Optimal monetary policy

Transcription

1 Opimal moneary policy Discreion versus commimen sabilizaion bias value of he loss funcion (weighed average of he variances of inflaion and he oupu gap) is smaller under commimen price level sabilizaion under commimen he cenral bank promises o sabilize he price level endogenous persisence even in he case of ransiory (whie-noise) shocks, oupu, inflaion and he ineres rae deviae from heir arge levels for many periods under commimen Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 1

2 Opimal moneary policy Discreion versus commimen sabilizaion bias 6 5 Discreion Commimen Variance Oupu Gap Discreion Commimen Variance Inflaion ρ u ρ u Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide

3 Opimal moneary policy Discreion versus commimen sabilizaion bias 3 5 Discreion Commimen Loss ρ u Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 3

4 Opimal moneary policy Discreion versus commimen sabilizaion bias 6 5 Discreion Commimen Variance Oupu Gap 5 4 Discreion Commimen Variance Inflaion θ θ Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 4

5 Opimal moneary policy Discreion versus commimen sabilizaion bias 5 4 Discreion Commimen Loss θ Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 5

6 Opimal moneary policy Discreion versus commimen price level sabilizaion under commimen impulse responses o a cos-push shock (ρ u = ) Inflaion Oupu gap doed lines: opimal discreionary policy Price level 4 Nominal ineres rae Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 6

7 Opimal moneary policy Discreion versus commimen opimal policy is idenical under commimen and discreion in he iniial period τ = consolidaed firs-order condiion in he iniial period κ κ x = π = ( p p 1) αx αx consolidaed firs-order condiion in all oher periods κ κ x = x 1 π = x 1 ( p p ) 1 α x α x κ x 1 = x ( p 1 p ) α x boh foc can be represened by a single equaion in levels (under he assumpion ha x 1 = ): κ κ x p p p ( ) ˆ = 1 = αx αx Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 7

8 Opimal moneary policy Discreion versus commimen p 1 is he price level arge, i.e. he seady-sae value ha prevailed one period before he cenral bank chooses is opimal plan under discreion he cenral bank keeps oupu above is efficien level as along as inflaion is negaive under commimen he cenral bank keeps oupu above is efficien level as along as he price level is below is implici arge hus he price level will always rever o is iniial level Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 8

9 Opimal moneary policy Discreion versus commimen using he soluion for he oupu gap a a 4β κ x = x + u β α βρ 4β and he foc x ( ) u a a 1 κ = α x pˆ x gives a saionary soluion for he deviaion of he price level from arge a a 4β pˆ = pˆ u 1 β βρu a a 4β Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 9

10 Opimal moneary policy Discreion versus commimen endogenous persisence under commimen impulse responses o a cos-push shock (ρ u = ) Inflaion Oupu gap doed lines: opimal discreionary policy Price level 4 Nominal ineres rae Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 1

11 Opimal moneary policy Discreion versus commimen endogenous persisence under commimen under discreion, boh, he oupu gap and inflaion reurn o heir zero iniial value one period afer he shock under commimen, he deviaions persis well beyond he life of he shock a a 4β κ x = x + u β α βρ 4β π ( ) u a a 1 x α a a 4β κ β βρ 4β x = 1 x 1 u ( ) u a a Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 11

12 Opimal moneary policy Discreion versus commimen endogenous persisence under commimen why does he cenral bank behave like his, given ha zero inflaion and a closed oupu gap would be opimal? because of he forward-looking behavior of inflaion afer forward-ieraion he NKPC can be wrien as π = βeπ + κx + u = + 1 k = κx+ κ β Ex + k+ u k= 1 no only he curren oupu gap, bu also fuure expeced oupu gaps have an impac on curren inflaion hus he inflaionary impac of a cos-push shock can also be lowered by commiing o lower fuure oupu gaps Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 1

13 Opimal moneary policy Discreion versus commimen endogenous persisence under commimen only allowing he curren oupu gap o become negaive as a consequence of policy acions (as under discreion) is very resricive (and of course a special case of he commimen case) commimen widens he specrum of combinaions of oupu gaps and inflaion so ha in general he inflaion/oupu gap radeoff is improved Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 13

