OPTIMAL INTEREST RATE RULES UNDER ONE-SIDED OUTPUT AND INFLATION TARGETS. Peter J. Stemp

Transcription

1 OPTIMAL INTEREST RATE RULES UNDER ONE-SIDED OUTPUT AND INFLATION TARGETS b Peter J. Stem Deartment of Economics Monash Uniersit PO Box 197, Calfield East Victoria, 3145 Astralia eter.stem@bseco.monash.ed.a Aril, 009

2 ABSTRACT We inestigate the deriation of otimal interest rate rles in a simle stochastic framework. The monetar athorit chooses to minimise an asmmetric loss fnction, where the monetar athorit laces ositie weight on negatie (ositie deiations of ott (inflation and ero weight on ositie (negatie deiations. Recent aroaches to monetar olic nder asmmetric references hae emhasised the adotion of a linear exonential (linex reference strctre. This aer resents a new and different analtic methodolog that is based on the exlicit calclation of semi-ariances. This aroach can be sed to derie recise coefficients of the otimal interest rate rles. Or focs is on deriing otimal interest rate rles in a series of secial cases where shocks come from onl one sector. We show that whether or not the otimal rles deried nder asmmetric loss fnctions will be different than the otimal rles deried nder smmetric loss fnctions is crciall deendent on the interest rate rle chosen. Under a fixed interest rate rle, the otimal rles differ. Howeer, nder a more flexible interest rate rle that incororates additional information abot contemoraneos shocks, the otimal rles are the same. JEL classification: C61; E43; E58. Kewords: Monetar Economics; Interest Rate Rle; Inflation Target; Ott Target; Asmmetric Loss Fnction; One-sided Target; Linex Preferences; Semi-Variance. ii

3 1. INTRODUCTION Following from the seminal aer b Poole (1970, an extensie literatre has deeloed which examines the roerties of otimal mone sl rles in economies that are faced with stochastic distrbances to different sectors. Poole considered otimal mone sl rles in a stochastic IS/LM framework. Otimal mone sl rles hae also been deried in models with fll market clearing when there is an asmmetr of information between the blic and riate sector; and when mone sl resonses are fll anticiated. Stdies for oen economies tend to emhasise the relationshi between monetar olic and exchange rate olic. More recent work has emhasised the distinction between anticiated, nanticiated, ermanent and temorar shocks and the relationshi between wage indexation, lagged feedback rles, and monetar olic. While this aroach has become standard in the literatre, een to the oint of being adoted as one aroach to monetar olic in the standard textbooks on monetar olic, standard monetar oerating rocedres adoted b central banks now tend to focs on a short-term interest rate as the rimar instrment of monetar olic rather than the manilation of monetar aggregates. As a conseqence, recent literatre on monetar olic rles has tended to focs on interest rate rles rather than mone sl rles (see Walsh, 1998, and Woodford, 003. All these aroaches to monetar olic tend to assme that monetar olic rles are chosen so as to minimise a qadratic loss fnction, made of a weighted sm of sqared terms, ticall inclding major macroeconomic indicators, sch as ott and inflation. Sch loss fnctions are characterised b smmetric roerties, in that the gie the same weight to ositie and negatie deiations abot some chosen ath. 1

4 In this aer we will focs on otimal interest rate rles that are chosen so as to minimise a loss fnction with asmmetric roerties, that is, a loss fnction that gies different weights to ositie a negatie deiations from a chosen ath. Intitiel this makes sense becase, for examle, agents in the econom are more concerned abot negatie deiations in ott (high nemloment than the are abot ositie deiations (oerfll emloment and the are more concerned abot ositie deiations of inflation from some desired ath than the are abot negatie deiations. Recent stdies (Noba and Peel, 003; Srico, 007 make a comelling case that, in ractice, olicmakers are likel to adot a loss fnction with asmmetric roerties. Aienman and Frenkel (1985, Eq. A8,. 40 show how the microfondations for a loss fnction that has asmmetric roerties can be deeloed. Their reslts demonstrate that b exanding the rodction fnction in a Talor series arond the general eqilibrim to second-order terms the loss fnction will be smmetric; bt b also inclding third-order terms of the Talor exansion, an asmmetric loss fnction reslts. Preios stdies hae sed comter simlation techniqes to examine the imlications of asmmetric loss fnctions for the roerties of otimal rles (Friedman, 1975; Knstman, Wad (1976 sed analtic techniqes to examine the roerties of otimal resonses nder asmmetric criteria. Ckierman and Melter (1986 considered otimal monetar olic nder an asmmetric criterion b considering the case when the olicmaker's loss fnction comrised both qadratic and linear comonents. Stem (1993 calclated the semi-ariance of a normall distribted ariable and showed how these theoretical reslts cold be sed to calclate otimal mone sl rles when references are asmmetric.

5 Recent stdies hae focsed on the secification of asmmetric references sing a linear-exonential (linex secification. This aroach to determining otimal monetar rles was first introdced in a theoretical framework b Noba and Peel (003. Analsis of asmmetric references sing the linex secification, or some generalisation of the linex secification, has also been emloed in recent emirical stdies of monetar olic (Rge-Mrcia, 003; Srico, 007; Boinet and Martin, 008. The linex secification has the adantage that its theoretical distribtion can be determined in a straightforward manner and that the qadratic distribtion is nested within the linex secification as a secial case. This makes this secification articlarl sefl for the emirical testing for asmmetric references erss smmetric (qadratic references. It has the disadantage that it onl roides an aroximation to an articlar form of asmmetric references and cannot be sed to recisel constrct a loss strctre that gies ositie weight to ositie (negatie deiations, bt ero weight to negatie (ositie deiations of a articlar ariable. This aer draws on and extends ideas first resented in Stem (1993. It is ossible to se this aroach to aste together different comonents comrising the semi-ariances of articlar ariables (sch as ott and inflation. The sm of ero weighted and different, bt ositie, weighted comonents to form a range of asmmetric loss fnctions more recisel reflects the objecties of a monetar athorit that has trl one-sided targets. As far as this athor is aware, the aroach to the constrction of asmmetric references adoted in this aer has not been emloed anwhere else, aart from the Stem (1993 aer. The rest of this aer roceeds as follows: Section roides a formla for calclating the semi-ariance comonents of an asmmetric loss fnction. Section 3 3

6 shows how each of these comonents can be minimised b aroriate choice of mean and ariance. A simle monetar model is introdced in Section 4. This model is sed in Sections 5 and 6 to calclate otimal (fixed and ariable interest rate rles nder asmmetric references when the econom faces searate demand-side and sl-side shocks. Section 7 comares the interest rate rles deried nder asmmetric references, as resented in Sections 5 and 6, with the interest rate rles nder the standard smmetric (qadratic reference strctre. Conclding comments are roided in Section 8.. CALCULATING THE COMPONENTS OF AN ASYMMETRIC LOSS FUNCTION Assme that X is a normall distribted random ariable with mean,, and ariance,. We will write this as: X N (,. We then define two random ariables, X + and X, as follows: X,if X > 0, X + = 0, otherwise, (1a And X,if X < 0, X = 0, otherwise. (1b It follows from these definitions that And + X + X = X (a + E( X + E( X = E X = +. (b In addition, assme that Z N(0,1. Then, 4

7 π 1 Pr( Z < = F = ex d. (3 We can then roe the following theorems: Theorem 1. + = ( + + ex = { } E X F H. π Where the fnction H is defined as follows: H = 1 + F + ex. π Proof. E X + Sbstitting ( x 1 x ex ( x = π dx. (4 =, 0 So that + 1 E X ( ex = + d π. (5 + E X ex = d π ex π d ex π + d. (6 Define F π = 1 ex d. (7 5

8 Then the associated moment generating fnction is gien b: φ 1 ex ex ( + s = d π. (8 ( s s Taking first and second deriaties of φ ( s with resect to s ields: 1 1 d φ '0 = ex = ex π π, (9a φ '' 0 ex π = 1 d = F 1 ex π (9b Letting = in Eqations (7, 9a, 9b and sbstitting into Eqation (6 ields: = + + π. (10 ( F( E X + ex Noting the definition of H( in the statement of Theorem 1 comletes the roof. ### Theorem. Using the same notation as Theorem 1. = ( + ex = { } E X F H. π Proof. Using eqation (b, ( = ( + E( X + E X = + 1 ex π. (11 ( F ( Bt as demonstrated in the Aendix, 6

