9. Simple Rules for Monetary Policy

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1 9. Smpl Ruls for Monar Polc John B. Talor, Ma 0, 03

2 Woodford, AR 00 ovrvw papr Purpos s o consdr o wha xn hs prscrpon rsmbls h sor of polc ha conomc hor would rcommnd Bu frs, l s rvw how hs sor of polc was orgnall drvd from wha conomc hor would rcommnd. Dnamc Sochasc Forward-lookng RUL Lucas crqu Tm nconssnc Sck prcs

3 Novmbr 99

4 So whr dd com from? Sarchng for opmal ruls B smulang alrnav ruls n dnamc sochasc monar modls wh raonal xpcaons & sck prcs Fndng ruls whch prformd wll: small Var & Varp xampl: Modl comparson a Brookngs 9 mul-counr modls 7 wh R ncludng Talor mul-counr modl Fndng ha good polc ruls had cran proprs r rahr han M, no forx, rac o and p, cof on p > -- Zro bound on nrs ra: swch o M-growh Thn roundd off h coffcns for praccal work Thn showng polc was clos n bu was no an smad rul was man o b normav, no posv

5 p Oupu Inflaon p r Inrs ra r From Talor 99

6 / / / / / / and whr n h form : modl xpcaons raonal lnar vcor Whn combnd wh scond quaon w oban a n h hrd quaon usng h frs quaon : Now subsu for ar posv and 0,,, whr * * * * n n n n n n g g g g A z Az z Woodford 00: Wha can w larn abou h rul from hs modl?

7 Solvng h Modl: Makng sur hr s a unqu soluon Obsrv ha hr ar no lags n hs modl, so h condons for unqunss wll b dffrn from h raonal xpcaons modl wh on lag and on lad consdrd arlr. - In hs cas hr ar wo lads. - In connuous m on would sa ha hr ar wo " jump" varabls. In h arlr cas w ndd on unsabl roo for unqunss. - Now w nd wo unsabl roos o pn down h soluon. - To s hs, assum ha s srall uncorrld wh zro man.

8 0 H ΛH H HΛH A A I A... A... Az z z z z ,,...,,,..., whch mpls ha rsuls n n h modl and Subsung n for... Look for a soluon usng undrmnd coffcns : - ach lmn s an quaon: 4 quaons n 8 unknowns ach of hs brngs 4 mor quaons, bu wh 4 mor unknowns Howvr, f h roos of A ar unsabl hn hr s onl on valu for ha wll no xplod: H - =0

9 Lsson : An mplcaon for h coffcns of opmal polc rul Th condon for a unqu saonar soluon s:

10 Lsson : Thr ma b nra n h opmal rul Obsrv ha hr s no laggd dpndn varabl n h Talor rul - Th nrs ra adjuss mmdal, rahr han graduall. -mprcall, a laggd dpndn varabl nrs ra appars n smad polc ruls -- ma b du o sral corrlaon of rrors or o paral adjusmn. Such a paral adjusmn mgh b opmal f polc makrs could comm o. Wh? - A rul whch ncrass h nrs ra lar would ncras xpcaons of such an nrs ra ncras n h fuur and bgn o lowr xpcd nflaon and hus acual nflaon whou an advrs ffcs on oupu oda. -- Ths rsul can b llusrad n a vr smpl modl

11 Illusraon of Gans From Inra Th da s o nclud rms n polc ruls ha xplo popls' xpcaons. Smpl xampl : r r g h mn var r var r rm srucur - wo prods - affcs spndng polc rul for nrs ra loss funcon In sochasc sad sa : var g var r g Opmal polc rul s hus : r h - No ha for 0 w hav g / and h 0, bu ohrws h 0. - No ha hs racon o vn hough h h - s opmal s no n h basc modl. Wh? B movng r n rspons o -, h polc sgnals ha r + wll rac o. Thus r + movs wh. Ths mans ha r + shars h work wh r Bu ou hav o kp wh.

12 Lsson 3: Raonal for h Loss funcon Can hs loss funcon b vwd as an approxmaon o maxmzng h wlfar of h rprsnav ndvdual n h conom? - Woodford s answr s s: --- Targ nflaon ra s 0 bcaus wh saggrd prcng, 0 mnmzs prc dsprson whch rducs ffcnc --- Th quadrac s an approxma cos of h dsprson. - Smlarl, h ul loss of movmns of awa from h flxbl prc lvl n can b approxmad b a quadrac. --- Bu arg s grar hn n du o monopolsc compon assumpon --- And n s clarl no a lnar rnd, or an smpl rnd --- I dpnds on chnolog changs, prfrnc changs, c. --- Should no rspond o all dvaons

Consider a system of 2 simultaneous first order linear equations

Consider a system of 2 simultaneous first order linear equations Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm