exercise1.Time-consistent aryWhenwe ysedthecredibilityproblemunderdiscretionarym ary,weassumedforsimplicitythatthemaker’stargetrateofinflation(p*)waszero.Nowweassumethat,insteadofbeinggivenbyEq.(3)inthemaintext,themaker’slossfunctionis:SL(yty*)2k(pt p* Discussreasonswhyitmaybereasonabletohaveapositivetargetrateofinflation(recalltheargumentsmade hapter19).Supposethattherearenostochasticdemandandsupplyshockssothattheeconomymaybedescribedbythefollowingequations:t,tAggregate pt ytt,tGoodsmarket ytya2(rt Announced ary rtrh(ptp*)b(yty Target y*yw,w Exinbrieflywhyitmaybereasonabletoassumethattargetoutputexceedstrendoutput,aswehavedoneinEq.(57).Whatlevelsofoutputandinflationwillemergeifprivateagentshaverationalexpectationsandiftheybelievethatthecentra willsticktotheannouncedm aryrule(56)?Willthem makeractuallywanttosticktothisruleifhecanactinadiscretionarymanner?Ex Derivethe‘cheating’solutionsforoutputandinflationifthepublicbelievesthatthecentralbankfollowstherule(56)whereasthebankactuallyfollowstheoptimaldiscretionary.CompareyourresultswithEq.(10)inthemaintextandexinthedifference.Exintheconceptoftimeconsistency.Derivethetime-consistentrationalexpectationsequilibriumvaluesforoutputandinflation.CompareyourresultswithEq.(12)inthemaintextandexinthedifference.Drawadiagram ogoustoFig.22.1andillustratetheTaylorruleequilibrium(wheretheprivatesectorbelievesthatthemakerwillfollow Chapter22Limitstostabilizationtherule(56)),the‘cheating’ e,andthetime-consistentequilibriuminthepresentsituation,wherep*0.exercise2.inflationbiasanddelegationof aryWhenwediscussedtheeffectsofdelegatingm arytoantcentralbank,weassumedthatthecentra erhadthesameoutputtargety*asthe However,centra ersoftenclaimthattheyarenottargetinganoutputlevelabovethenaturalrate,s etheyknowthatthiswillonlyleadtoinflation.thisbackgroundyouarenowaskedtoysetheeffectsofdelegatingmaryinasettingwherethe targetstrendoutputy,whereaselectedhaveamoreambitiousoutputtargety*y.Thusweassumethat ernment’sfunctiontSL(yty*)2 k ty*y w whilethe ’slossfunctionisgiventSLB(yty)2 tNotethatthe ernmentandthecentra areassumedtohavethesamedegreeofaversiontoinflation(k).The hasbeengrantedadegreeofcebfrom Hencem aryisdeterminedbyminimizationofthefollowinglossfunction,representingacompromis weenthepreferencesofpoliticiansandthepreferencesofthe SL(1b)SLbSLB 0b Finally,therateofinflationisgivenby ptptt,t

yt Bycontrollingthe(real)interestrate,m cancontroltheoutputgapytyandhenceindirectlytherateofinflation.UsetheproceduredescribedinSection22.1toderivethetime-consistentrationalexpectationsequilibriumemergingwhen aryisdeterminedby(61).ComparethesolutionsforoutputandinflationwithEq.(12)inSection1andcommentonsimilaritiesanddifferences.Whatisthesociallyoptimaldegreeofceforthecentra (theoptimalvalueofb)?Givereasonsforyouranswer.Howmaycentra cebeachievedinexercise3.TheimplicationsofmacroeconomicmeasurementIn(43)and(44)inthemaintextwederivedthefollowingformulaefortheoptimalvaluesofthem aryparametersbandhwhenthereareerrorsinthemeasurementoftheoutputandinflations2sb

22.5 2h z Giveanintuitiveinterpretationoftheseresultsinwhichyouexintheroleofalltheparameterss2,s2,s2, exercise4.OptimalstabilizationwithmeasurementInSection22.4wenotedthatwhenmakerserrinmeasuringtheoutputgap,theyarealsolikelytomismeasuretheequilibrium(‘natural’)realinterestrate.Supposethereforethattheeconomyisdescribedbythefollowingsystem,wherereistheestimatedequilibriumrealinterestrate,eand3earetheestimatedoutputandinflationgaps, wherethemeasurementerrorsat,mtandetarestochasticwhitenoiseAggregatesupply: 3tgt,Goodsmarket

3tptzta2(rtr

tyt E[zt] E[z2]

t rtreh3teb t Estimateofr rerat E[at] E[a2] Estimateofy e m E[m] E[m2]s2 Estimateof3 3e3e E[e] E[e2] Equation(65)assumesthatthecentra’sinflationtargetiscrediblesothatrationalagentshavetheexpectedinflationratepep*.Forsimplicity,themodel sfromsupplyshocksandfrompersistenceinthemeasurementerrors.Notethats e(65)impliesthatthevarianceoftheinflationgap(s2)isproportionaltothevarianceoftheoutputgap(s2),anoptimal inimizesthesociallossfunctionSLs2ks2mustminimizethevarianceoftheoutputgaps2E[2 Solvethemodel(65)–(70)fortheoutputgapandshowthatthevariabilityofoutputisgiven z ss2a2(h2s2b z [1a2(bCommentonthi pressionandexintheeffectsofthemeasurementerrors,inparticulartheeffectoftheerrorinmeasuringthenaturalrateofinterest.UsetheproceduredescribedinSection22.3toshowthattheoptimalvaluesofbandharegivenbytheexpressionss2b 2g(s2a2s2h 2aComparetheseexpressionstothosegiveninEqs(43)and(44)inthetextandgiveintuitiveexnationforthe1(p*)(3)正文中，決SL=(yt-y*)2+kp(t-p*)2，p*>0討論為什么設定一個正的通貨膨脹目標率可能是合理的（19o總供給：pt=ptte,-1+yt-y,(54)yta2(rtrrrh(pp*)+by(y(56)tttww目標輸出：y*=+y,>0,(57)簡要解釋為什么假設目標輸出超過趨勢輸出可能是合理的，正如我們在等式中所做的那樣。脹的“作弊”解決方案。將您的結果與方程式進行比較。(10)在正文中并解釋其區別。方程式進行比較。(12)在正文中并解釋其區別。畫一個類似于圖22.1的圖并說明規則均衡68022（規則（56））、“作弊”結果以及當前情況下的時間一致均衡，其中p*>0。目標y*>y的情況下，授權貨幣政策的影響。因此我們假設政府的損失函數為：SL=(yt-y*)2+kpt2,k