9 F + F( = 1, for all, (1 so that sbstitting = into Eqation (1 ields: F ( F ( 1 + = (13 Hence, sbstitting Eqation (13 into Eqation (11 ields = ( + ex π. (14 F ( E X Noting the definition of H(, this comletes the roof of Theorem. 3. MINIMISING COMPONENTS OF ASYMMETRIC LOSS FUNCTION For fixed ariance, the methodolog for choosing so as to minimise E X + essentiall inoles finding the ale of that minimises H and then deriing the ### otimal ale of b noting that =. In this section, we resent a methodolog for minimising H(. We begin b first establishing roerties of the fnction, H. Hence, So that π F( 1 ex = d 1 F( = (15a F 1 = ex ( = F( (15b π F F F π = ex = = ( (15c Since 7

10 It follows that Also, H( = 1 + F( + ex π ( 1 F F = + + (16a = (1 + + H F F + 1 ex ex π π (1 F F F F (16b = (16c = F + F. (16d = + + H F F F (16e = + + F F F (16f = F From Eqation (16g, since F( > 0 (16g, for all finite ales of, it follows that the fnction H satisfies the second-order conditions for a maximm. From Eqation (16d, the first-order conditions for a maximm are satisfied when H = F + F = 0. (17. Figre 1 lots H (Figre 1 abot here. On the basis of the lot we can determine that the fnction H takes its minimm ale when. Bt =. On the basis of this we can assert that, for fixed ariance, E( X + takes its minimm ale when. 8

11 A similar methodolog can be sed, for fixed ariance, to choose so as to minimise E( X. Note that { } E X H =, where =. (18 In order to determine the minimm ale of E( X, for fixed ariance, we need to find the ale of that minimises H (. As reiosl, we begin b first. establishing roerties of the fnction, H H( = 1 + F( ex π ( 1 ( 1 F F (19a = + (19b ( 1 H = + (19c It follows that And that H ( = F F (19d = H (19e ( = 1 H F (19f From Eqation (19f, since 1 F >0, for all finite ales of, it follows that the fnction H ( satisfies the second-order conditions for a maximm. From Eqations (19d, 19e, the first-order conditions for a maximm are satisfied when H ( = 0. That is, when H = F + F =. (0. 9

12 Again, sing Figre 1, we can lot H ( and the fnction h = on the same diagram. The fnction H ( takes its minimm ale when these two fnctions intersect. This occrs when +. Bt =. On the basis of this we can assert that, for fixed ariance, E( X takes its minimm ale when A SIMPLE MONETARY MODEL Or rose in deriing the theorems of Section was to se those reslts as comonents of an asmmetric loss fnctions. To this end, we consider a simle model of the following form: = β r+ (1a = α + (1b r r γ = + (1c where = real ott (exressed in deiations abot some target leel; = inflation (exressed in deiations abot some target leel; r = nominal interest rate;, are indeendent normall distribted ariables, both with mean ero and with resectie ariances gien b,. α, β are exogenosl fixed constants; r, γ are choice ariables. Eqation (1a defines the demand-side of the econom throgh a simle ctdown secification of an IS cre. Eqation (1b defines the sl-side of the econom throgh a Phillis cre relationshi. Eqation (1c defines a simle interest rate rle. The model can be soled to gie soltions for, as follows: 10

13 αβ α βγ ( ( α + βγ ( ( α + βγ ( ( α + βγ = r + + β 1 1 ( ( α + βγ ( ( α + βγ ( ( α + βγ = r + (a (b Next, in the sirit of Section of this aer, we will define: =, if <0 0, otherwise (3a And + =, if >0 0, otherwise (3b And we will choose an interest rate rle (gien b some ariant of Eqation 1c to minimise an asmmetric loss fnction of the following form: A + (1 δ L = δe + E, where 0< δ < 1. (4 We consider two tes of interest rate rles: A fixed interest rle of the form: r = r (5a This is deried b fixing γ = 0, keeing r as a choice ariable in Eqations (a, b, and choosing r so as to minimise Eqation (4. A flexible interest rate rle of the form: r r γ = + (5b This is obtained b keeing both r and γ as choice ariables in Eqations (a, b, and choosing r, γ so as to minimise Eqation (4. We consider the imlications of each of these rle tes in trn. 11