14 Simple opimal policy rules assume he cenral bank adops a simple Taylor rule of he form i = ρ+ φπ iˆ = φπ π π he economy evolves according o (NEW) π = βeπ + κx + u x = i E + Ex + 1 e x = ( i Eπ r ) + Ex σ e e r = ρ + σe y+ 1 + (1 ρd ) d = = ρ a σ α α ϕ ( π + 1 ρ) + 1 v σ ( 1+ ϕ)( 1 ρa ) σ ( 1 ) + + σ ( 1 α ) ( α + ϕ) + e ( 1 ) σ α + α + ϕ + e Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 14

15 Simple opimal policy rules (NEW) u and v are independen and idenically disribued (i.i.d.), muually uncorrelaed, supply and demand disurbances wih variances given by σ u = σ v = 1; demand disurbances are serially uncorrelaed he cenral bank minimizes he following loss funcion W = α var + ( x ) var ( π ) x wha is he value of he inflaion coefficien φ π ha minimizes he cenral bank s loss funcion? analyically: solve for he equilibrium processes for x and π as a funcion of he shocks, calculae he variances as a funcion of he shocks, find φ π ha minimizes W numerically: NKM_simplepolicy_.m Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 15

16 Simple opimal policy rules (NEW) value of he loss funcion as funcion of he inflaion coefficien (serially correlaed supply shock, ρ u =.8) 3 5 simple rule discreion commimen α x var(x ) + var(π ) minimum a φ π Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 16

17 Simple opimal policy rules (NEW) value of he loss funcion as funcion of he inflaion coefficien (serially uncorrelaed supply shock, ρ u =.) simple rule discreion commimen α x var(x ) + var(π ) minimum a ( v ) ( u ) var φπ = σε 1 + ( α x + κ ) = var = φ π Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 17

18 Uncerainy in moneary policy The conrol of moneary policy is carried ou in a conex of grea uncerainy In he sandard approach o analyzing moneary policy (i.e. seing-up a model of he economy, specifying an objecive for he moneary policy maker, and deermining how moneary policy should respond o disurbances o he economy) uncerainy occurs in wo respecs: 1. uncerainy abou wha happens in he fuure and. uncerainy abou he curren descripion of he economy Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 18

19 Uncerainy in moneary policy Uncerainy abou wha happens in he fuure refers o he sochasic naure of he course of an economy under he assumpion ha he policy maker possesses perfec knowledge abou he model describing he macroeconomic relaionships, he is unable o predic fuure evens in our model se-up he sochasic fuure disurbances ener in he form of addiive shocks which are independenly and idenically disribued wih known sochasic properies (in paricular, wih zero mean and a consan variance) on a echnical level, i can be shown ha, for he choice of opimal policy rule for he policy insrumen, he oucome under uncerainy is idenical wih ha occurring under a cenral bank ha is assumed o be compleely cerain abou how he economy is developing Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 19

20 Uncerainy in moneary policy Uncerainy abou wha happens in he fuure his so-called cerainy equivalence principle (or someimes separaion principle) follows from he paricular srucure of he opimizaion problem according o which he cenral bank s objecive funcion is quadraic, he model of he economy is linear, and he expeced fuure value of he disurbances is zero for a echnical reamen of he cerainy equivalence principle see Sargen, Thomas J. (1997), Dynamic Macroeconomic Theory, Cambridge, Mass., and Ljungqvis, Lars and Thomas J. Sargen (), Recursive Macroeconomic Theory, Cambridge, Mass. Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide

21 Uncerainy in moneary policy Uncerainy abou he curren descripion of he economy 1. uncerainy abou he model used by he moneary policy maker (model uncerainy). uncerainy abou he daa on he basis of which decisions are aken (daa uncerainy) in conras o he case of addiive shocks, now he formulaion of moneary policy is affeced by he degree and he ype of uncerainy wih which he cenral banks are confroned Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 1

22 Uncerainy in moneary policy Model uncerainy no cenral bank is endowed wih full knowledge of how he economy funcions Milon Friedman s long and variable lags for he ransmission of moneary policy acions are probably one of he mosly cied argumens o demonsrae he lile knowledge abou he conen of he black box. model uncerainy concerns in paricular wo aspecs: 1. he specificaion of he macroeconomic model (specificaion uncerainy). he esimaion of he parameers of a specified model (parameer uncerainy) Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide

23 Uncerainy in moneary policy Specificaion uncerainy lack of agreemen concerning he appropriae specificaion of a model suiable for he analysis of moneary policy his uncerainy abou he rue srucure of he economy, or he funcional form of he economy, eners he model hrough he lack of agreemen abou he composiion of he vecor of sae variables z he differences refer o he lag srucure of he variables, he degree of forward-looking behavior of privae agens and he omission or he inclusion of paricular variables Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 3

24 Uncerainy in moneary policy Specificaion uncerainy one way o approach his uncerainy is o search for a policy rule ha possesses robusness in he sense of yielding reasonably desirable oucomes in policy simulaion experimens conduced wih a wide variey of models a cenral resuls of many sudies is ha in paricular simple policy rules are quie robus agains uncerainy abou he rue srucure of he economy Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 4

25 Uncerainy in moneary policy Parameer uncerainy uncerainy occurs wih respec o he parameers of he model which are esimaed on samples of small size and, hence, are prone o errors of esimaion insead of defining uncerainy over a muliude of classes of models, parameer uncerainy usually occurs in he conex of a single model now he policy maker is assumed o know ha he model equaions are rue on average on an ex ane basis, bu he is uncerain abou he values of he parameers in any paricular period Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 5

26 Uncerainy in moneary policy Parameer uncerainy consider he following example (aken from Walsh, 3, Ch ) NKPC π = βeπ+ 1 + κx + u where κ = κ + ζ and ζ is a whie noise sochasic process wih ζ ~N(, σ ζ ) in his example he cenral bank is uncerain abou he rue impac of he oupu gap on inflaion he cenral bank s bes guess of his coefficien is κ, bu is acual realizaion is κ he cenral bank mus choose is policy before observing he realizaion of ζ assume ha he cenral bank s loss funcion is W = α var + ( x ) var ( π ) x Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 6

27 Uncerainy in moneary policy Parameer uncerainy assume furher ha policy is conduced wih discreion and ha he cos-push shock u is serially-uncorrelaed he firs-order condiion for he opimal choice of he oupu gap is E { αxx + κπ } = afer plugging in he NKPC his equaion can be rewrien as E{ αxx + κ ( βeπ+ + κx + u) } = αxx + κβeπ+ + κu + E { κ x} = expecaions abou squared variables can be rewrien in erms of variances using var x = E x x = E x x x+ x = E x xe x + x = ( ) ( ) 1 1 { } { } { } { } { } { } = E x x + x = E x x Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 7

28 Uncerainy in moneary policy Parameer uncerainy hus { κ } = { κ } { } = { κ } = ( var{ κ } + κ ) = ( σζ + κ ) E x E E x E x x x and he firs order condiion reads α κβ π + ( ζ ) xx + E 1 + κu + σ + κ x = solving for x and applying he mehod of undeermined coefficiens (since all sochasic disurbances are serially uncorrelaed, expeced inflaion will be zero) gives he law of moion for he oupu gap x κ = κ α x σ + + ζ u Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 8

29 Uncerainy in moneary policy Parameer uncerainy if here is no parameer uncerainy (i.e. σ ζ = ), he oupu gap behaves as in he sandard case wih serially uncorrelaed shocks (see Chaper 5 Par III slide 9 wih ρ u = ) κ x = u κ + α x if parameer uncerainy increases, he variance of he oupu gap decreases for a given variance of he cos-push shock var x κ = κ α x σ + + ζ σ u as uncerainy increases (σ ζ ), i becomes opimal o respond less o cos push shocks Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 9

