Chapter 10 Review Questions

Size: px
Start display at page:

Download "Chapter 10 Review Questions"

Transcription

1 Chaptr 0 Rviw Qstios. How ca o accot for a short-r trad-off btw iflatio ad mploymt? What is th diffrc btw th atral rat of mploymt ad th NAIRU? Th NAIRU is dtrmid as th qilibrim lvl of mploymt i th imprfct comptitio modl. This rprsts a bargai ovr th lvl of ral wags, whr firms st prics ad workrs st omial wags. w w RW w BRW air Th wag sttig rlatio is W ( Z β) = whr is th xpctd pric lvl, Z a st of factors dtrmiig th bargaiig powr of labor, ad th rat of mploymt. W rics ar st as a mark-p ovr margial costs, = ( + μ ), whr μ is th markp, W th omial wag ad L th lvl of labor L prodctivity. Th NAIRU is dfid as th poit whr th ral wag dmads of both sids of th bargai ar cosistt with ach othr. At this qilibrim positio thr is o prssr o ithr prics or wags to chag. If mploymt fll blow th NAIRU, th workrs will attmpt to icras th ral wag by bargaiig for highr omial wags. This howvr will th fd ito th firm s pric sttig rlatioship, ladig to a icras i prics if th mark-p ad labor prodctivity rmai chagd. Likwis, if mploymt ros abov th NAIRU, th workrs will modrat ral wag claims by rdcig th omial wagwhich shold th fd ito prics throgh th pric-sttig rlatioship.

2 Hc, prics rspod positivly to th dviatio of mploymt from th NAIRU. ΔW = = f air W ( ) This givs ris to a trad-off btw mploymt ad iflatio dscribd by th hillips crv. 0 = α( air ) = α( r ) r air Th NAIRU ad th atral rat of mploymt ar oft sd itrchagably, althogh thr is a sbtl diffrc. Th atral rat is th qilibrim mploymt rat is a comptitiv labor markt, whras th NAIRU rfrs to a imprfctly comptitiv o. Th atral rat of mploymt lis blow th NAIRU. This is bcas i comptitiv markts th pric lvl is qal to th margial cost, whr imprfct comptitio lads to a wdg btw margial costs ad prics. Bcas of th highr pric lvl, th fasibl ral wag that firms ca afford at ach lvl of mploymt is lowr, hc qilibrim i th labor markt rqirs a highr rat of mploymt, air > r. Imprfct comptitio lads to th prsc of a iflatio wdg i th hillip crv. This is th lvl of iflatio that wold aris if a coomy charactrisd by imprfct comptitio tris to assrt th atral rat of mploymt.

3 Iflatio wdg = α( ) =. air r. Th hillips crv is writt as: t t ( 0. ) = a. What is th atral rat of mploymt? Th atral rat of mploymt is whr iflatio is costat, Δ t = t t = 0. ( ) = = 0 t t = 0.05 or 5%. r b. Graph th rlatioship btw iflatio ad mploymt; graph th rlatioship btw th acclratio i iflatio ad mploymt. t t t t ( 0. ) = Th hillips crv, wh mploymt is qal to th atral rat iflatio rmais chagd at its prvios lvl. Th rat of iflatio riss at ay lvl of mploymt blow 5%, ad falls at ay rat of mploymt blow 5%.

4 Th acclratio i iflatio roghly corrspods to th chag i iflatio, which qals: t t t ( 0. ) Δ = = Δ t ( 0. ) Δ t = Th plot of th acclratioist hillips crv highlights that th rat of iflatio is oly costat wh mploymt is at its atral rat of 5%. If mploymt wr to xcd this rat th iflatio wold dclrat, if mploymt wr to fall blow 5% iflatio wold acclrat. c. What lvl of mploymt is rqird to rdc iflatio by 3%? From th acclratioist hillips crv w rqir = 0. 03, hc ( 0. ) 0.03 = Δ t 0.03 = = = = or 8.75%. 0.8

5 To rdc iflatio by 3% a tmporary icras i mploymt to 8.75% is rqird. d. What policis ca th govrmt s to achiv a 3% fall i iflatio? Umploymt ca b raisd by 3.75% throgh a dmad cotractio ithr tightig motary policy by icrasig itrst rats or limitig crdit cratio, or throgh fiscal policy by raisig taxs or cttig govrmt spdig. This cotractio is oly rqird for o priod, as oc iflatio has fall to its rqird lvl it ca b hld thr by movig mploymt back to th atral rat. If this dos t happ iflatio will coti to fall by this rat ach priod. A short sharp cotractio will thrfor hit th iflatio targt. If thogh th govrmt wishd th iflatio adjstmt to b lss svr it cold hav achiv th sam targt throgh a smallr bt mor protractd cotractio. For xampl, if mploymt was raisd to 6% th iflatio wold fall by 0.8% pr yar. This is ddcd from th hilips crv: ( ) Δ = = t Hc, at this rat iflatio will hav fall by 3% withi 0.03/0.008 = 3.75 priods. Altrativly, iflatio ca b lowrd by rdcig th atral rat of mploymt by 3.75% to.5%. Howvr, aftr o priod th coomy ca b xpadd to kp mploymt at this lvl othrwis iflatio will coti to dclrat frthr. Thr ar svral spply-sid policis that might b sd to achiv this, bt all ar coctd with ithr improvig th prodctivity or fficicy of labor ad capital, or by icrasig th total spply throgh ictivs to work or ivst. Covtioal wisdom thogh args that spply-sid (shiftig th log r aggrgat spply crv of th coomy) is part of logr-trm policy makig. 3. Sppos a govrmt ovrstimats th NAIRU ad attmpts to prvt iflatio from risig by cotractig aggrgat dmad. Show th likly otcom of this policy i th short r ad th log r. This dscribs a sitatio whr th coomy is at th NAIRU, bt th govrmt mistakly blivs that th NAIRU is i fact highr. Th coomy is actally at poit a, bt if th govrmt prcivs th NAIRU to b at thy bliv th coomy will mov to a poit sch as c. Bcas crrt mploymt is blow th prcivd NAIRU, iflatio wold acclrat. I ordr to prvt this thy may drtak a cotractioary policy with th viw of movig th coomy to poit b. Umploymt will rtr to th stimatd NAIRU, bt iflatio will b prvtd from acclratig.

6 Log R hillips Crv c a b d ( ) = α + 3 ( ) = α + ( ) 3 = α + If th govrmt blivs th NAIRU to b at. th poit b is a stabl qilibrim i th coomy. Bt bcas thy ar wrog i thir stimats of th NAIRU, th coomy was actally i a stabl positio at th origial poit a. Th policy of icrasig mploymt to prvt th acclratio of iflatio jst movs th coomy ito a rcssio with mploymt xcdig th NAIRU. At poit b, th ris i mploymt, via a rdctio i omial wag growth will lad to a rdctio i iflatio. As iflatio xpctatios ar pdatd this will lad to a frthr dclratio i wag ad pric iflatio. Th coomy will rtr to th NAIRU, bt with lowr qilibrim iflatio at poit d. Umploymt falls bcas as log as th actal rat of mploymt xcds th NAIRU, th ral wag dmadd by workrs will fall blow th fasibl ral wag affordd by firms giv labor prodctivity ad prodct markt coditios. 4. Udr what circmstacs is it possibl to rdc iflatio withot icrasig mploymt? Th hillips crv posits a short r trad-off btw iflatio ad mploymt, so iflatio ca oly b rdcd by icrasig mploymt abov th NAIRU. Howvr, th log r hillips crv is vrtical at th NAIRU ad args that ( ) thr is o log-r trad-off btw iflatio ad mploymt. Each poit o th log r hillips crv plots mploymt at th NAIRU, bt th prvailig rat of iflatio dpds o iflatio xpctatios.

7 A pailss iflatio, i trms of ot icrasig mploymt abov th NAIRU, ca b achivd if iflatio xpctatios ca b rdcd. This simply lads to a dowward shift i th short r hillips crv, ad th rat of iflatio, whilst stayig o th log r hillips crv. No icras i mploymt is rqird. A drstadig of th importac of xpctatios ca b s from th itractio of th wag-sttig ad pric-sttig modls W = ( Z β) A icras i pric xpctatios lads to a icras i omial wag dmads. = W ( + μ) L A icras i omial wag dmads fds dirctly ito a icras i prics. Thrfor thr is a chai likig pric xpctatios to actal prics i this bargaiig framwork. ΔW W Log R hillips Crv ( ) = α + ( ) = α + A fall i iflatio xpctatios simply acts to rdc actal iflatio from. Thr is a ititiv whilst mploymt rmais at th NAIRU ( )

8 raso as to why th NAIRU rmais chagd. Bcas wags ad prics ar growig at th sam rat, th ral wag i th coomy ( w = W/ ) rmais chagd. Thrfor, thr is o chag i or movmt alog ithr th RW or th BRW schdls so th labor markt rmais i qilibrim. This also accots for how thr ca b a costat ral wag at th NAIRU, bt still a positiv bt stabl rat of iflatio. If wags ad prics icras i th sam proportios th th ral wag will rmai costat. If workrs bliv pric iflatio to hav slowd, th maitaiig th sam ral wag will rqir a smallr icras i th rat of growth of omial wags- which th fd throgh ito actal pric iflatio. If this xpctatios rot dos ot work, th policy-makrs will hav to iflc omial wag growth dirctly throgh mploymt. From th wag sttig schdl, a icras i mploymt wold lad to workrs cttig omial ad ral wags. Lowr omial wags will th fd throgh to pric sttig. This procss highlights two importat factors cocrig iflatio xpctatios. First, if xpctatios ar pdatd slowly i a backward-lookig procss it will b impossibl to rdc iflatio withot icrasig mploymt. Scod, v if xpctatios adjst qickly th prsc of omial rigiditis prvtig th rapid adjstmt of wags ad prics might prvt iflatio from fallig. Mor advacd problms 5. What wold b th ffcts o iflatio ad mploymt of: a. A sbstatial ris i oil prics. LRC LR C

9 W If th bargaid ral wag is BRW = = Z β ad th pric dtrmid ral wag W L is RW = =, th th NAIRU ca b fod whr BRW = RW. + μ Z β = = Z β L ( + μ ) ( ) L ( + μ ) Thrfor a fall i labor prodctivity idcd by a ris i oil prics implis a ris i th NAIRU. This will shift th log r hillips crv to th right so crrt mploymt ow lis blow th w NAIRU. At th crrt lvl of iflatio xpctatios, th ris i th NAIRU will idc acclratig iflatio to th rat. Iflatio will coti to ris as log as th mploymt rat rmais blow th NAIRU, ad that iflatio xpctatios ar adjstd pwards i li with actal iflatio. Th w qilibrim positio i th coomy will b whr mploymt qals th NAIRU, ad iflatio xpctatios ar cosistt with actal iflatio at th rat. b. a major improvmt i prodctivity This wold hav tirly th opposit impact as that dscribd i part a. LR C LRC

10 A improvmt i prodctivity will rdc th NAIRU. Thrfor, at th crrt rat of iflatio xpctatios thr wold b a fall i th rat of iflatio, rflctig th fact that mploymt is abov its w NAIRU. Iflatio will coti to dclrat as log as iflatio xpctatios ar adjstd dowwards, ad that mploymt rmais abov th w NAIRU. Th w qilibrim positio will xhibit both lowr mploymt ad iflatio. c. A rform of labor markt istittios This wold b xpctd to hav th sam cosqcs as a ris i labor prodctivity. By rdcig th bargaiig strgth of labor, dirctly throgh rglatio o trad ios powr tc. or idirctly throgh attmpts to icras comptitio, th bargaid ral wag will fall at all lvls of mploymt. Hc th log r NAIRU will also fall. Giv that workrs ow targt lowr ral wags at ach lvl of mploymt, firms ca afford to hir mor labor giv th lvls of th mark-p ad prodctivity. ric iflatio will fall bcas workrs will psh for a lowr rat of omial wag growth i ordr to rdc th ral wag, ad also bcas iflatio xpctatios will fall. 6. Th govrmt ca do othig abot th NAIRU, so shold jst targt iflatio ad lt mploymt sttl at th lowst rat possibl? I I TC = α( )

11 Th govrmt has prfrcs ovr iflatio ad mploymt that ca b rprstd by a sit of idiffrc crvs (I). Bcas iflatio ad mploymt both giv distility, ths ar cocav to th origi ad wlfar is hacd by movig oto lowr crvs. Th NAIRU dtrmis th lvl of mploymt whr th rat of iflatio is costat. Ay positio away from this lvl ca oly b hld tmporally, ad adjstmt back to th NAIRU will lad to a prmat chag i th lvl of iflatio. This dtrmis th positio of th log r hillips crv. Thrfor, i th log r it is likly that th govrmt shold maximis tility sbjct to this costrait. This will ivariably lad to a otcom of tryig to mov th coomy to a poit towards th bottom of th log r hillips crv. I th short r a prfrrd positio ca b achivd by xploitig th short-r hillips crv trad-off. This abls mploymt to b tradd off for highr iflatio, bt oly i th short r. Rdcig mploymt blow th NAIRU will idc a acclratio i iflatio that wold lad to a wors log r positio, i.. highr p th log r hillips crv. For this raso, th govrmt may b bst advisd to allow mploymt to sttl at th NAIRU ad th jst targt a low rat of iflatio. Howvr, i th log r th govrmt ca prmatly mov to a lowr idiffrc crv by shiftig th log r hillips crv to th lft. This howvr wold rqir policis to rdc th NAIRU throgh spply sid policis. From qstio 5, th NAIRU is dfid as: = Z β L ( + μ ) Hc a rdctio i th NAIRU ca b achivd by: Icrasig β : This raiss th ssitivity of ral wag aspiratios to th lvl of mploymt. Thrfor th bargaid ral wag crv will pivot dowwards ad th NAIRU will fall. This cold b achivd throgh policis to icras th job sarch itsity of mployd workrs. Th log-trm mployd ar spcifically pro to los motivatio ad sffr aliatio. Attmpt to icras labor mobility may also act to crb isidr powr i labor markts, maig that mployd workrs ca compt mor ffctivly with xistig mployd workrs. Rdcig μ : A lowr mark-p abls a highr pric dtrmid ral wag ad a fall i th NAIRU. This cold b achivd by icrasig comptitio i prodct markts, prhaps throgh comptitio policy.

