in the backward-looking model as a permanent negative demand shock. The sequence of dynamic multipliers are therefore negative. but diminishing in magnitude. Inflation is gradually reduced down to the new target, the long run-multiplier for inflation is as we know it should be. δ π long run = γαβ 1 β = 1 The critique is that this gives a misleading analysis of the effects of a lower inflation target, if expectations are in fact rational. As we have seen, in the RE solution, expectations adjust fully in period t if the change was announced in period t 1.

27 The Lucas critique V GDP is unaffected, meaning that there is less welfare loss under RE. Today, the validity of Lucas critique is taken for granted in large parts of the economist profession. But the relevance of the critique hinges on the REH being a suffi ciently good approximation to real world expectations.