14 5. OPTIMAL POLICY UNDER A FIXED INTEREST RATE RULE Under a fixed interest rate rle the monetar athorit is nable to inflence the imact of stochastic shocks in the econom and can onl inflence the means of,. Demand-side shocks We first consider the case when there are onl demand-side shocks so that > 0 and = 0. Then N and (, N where (, = r (6a β = (6b In articlar, β r α = (6c α 1 = (6d β = = r (7a and = α = (7b So that the loss fnction, gien b Eqation (18, can be written in the form: L = δ H + δ H A (1 ( 8a This can be redced to: L A δα H r (1 r = β δ H β + α (8b The first comonent of this loss fnction is of the form: 1

15 δα { } (1 δ { } J = H + H, where = β r (9a ( 1 δ( 1 δα (1 δ = + + H (9b It follows that ( δ δα ( δ J = H (9c = + δα 1 δ θ H, ( 1 δ where θ = δα ( 1 δ (9d And that = 1 ( + ( 1 J δ δα δ H (9e ( δ δα ( δ = F (9f ( δ ( δα = 1 1 F + F (9g Since 0< δ < 1, note that θ ranges monotonicall from -1 (when δ = 0 to 0 (when δ = 1. Conseqentl, from Eqation (9g, since 0< F < 1 for all finite and since 0 δ 1 < <, it follows that 0 J > so that the fnction J satisfies the second-order conditions for a minimm. From Eqation (9d, the first-order conditions for a minimm are satisfied when θ + H = 0 (30 Hence, the first-order conditions are satisfied when H = θ (31 Figre lots the ale of the fnction h = θ as the ale of δ and hence the ale of θ is allowed to ar. When δ = 0, θ = 1, so that = h. When 13

16 δ = 1 ( 1 + α Ths, as Figre., θ =, so that h( =. When δ = 1, 0 0 θ =, so that h =. δ ranges from 0 to 1, the lot of h( moes conter-clockwise as shown in (Figre abot here Figre also lots H (. The fnction J takes its minimm ale when the two fnctions, H ( and h = θ, intersect. This occrs for finite ales of, ranging from =+, when δ = 0, to = 0, when δ = 1, to =, ( 1 + α when δ = 1. Bt = β r. On the basis of the aboe reslts, we can assert that, nder a demand-side shock, the ale of r for which L takes its minimm ale, aries A along a continos range as δ is allowed to ar, from r + (when δ = 0, to r = 0 (when δ = 1 ( 1 + α, to r (when δ = 1. The interest rate rle can be smmarised b the following: 0 r = f δ, where f δ δ < (3a With the additional roerties that r r > 0, when δ < = = < 0, when δ > 1 ( 1+ α ( α (3b r =+, when δ = 0. (3c r =, when δ = 1. (3d 14

17 Sl-side shocks We next consider the case when there are onl sl-side shocks so that = 0 and > 0. Then N and (, N where (, = r (33a β = 0 (33b If X β r α = (33c α 1 = (33d N (, and we allow to aroach 0 from aboe, then,if 0 E( X = 0, otherwise (34 Hence, the asmmetric loss fnction redces to: L A δ + (1 δ H, if 0 = (1 δ H, if 0 (35a This can be frther simlified to: βr (1 H βr δα δ +, if r 0 α A = (1 r δ H β, if r 0 α L (35b When we let β r =, the first comonent of this loss fnction is gien b 15

18 J { } { H } δ H δ α + (1, if 0, = (1 δ, if 0. (36a Proceeding as reiosl, J { } { } δ H δ H δ α + (1, if = 0, (1, if 0. (36b J F αδ + (1 δ, if 0, = (1 δ F, if 0. (36c From Eqation (36c, both arts of the exression satisf the second-order conditions for a minimm. From Eqation (36b, the first order conditions for an nconstrained minimm (that is, ignoring for the time being, the constraints on of the first term are gien b: δ { H } δ α + (1 = 0 (37 So that H = θ, where θ = δα (1 δ (38 For the nconstrained roblem, the first term in Eqation (36a takes its minimm when the two fnctions, H ( and h = θ, intersect. Essentiall the constrained roblem will be minimised b this soltion, wheneer θ > 0. This condition is met for all ales of δ, oer the range, 0< δ < 1. For the constrained roblem defined b the second term in Eqation (36a, the minimm soltion will be gien b = 0, since from Eqation (36b, the second term in Eqation (36a is greater for all ales of aboe 0. 16