12 Icras L: Highr labor prodctivity also icrass th pric dtrmid ral wag schdl, ablig a highr lvl of ral wags to b paid i th coomy. Thrfor mor workrs ca b hird withot placig pward prssr o prics. Typical policis to improv prodctivity may cosist of coragig ivstmt i skills ad iovatio. Rdc Z: This is a catch-all variabl that picks p factors affctig th bargaiig powr of labor. A lowr Z rdcs ral wag aspiratios, ad thrfor mploymt ca xpad withot idcig iflatio. olicis that might achiv cold icld dirct cotrols o labor powr sch as trad io rform; rdctios i th altrativ wag which is affctd by th grosity of mploymt bfits, rddacy paymts ad lowr miimm wags; ad by makig th labor markt mor comptitiv- both domstically ad itratioally. 7. Explai how a rcssio might rais th NAIRU. Hystrsis rfrs to th cas whr tmporary or short r movmts i th NAIRU ca b propagatd ito mdim or log trm ffcts. As a rslt, chags i mploymt that aris i th short r ca bcom vry prsistt or v prmat. LRC LR C w (a) RW w RW BRW (b) SRC w RW BRW BRW For xampl, th coomy starts off at th NAIRU, bt followig a rcssio mploymt riss to. If th sal dyamics play ot, th coomy will rtr to th NAIRU bt iflatio will dclrat. This is bcas at high mploymt ral wag aspiratios of workrs fall, which both prics thm back ito mploymt bt also throgh a wag-pric spiral lowrs th xpctd ad actal rat of iflatio prvailig i th coomy.

13 This mchaism thogh will b sht off if th NAIRU also icrass to. Bcas bargaid ad pric dtrmid ral wags ar ow cosistt at this lvl of mploymt, thr will b o frthr prssr o ithr mploymt or iflatio to fall. A hystrsis mchaism is aythig that lads to a tmporary chag i mploymt bcomig prmat. This wold aris throgh somthig iflcig th compots of th wag bargaiig procss. For xampl, (s pal (a)) a ris i mploymt may act to rdc labor prodctivity ad shift th pric dtrmid ral wag pwards. This cold b th rslt of skill dgradatio from mploymt, or lowr capital ivstmt d to a dprssd coomy. Altrativly, th bargaid ral wag may shift or pivot pwards (s pal (b)). Thr ar mros ffcts which may cas this to happ. A icras i mploymt may rais th lvl of log-trm mployd ad rdc th sarch itsity of mployd workrs, so ral wags bcom lss ssitiv to mploymt. If powrfl isidr-otsidr ffcts occr a similar sitatio wold aris if th highr mploymt displacs prvios isidrs ito th pool of otsidrs- who hav littl powr to iflc th wag bargai. 8. What ar th costs of iflatio? Dos th cotrol of iflatio dsrv its prmit positio i policy circls? If iflatio is aticipatd, th it may grat rdistribtio ffcts if cotracts ar ot sfficitly flxibl. A xpctd ris i iflatio will most sigificatly rdistribt from savrs/ldrs to borrowr. This is bcas th ral itrst rat o dbt is rdcd, bt also highr iflatio rods th val of omial liabilitis. It also bfits pric-sttrs (firms) ovr wag-sttrs (wags) by rdcig th ral wag. Fiscal drag occrs wh omial wags ar pdatd for iflatio, bt icom tax thrsholds rmai chagd. This has th tdcy to drag hosholds ito highr tax brackts- thrfor gratig rdistribtio from tax payrs to govrmt. Also, thos o fixd icoms, which ar prdomiatly thos that liv o social scrity paymts, may s th ral val of this icom dcli. Almost all of ths r-distribtiv ffcts cold b cotrd by spcifyig cotracts that tak accot of iflatio. If ths variabls wr idxd, so adjst atomatically to iflatio, th th ral val of ths variabls cold b maitaid v if iflatio was xpctd. Wh iflatio is aticipatd it ca still prodc m ad sho lathr costs. Ths ar th ral rsorcs that ar cosmd i dalig with chags i prics. M costs rfr to th costs of rpritig ms- bt cold apply to ay rcord of sals prics sch as brochrs tc. If prics ris, th mor moy is rqird to drtak th