30 Uncerainy in moneary policy Parameer uncerainy his is also refleced in he ineres rae rule o obain he behavior of he ineres rae solve he DIS for he policy insrumen and assume ha expecaions abou fuure variables are zero (noe ha we have assumed ha here are no echnology and preference shocks so ha r e = ρ) e e σκ i = r + Eπ+ 1 + σ ( Ex+ 1 x) = r σx = ρ + u κ α x σ + + ζ as uncerainy increases (σ ζ ), he cenral bank behaves more cauiously in seing is policy insrumen Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 3

31 Uncerainy in moneary policy Parameer uncerainy The resul ha muliplicaive parameer uncerainy provides a raionale for a cauious and gradualis approach o policy-making goes back o a seminal paper of William Brainard (1967), Uncerainy and he Effeciveness of Policy, in: The American Economic Review, Papers and Proceedings 57, The lieraure refers o his resul as Brainard conservaism. Noe: As he sochasic parameer κ is muliplied by an elemen of he sae vecor, his kind of uncerainy is ofen labeled muliplicaive uncerainy (in order o emphasize he difference o addiive uncerainy). Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 31

32 Uncerainy in moneary policy Uncerainy abou he curren descripion of he economy 1. uncerainy abou he model used by he moneary policy maker (model uncerainy) a. specificaion uncerainy b. parameer uncerainy. uncerainy abou he daa on he basis of which decisions are aken (daa uncerainy) Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 3

33 Uncerainy in moneary policy Daa uncerainy Even if he cenral bank knew he rue model of he economy, here would noneheless remain a significan degree of uncerainy regarding he real ime daa which describes he curren sae of he economy. Decisions of policy makers ofen have o rely on daa ha are provisional and prone o revision, and some conceps of calculaing daa (i.e. daa which canno be direcly observed by he cenral bank) are subjec of dissension beween economiss. Applied o our model of he economy he policy maker only observes (or measures) a noisy conemporaneous esimae of (pars of) he sae vecor. Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 33

34 Uncerainy in moneary policy Daa uncerainy In pracice, he variables ha are mosly affeced by his kind of uncerainy are he oupu gap (Rudebusch, 1, ), he inflaion rae (Rudebusch, 1): Rudebusch, Glenn D. (1), Is he Fed oo Timid? Moneary Policy in an Uncerain World, in: The Review of Economics and Saisics, 83, Rudebusch, Glenn D. (), Assessing Nominal Income Rules for Moneary Policy wih Model and Daa Uncerainy, in: The Economic Journal, 11, Example from he Gali exbook (exercise 4.1): measuremen errors relaed o he rae of inflaion π = π + ξ where π denoes measured inflaion by he cenral bank, π is he acual inflaion rae, and ξ ~N(, σ ξ ) is he measuremen error Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 34

35 Uncerainy in moneary policy Daa uncerainy consider he following model of he economy π = βeπ + κx + 1 e ( π ) x = Ex σ i E r i = ρ+ φπ π he equilibrium processes for inflaion, he oupu gap and he policy insrumen are given by κ e π ( ˆ = r φπ ξ) σ + κφ x i π 1 = σ + κφ π e ( rˆ φπ ξ) κφπ e σφπ = ρ + rˆ + σ + κφ σ + κφ ξ π π Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 35

36 Uncerainy in moneary policy Daa uncerainy if φ π approaches infiniy, hese processes converge o π = ξ 1 x = ξ κ e σ i ˆ = ρ + r + κ ξ hus, by reacing very srongly o perceived inflaion, he cenral bank acually causes inflaion, and he oupu gap becomes a sochasic process proporional o he measuremen error shocks o he efficien real rae (echnology shocks) are fully sabilized (similar o our resuls in Chaper 5, Par I, slides 75+76) in he absence of any measuremen error (ξ = ), moneary policy achieves he social opimum Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 36

37 Uncerainy in moneary policy Daa uncerainy wha is he size of φ π ha minimizes he variance of acual inflaion? κ e π ( ˆ = r φπ ξ) σ + κφ π π π κ var π = σ + κφπ var π = φ φ = κ σ σ σ r ξ ( σ r φπ σξ ) if measuremen errors are large, he cenral banks should reac less (i.e. more cauiously) o perceived inflaion Carsensen / Wollmershäuser, New Keynesian Macroeconomics, Slide 37