19 Bt the two constrained roblems oerla at = 0. And we know that the soltion for the first term in Eqation (36a is better than = 0, which is the otimal soltion for the second term in Eqation (36a Ths the otimal soltion is satisfied b the intersection of the two fnctions, H ( and h = θ. When δ 0, θ = 0 and. When δ 1, θ and = 0. Bt β r =. Ths r ranges from + (when δ = 0 to 0 (when δ = 1. The interest rate rle can be smmarised b the following: r = g( δ, where 0 With the additional roerties that g δ δ < (39a r = r > 0, for all ales of δ ( 0< δ < 1. (39b r =+, when δ = 0. (39c r = 0, when δ = 1. (39d 6. OPTIMAL POLICY UNDER A FLEXIBLE INTEREST RATE RULE Under a flexible interest rate rle the monetar athorit is also able to inflence the imact of stochastic shocks in the econom and so can inflence both the means and ariances of,. Demand-side shocks This time we first consider the case when there are onl demand-side shocks so that > 0 and = 0. Then N and (, N (, where = ( αβ ( r ( α βγ α = ( α + βγ + (40a (40b 17

20 = In articlar, ( β ( r ( α βγ + (40c 1 = α + βγ (40d βr = = (41a Bt this time = α α = ( α + βγ (41b So that the loss fnction can be written in the form: L (1 A H βr H βr = δα δ + ( α + β γ (4 This loss fnction can be slit into two comonents, J, K, where: δα { } (1 δ { } J = H + H, where K ( γ = ( α + βγ. = β r, (43a (43b The second comonent, gien b Eqation (43b, can be minimised b letting γ, so that 0 K =. Then the loss fnction, L A, will take its minimm ossible ale of ero roided is chosen so that J takes an finite ale. There are mltile soltions for that fit this reqirement, so the soltion is not niqe. One sch soltion is gien b = 0. This imlies that r = 0. Ths, one sitable interest rate rle in this case, is to choose γ. This is eqialent to choosing an interest rate rle so that: r = 0 = 0. (44 and to let 18

21 Sl-side shocks We finall consider the case when there are onl sl-side shocks so that = 0 and > 0. Then N and (, N where (, = ( αβ ( r ( α βγ βγ = = In articlar, ( ( α + βγ + (45a β ( r ( α βγ (45b + (45c 1 = α + βγ (45d αr = (46a γ βr = (46b And this time = βγ βγ = ( α + βγ (46c So that the asmmetric loss fnction can be written in the form: LA = δ H + δ H (1 ( 47a This can be redced to: L (1 r A H H β = δ βγ δ γ + ( α + βγ αr { } (47b 19

22 The first comonent of this loss fnction is of the form: δ ( βγ α { } (1 δ { ( βγ J = H + H }, where = β r. (48a This can be rewritten in the form: δ α { } (1 δ { J = H + H }, where = βγ. (48b ( 1 δ( 1 δ { H α } ( 1 δ H( { } = + + (48c The first and second deriaties of J(, with resect to are gien b: 1 ( δ δα α { 1 } ( 1 δ { J = + H H } (48d And 1 ( δ δα α { 11 } ( 1 δ { J = + H H } (48e ( δ ( δ F δα F α ( = (48f ( δ F δα F α ( = (48g > 0, if 0< δ < 1. (48h Hence, b Eqation (48h, the second-order conditions for a minimm are satisfied. We can also se Eqation (48d to show that: = 0 is a minimm δα ( 1 δ =0 (49a ( 1 δ = (49b αδ δ = 1 1 ( + α (49c Bt 0

23 = r = = =0 0 0 (50 And, if = 1 0, then H 0 =, and, from Eqation (47a, 1 L A = + (1 δ δ = ( δ ( βγ + (1 δ 1 α + βγ (51a (51b ( δ + (1 δ = α + 1 Then, from manilation of Eqation (51c,, since = βγ (51c L is minimised δα ( 1 δ = 0 (5a A ( 1 δ = (5b αδ δ = 1 1 ( + α (5c Ths, identical conditions ensre both that = 0 and that is minimised. These conditions are that: or, eqialentl, that ( 1 δ = (53a αδ L A δ = 1 1 ( + α (53b Since = βγ, these two eqialent conditions redce to: ( 1 δ γ = and δ = αβδ 1 1 ( + αβγ (54 Then the otimal flexible interest rate rle is gien b: r = γ δ = 1 1 ( + αβγ ( 1 δ γ = (55a αβδ 1

24 In articlar, 0< γ < 0< δ < 1 (55b γ = 0 δ = 1 (55c γ = δ = 0 (55d 7. COMPARISON WITH OPTIMAL POLICY RULES UNDER A SYMMETRIC LOSS FUNCTION It is informatie to comare the reslts for otimal interest rate rles deried reiosl nder an asmmetric fnction with those deried nder the more commonl sed smmetric qadratic loss fnction. In this section we examine otimal interest rles deried nder the following smmetric loss fnction: S (1 δ L = δe + E (56a When N and (, N (,, this loss fnction can be rewritten as: L (1 ( = δ + + δ + (56b S The reslts deried nder this aroach are standard in the literatre and, in this case, are gien b the following: Under the fixed interest rle, the otimal interest rate rle is gien b: r = 0, irresectie of the relatie magnitde of the shocks (57 Under the flexible interest rate rle, the otimal interest rate rle is gien: = 0, in the case of a demand-side shock (so that > 0, = 0, (58a r = γ δ = 1 1 ( + αβγ ( 1 δ γ =, in the case of a sl-side αβδ shock (so that = 0, > 0 (58b (Table 1 abot here

25 Table 1 smmarises the reslts of this aer, b comaring alternatie interest rate rles nder asmmetric and smmetric loss fnctions. For the fixed interest rate rle, where the monetar athorit is nable to hae an imact on the ariance of contemoraneos shocks, the otimal rles nder the asmmetric and smmetric loss fnctions are qite distinctl different. Under the flexible interest rate rle, the monetar athorit is able to inflence the ariance of contemoraneos shocks. Then, it is ossible to se the same otimal interest rate rle nder both asmmetric and smmetric objecties. For the model considered in this aer, this is tre, irresectie of the sorce of shocks. 8. CONCLUDING COMMENTS This aer has inestigated the deriation of otimal interest rate rles in a simle stochastic framework where the monetar athorit chooses to minimise an asmmetric loss fnction. We hae focsed on deriing otimal interest rate rles in a series of secial cases where shocks come from onl one sector at a time. The aer has resented an analtic methodolog that cold be sed, in conjnction with comting techniqes, in a range of these secial cases, to derie recise coefficients of the otimal interest rate rles. We hae shown that whether or not the otimal rles deried nder asmmetric loss fnctions will be different than the otimal rles deried nder smmetric loss fnctions is crciall deendent on the interest rate rle chosen. Under the fixed interest rate rle, the otimal rles differ. Howeer, nder the flexible interest rate rle, the otimal rles are the same. This sggests that the otimal rles nder both asmmetric and smmetric criteria ma be identical across a broad range of interest rate rles when the chosen rles are sch that the are based on 3

26 sfficient information that the hae an imact on the ariance of contemoraneos shocks. As discssed in the introdction, the crrent aroach to the literatre on otimal monetar olic nder asmmetric references has emhasised sing the linex (linear exonential reference strctre, which has the case of smmetric references embedded as a secial case. Sch an aroach wold not be able to distingish between asmmetric and smmetric references, if the otimal rles nder the two tes of references are identical. In this aer, we hae shown that this is the case for a significant set of interest rate rles. The reslts of this aer cold be extended in a ariet of was. It wold be interesting to determine the otimal interest rate rle nder an asmmetric loss fnction, when shocks hit the econom from mltile sectors simltaneosl. Also, in this aer and in Stem (1993, we considered simle monetar models where it was ossible to choose an interest rate rle (in this aer, where r = 0 so that the means of both ariables in the loss fnction were drien to ero simltaneosl. This is a secial assmtion that greatl simlifies or analsis. In general, this assmtion is not tre. For examle, consider the case of an exectations agmented Phillis cre with exogenosl fixed exectations. It wold be interesting to consider otimal interest rate rles in the more general case. The aer cold also be extended along the lines of extensions to the Poole (1970 literatre. For examle, or simle monetar model cold be extended to incororate different informational assmtions and different exectations assmtions. It wold also be ossible to consider asmmetric loss fnctions in conjnction with a dnamic monetar modelling framework. 4

27 APPENDIX Assme that F( is the cmlatie distribtion fnction of a N (0,1 ariable as gien b Eqation (3. In this Aendix we demonstrate that for an constant,, consider Figre A1. F F 1. + = Withot loss of generalit, assme that 0. Then, (Figre A1 abot here Using Figre A1, F can be reresented b Area A ls Area B. Also, F( can be reresented b Area A. Bt b smmetr of the robabilit densit fnction of a normal distribtion, Area A eqals Area C. Hence, F( can be reresented b Area C. As a conseqence, F + F can be reresented b Area A ls Area B ls Area C. Bt since the total area nder a robabilit densit fnction is eqal to 1, hence the sm of the three areas is 1 and hence F + F( = 1. 5

28 REFERENCES Aienman, J., and Frankel, J. A. (1985, Otimal Wage Indexation, Foreign Exchange Interention, and Monetar Polic, American Economic Reiew, 75, Boinet, V., and C. Martin (008, "Targets, Zones, and Asmmetries: A Flexible Nonlinear Model of Recent UK Monetar Polic," Oxford Economic Paers, 60, Ckierman, A., and A. H. Melter (1986, "A Theor of Ambigit, Credibilit, and Inflation nder Discretion and Asmmetric Information," Econometrica, 54, Friedman, B. M. (1975, Economic Stabilisation Polic: Methods in Otimiation, North Holland, Amsterdam. Knstman, A. (1984, "Controlling a Linear Dnamic Sstem According to Asmmetric Preferences," Jornal of Economic Dnamics and Control, 7, Noba, A. R., and D. A. Peel (003, "Otimal Discretionar Monetar Polic in a Model of Asmmetric Central Bank Preferences," Economic Jornal, 113, Poole, W. (1970, Otimal Choice of Monetar Instrments in a Simle Stochastic Macro Model, Qarterl Jornal of Economics, 84, Rge-Mrcia, F. J. (003, "Inflation Targeting nder Asmmetric Preferences," Jornal of Mone, Credit and Banking, 35, Stem, P. J. (1993, Otimal Mone Sl Rles nder Asmmetric Objectie Criteria, Jornal of Economics (Zeitschrift fr Nationalokonomie, 57, Srico, P. (007, "The Fed's Monetar Polic Rle and U. S. Inflation: The Case of Asmmetric Preferences," Jornal of Economic Dnamics and Control, 31, Walsh, C. E. (1998, Monetar Theor and Polic, MIT Press, Cambridge, Massachsetts, and London, England, 58. Wad, R. N. (1976, Asmmetric Policmaker Utilit Fnctions and Otimal Polic nder Uncertaint, Econometrica, 44, Woodford, M. (003, Interest and Prices: Fondations of a Theor of Monetar Polic, Princeton Uniersit Press, Princeton and Oxford, x

29 Figre 1 + Minimising E( X and E( X H h ( H ( h = 60-0 Minimiing Figre L nder Fixed Interest Rate Rle with a Demand-Side Shock A Increasing ales of δ H h ( H h when δ = 1 1 ( + α h when δ = 0 h when δ = 1-0 7

30 Figre A1 Areas Under Probabilit Densit Fnction of a Normal Distribtion Area B Area C Area A - 0 8

31 Table 1 Comarison of Otimal Interest Rate Rles nder Smmetric and Asmmetric Loss Fnctions Sorce of Shocks Asmmetric Loss Fnction L A Smmetric Loss Fnction L S Fixed Interest Rate Rle r = r Demand-Side Shock Sl-Side Shock f ( δ f δ ( δ < 0 r =, where < f ( δ <+ g( δ g δ ( δ < 0 r =, where 0 < g( δ <+ r = 0 r = 0 Demand-Side Shock = 0 = 0 Flexible Interest Rate Rle r = r + γ Sl-Side Shock r δ = γ = = γ αβγ ( 1 δ αβδ r δ = γ = = γ αβγ ( 1 δ αβδ 9