14 sam trasactios. h cost of dalig with this is kow as sho-lathr costs, bcas it is symbolisd with popl havig to mak mor trips to th bak. Sic th 970s coomic policy-makig has shiftd away from th targt of fll mploymt towards maitaiig pric stability, or low ad stabl iflatio. This chag i thos partly rflcts th poor prformac of dmad-maagmt programs dsigd to maitai fll mploymt- bt largly rsposibl for stop-go cycls i th coomy. Also, prior to th 970s high ad prsistt iflatio was ot a fatr of dvlopd cotris, ad thrfor th cotrol of iflatio was t a major macrocoomic policy objctiv. Th ris of iflatio was dmd to crat istabilitis i th coomy that might b dtrimtal to its log trm growth prospcts. Sharp flctatios i prics mak it difficlt for firms to valat th ral val of ftr cash flows. ric stability is also likly to b trasmittd ito omial ad ral itrst rat istability. Thrfor, i a ra of iflatio, policy-makrs saw thir rsposibility as cratig a stabl macrocoomic viromt charactrisd by low iflatio ad prdictabl itrst rats. This chag i mphasis has ld to motary policy risig i sigificac to fiscal policy. I trms of maitaiig iflatio targts fiscal policy is hamprd by a lack of flxibility (it ca oly b adjstd from tim to tim), ad is also affctd by Lcas critiq isss, as wll as havig th addd complicatio of affctig th compositio as wll as th lvl of otpt. 9. olicymakrs wold bttr attai thir macrocoomic objctivs if thy had thir discrtio tak away from thm. Discss TC I TC ( ) = α + TC I ( ) = α + I

15 olicy-makrs fac difficltis bcas low iflatio aocmts ar timicosistt. Sppos iflatio was at its tim cosistt lvl, TC, yt th govrmt wishd to rdc iflatio to a targt of. O aocmt, if this targt was crdibl th privat sctor wold rdc thir iflatio xpctatios to this lvl ad th coomy wold xpric a pailss disiflatio. Howvr, this policy is likly to work. If th privat sctor sts iflatio xpctatios qal to th govrmt facs a ictiv to lash a iflatio srpris, ad trad-off highr iflatio for lowr mploymt. This is bcas th govrmt ca mov oto a lowr idiffrc crv as a rslt. I fact, th lowst lvl of iflatio whr this ictiv dos ot xist is TC - this wold b a tim cosistt iflatio aocmt. Th difficlty, i trms of crdibility, ariss bcas th govrmt has prfrcs for both low iflatio ad mploymt ad might b prpard to xploit a short r hillips crv trad-off. If motary policy is dlgatd to a body that has o prfrc for mploymt, th th tim icosistcy problm disappars. Thr ar howvr som costs to dlgatig motary policy to a body sch as a idpdt ctral bak. First, th govrmt loss cotrol of a lvr it cold othrwis s to cotrol th coomy, so dlgatio may ivolv stabilisatio costs for otpt. Scodly, if motary ad fiscal policis hav comptig objctivs, th a lack of coordiatio cold sr that pshs th coomy towards a ifrior qilibrim positio. 0. Wold a dcras i th ctral bak s iflatio targt affct th lvl of mploymt? If motary policy is crdibl, th low iflatio aocmts wold lad dirctly to a rdctio i iflatio xpctatios. Ths will th fd throgh th pric-sttig procss ito actal iflatio. As a rslt thr is a pailss disiflatio to th w targt. For policy to b crdibl, th privat sctor has to bliv that th ctral bak wold b prpard to rais itrst rats ad psh mploymt abov th NAIRU to forc iflatio dow to targt- if cssary. Thrfor, th privat sctor wold ot s fit to try ad call th blff of th ctral bakrs. If ctral bakrs wr cocrd abot iflatio, th a ris i mploymt wold lad to a shift o a highr idiffrc crv. Udr ths coditios th rsolv of th ctral bakrs to actally tak th paifl policy actio if rqird may b wak. Thrfor, th privat sctor may b prpard to qstio whthr th w iflatio targt will b forcd.

16 I I ( ) = α + ( ) = α + Crdibility ovr iflatio aocmts thrfor ariss from th sigl-middss of ctral bakrs to targt iflatio ad igor othr possibl macrocoomic objctivs. Thr ar varios ways i which crdibility ca b achivd: - dlgatio of policy to a idpdt ctral bakr with a strog rptatio for big iflatio advrs. - dsigig cotracts to corag ctral bakrs to bhav i a particlar way - allowig th ctral bak to bild a rptatio for big togh o iflatio ovr tim.. Drig World War II, both Grmay ad Britai pritd larg amots of th othrs crrcy. Why might droppig this o a my city cas mor damag tha high xplosivs? Th basic ratioal is rlatd to th qatity thory of moy. A dramatic icras i th moy spply wold grat high lvls of iflatio, which may drmi th domstic crrcy as a mdim of xchag. This cold th lad to coomic ad political disrptio. If th crrcy bcoms worthlss th agts may hav o othr choic bt to bartr.

17 . Explai th rols of motary ad fiscal policy is dig hypriflatio. A priod of hypriflatio oft ariss wh a govrmt rs a larg dbt which is pays for by pritig moy. It ca do this by sllig bods to th ctral baks which prit moy to pay for thm- th govrmt th spds th procds. Sigorag, or th iflatio tax, rprsts th ral rsorcs accrd to th govrmt by gratig iflatio. Howvr, raisig a giv amot of sigorag rv wold rqir highr ad highr iflatio which cold lad to hypriflatio. O way to d th priod of hypriflatio is to rplac th iflatio tax with othr taxs, or throgh a rdctio i govrmt spdig. This will stabilis th lvl of dbt ad thrfor th d to motaris it. I trms of motary policy, th govrmt ds to mak a crdibl commitmt ot to fiac dficit spdig by pritig moy. Thr ar a mbr of ways i which this crdibility might b achivd. Th ctral bak cold b prvtd from byig govrmt bods. Th omial xchag rat cold b pggd to a low iflatio crrcy- a attmpt to achor xpctatios ad import disiflatio. Altrativly, th govrmt cold dollaris- thrfor ay sigorag rvs wold prtai to th US so thr is o ictiv to motaris fiscal dficits.

Learning objectives. Three models of aggregate supply. 1. The sticky-wage model 2. The imperfect-information model 3. The sticky-price model

Learning objectives. Three models of aggregate supply. 1. The sticky-wage model 2. The imperfect-information model 3. The sticky-price model Larig objctivs thr modls of aggrgat supply i which output dpds positivly o th pric lvl i th short ru th short-ru tradoff btw iflatio ad umploymt kow as th Phillips curv Aggrgat Supply slid 1 Thr modls

More information

macro Road map to this lecture Objectives Aggregate Supply and the Phillips Curve Three models of aggregate supply W = ω P The sticky-wage model

macro Road map to this lecture Objectives Aggregate Supply and the Phillips Curve Three models of aggregate supply W = ω P The sticky-wage model Road map to this lctur macro Aggrgat Supply ad th Phillips Curv W rlax th assumptio that th aggrgat supply curv is vrtical A vrsio of th aggrgat supply i trms of iflatio (rathr tha th pric lvl is calld

More information

Chapter 13 Aggregate Supply

Chapter 13 Aggregate Supply Chaptr 13 Aggrgat Supply 0 1 Larning Objctivs thr modls of aggrgat supply in which output dpnds positivly on th pric lvl in th short run th short-run tradoff btwn inflation and unmploymnt known as th Phillips

More information

10. Joint Moments and Joint Characteristic Functions

10. Joint Moments and Joint Characteristic Functions 0 Joit Momts ad Joit Charactristic Fctios Followig sctio 6 i this sctio w shall itrodc varios paramtrs to compactly rprst th iformatio cotaid i th joit pdf of two rvs Giv two rvs ad ad a fctio g x y dfi

More information

Chapter 14 Aggregate Supply and the Short-run Tradeoff Between Inflation and Unemployment

Chapter 14 Aggregate Supply and the Short-run Tradeoff Between Inflation and Unemployment Chaptr 14 Aggrgat Supply and th Short-run Tradoff Btwn Inflation and Unmploymnt Modifid by Yun Wang Eco 3203 Intrmdiat Macroconomics Florida Intrnational Univrsity Summr 2017 2016 Worth Publishrs, all

More information

Inflation and Unemployment

Inflation and Unemployment C H A P T E R 13 Aggrgat Supply and th Short-run Tradoff Btwn Inflation and Unmploymnt MACROECONOMICS SIXTH EDITION N. GREGORY MANKIW PowrPoint Slids by Ron Cronovich 2008 Worth Publishrs, all rights rsrvd

More information

Diploma Macro Paper 2

Diploma Macro Paper 2 Diploma Macro Papr 2 Montary Macroconomics Lctur 6 Aggrgat supply and putting AD and AS togthr Mark Hays 1 Exognous: M, G, T, i*, π Goods markt KX and IS (Y, C, I) Mony markt (LM) (i, Y) Labour markt (P,

More information

EMPLOYMENT AND THE DISTRIBUTION OF INCOME. Andrés Velasco

EMPLOYMENT AND THE DISTRIBUTION OF INCOME. Andrés Velasco EMPLOYMENT AND THE DISTRIBUTION OF INCOME Adrés Vlasco Ju 2011 Motivatio My xpric as Fiac Miistr Chil: hatd discussios o iquality. But... Focus oly o th wag distributio Discussios o th shap of th wag distributio

More information

8. Barro Gordon Model

8. Barro Gordon Model 8. Barro Gordo Modl, -.. mi s t L mi goals of motary poliy:. Miimiz dviatios of iflatio from its optimal rat π. Try to ahiv ffiit mploymt, > Stati Phillips urv: = + π π, Loss futio L = π π + Rspos of th

More information

H2 Mathematics Arithmetic & Geometric Series ( )

H2 Mathematics Arithmetic & Geometric Series ( ) H Mathmatics Arithmtic & Gomtric Sris (08 09) Basic Mastry Qustios Arithmtic Progrssio ad Sris. Th rth trm of a squc is 4r 7. (i) Stat th first four trms ad th 0th trm. (ii) Show that th squc is a arithmtic

More information

1985 AP Calculus BC: Section I

1985 AP Calculus BC: Section I 985 AP Calculus BC: Sctio I 9 Miuts No Calculator Nots: () I this amiatio, l dots th atural logarithm of (that is, logarithm to th bas ). () Ulss othrwis spcifid, th domai of a fuctio f is assumd to b

More information

z 1+ 3 z = Π n =1 z f() z = n e - z = ( 1-z) e z e n z z 1- n = ( 1-z/2) 1+ 2n z e 2n e n -1 ( 1-z )/2 e 2n-1 1-2n -1 1 () z

z 1+ 3 z = Π n =1 z f() z = n e - z = ( 1-z) e z e n z z 1- n = ( 1-z/2) 1+ 2n z e 2n e n -1 ( 1-z )/2 e 2n-1 1-2n -1 1 () z Sris Expasio of Rciprocal of Gamma Fuctio. Fuctios with Itgrs as Roots Fuctio f with gativ itgrs as roots ca b dscribd as follows. f() Howvr, this ifiit product divrgs. That is, such a fuctio caot xist

More information

4. Money cannot be neutral in the short-run the neutrality of money is exclusively a medium run phenomenon.

4. Money cannot be neutral in the short-run the neutrality of money is exclusively a medium run phenomenon. PART I TRUE/FALSE/UNCERTAIN (5 points ach) 1. Lik xpansionary montary policy, xpansionary fiscal policy rturns output in th mdium run to its natural lvl, and incrass prics. Thrfor, fiscal policy is also

More information

APPENDIX: STATISTICAL TOOLS

APPENDIX: STATISTICAL TOOLS I. Nots o radom samplig Why do you d to sampl radomly? APPENDI: STATISTICAL TOOLS I ordr to masur som valu o a populatio of orgaisms, you usually caot masur all orgaisms, so you sampl a subst of th populatio.

More information

Introduction - the economics of incomplete information

Introduction - the economics of incomplete information Introdction - th conomics of incomplt information Backgrond: Noclassical thory of labor spply: No nmploymnt, individals ithr mployd or nonparticipants. Altrnativs: Job sarch Workrs hav incomplt info on

More information

11/11/2018. Chapter 14 8 th and 9 th edition. Aggregate Supply and the Short-run Tradeoff Between Inflation and Unemployment.

11/11/2018. Chapter 14 8 th and 9 th edition. Aggregate Supply and the Short-run Tradeoff Between Inflation and Unemployment. Chaptr 14 8 th and 9 th dition Aggrgat Supply and th Short-run Tradoff Btwn Inflation and Unmploymnt W covr modls of aggrgat supply in which output dpnds positivly on th pric lvl in th short run. th short-run

More information

They must have different numbers of electrons orbiting their nuclei. They must have the same number of neutrons in their nuclei.

They must have different numbers of electrons orbiting their nuclei. They must have the same number of neutrons in their nuclei. 37 1 How may utros ar i a uclus of th uclid l? 20 37 54 2 crtai lmt has svral isotops. Which statmt about ths isotops is corrct? Thy must hav diffrt umbrs of lctros orbitig thir ucli. Thy must hav th sam

More information

INTRODUCTION TO SAMPLING DISTRIBUTIONS

INTRODUCTION TO SAMPLING DISTRIBUTIONS http://wiki.stat.ucla.du/socr/id.php/socr_courss_2008_thomso_econ261 INTRODUCTION TO SAMPLING DISTRIBUTIONS By Grac Thomso INTRODUCTION TO SAMPLING DISTRIBUTIONS Itro to Samplig 2 I this chaptr w will

More information

Chapter 2 Infinite Series Page 1 of 11. Chapter 2 : Infinite Series

Chapter 2 Infinite Series Page 1 of 11. Chapter 2 : Infinite Series Chatr Ifiit Sris Pag of Sctio F Itgral Tst Chatr : Ifiit Sris By th d of this sctio you will b abl to valuat imror itgrals tst a sris for covrgc by alyig th itgral tst aly th itgral tst to rov th -sris

More information

Fooling Newton s Method a) Find a formula for the Newton sequence, and verify that it converges to a nonzero of f. A Stirling-like Inequality

Fooling Newton s Method a) Find a formula for the Newton sequence, and verify that it converges to a nonzero of f. A Stirling-like Inequality Foolig Nwto s Mthod a Fid a formla for th Nwto sqc, ad vrify that it covrgs to a ozro of f. ( si si + cos 4 4 3 4 8 8 bt f. b Fid a formla for f ( ad dtrmi its bhavior as. f ( cos si + as A Stirlig-li

More information

Recounting the Rationals

Recounting the Rationals Rconting th Rationals Nil Calkin and Hrbrt S. Wilf pril, 000 It is wll known (indd, as Pal Erd}os might hav said, vry child knows) that th rationals ar contabl. Howvr, th standard prsntations of this fact

More information

Discrete Fourier Transform. Nuno Vasconcelos UCSD

Discrete Fourier Transform. Nuno Vasconcelos UCSD Discrt Fourir Trasform uo Vascoclos UCSD Liar Shift Ivariat (LSI) systms o of th most importat cocpts i liar systms thory is that of a LSI systm Dfiitio: a systm T that maps [ ito y[ is LSI if ad oly if

More information

A Simple Proof that e is Irrational

A Simple Proof that e is Irrational Two of th most bautiful ad sigificat umbrs i mathmatics ar π ad. π (approximatly qual to 3.459) rprsts th ratio of th circumfrc of a circl to its diamtr. (approximatly qual to.788) is th bas of th atural

More information

ph People Grade Level: basic Duration: minutes Setting: classroom or field site

ph People Grade Level: basic Duration: minutes Setting: classroom or field site ph Popl Adaptd from: Whr Ar th Frogs? in Projct WET: Curriculum & Activity Guid. Bozman: Th Watrcours and th Council for Environmntal Education, 1995. ph Grad Lvl: basic Duration: 10 15 minuts Stting:

More information

Review Exercises. 1. Evaluate using the definition of the definite integral as a Riemann Sum. Does the answer represent an area? 2

Review Exercises. 1. Evaluate using the definition of the definite integral as a Riemann Sum. Does the answer represent an area? 2 MATHEMATIS --RE Itgral alculus Marti Huard Witr 9 Rviw Erciss. Evaluat usig th dfiitio of th dfiit itgral as a Rima Sum. Dos th aswr rprst a ara? a ( d b ( d c ( ( d d ( d. Fid f ( usig th Fudamtal Thorm

More information

The Open Economy in the Short Run

The Open Economy in the Short Run Economics 442 Mnzi D. Chinn Spring 208 Social Scincs 748 Univrsity of Wisconsin-Madison Th Opn Economy in th Short Run This st of nots outlins th IS-LM modl of th opn conomy. First, it covrs an accounting

More information

Chapter (8) Estimation and Confedence Intervals Examples

Chapter (8) Estimation and Confedence Intervals Examples Chaptr (8) Estimatio ad Cofdc Itrvals Exampls Typs of stimatio: i. Poit stimatio: Exampl (1): Cosidr th sampl obsrvatios, 17,3,5,1,18,6,16,10 8 X i i1 17 3 5 118 6 16 10 116 X 14.5 8 8 8 14.5 is a poit

More information

Worksheet: Taylor Series, Lagrange Error Bound ilearnmath.net

Worksheet: Taylor Series, Lagrange Error Bound ilearnmath.net Taylor s Thorm & Lagrag Error Bouds Actual Error This is th ral amout o rror, ot th rror boud (worst cas scario). It is th dirc btw th actual () ad th polyomial. Stps:. Plug -valu ito () to gt a valu.

More information

Two Products Manufacturer s Production Decisions with Carbon Constraint

Two Products Manufacturer s Production Decisions with Carbon Constraint Managmnt Scinc and Enginring Vol 7 No 3 pp 3-34 DOI:3968/jms9335X374 ISSN 93-34 [Print] ISSN 93-35X [Onlin] wwwcscanadant wwwcscanadaorg Two Products Manufacturr s Production Dcisions with Carbon Constraint

More information

Solution to 1223 The Evil Warden.

Solution to 1223 The Evil Warden. Solutio to 1 Th Evil Ward. This is o of thos vry rar PoWs (I caot thik of aothr cas) that o o solvd. About 10 of you submittd th basic approach, which givs a probability of 47%. I was shockd wh I foud

More information

dr Bartłomiej Rokicki Chair of Macroeconomics and International Trade Theory Faculty of Economic Sciences, University of Warsaw

dr Bartłomiej Rokicki Chair of Macroeconomics and International Trade Theory Faculty of Economic Sciences, University of Warsaw dr Bartłomij Rokicki Chair of Macroconomics and Intrnational Trad Thory Faculty of Economic Scincs, Univrsity of Warsaw dr Bartłomij Rokicki Opn Economy Macroconomics Small opn conomy. Main assumptions

More information

2617 Mark Scheme June 2005 Mark Scheme 2617 June 2005

2617 Mark Scheme June 2005 Mark Scheme 2617 June 2005 Mark Schm 67 Ju 5 GENERAL INSTRUCTIONS Marks i th mark schm ar plicitly dsigatd as M, A, B, E or G. M marks ("mthod" ar for a attmpt to us a corrct mthod (ot mrly for statig th mthod. A marks ("accuracy"

More information

Motivation. We talk today for a more flexible approach for modeling the conditional probabilities.

Motivation. We talk today for a more flexible approach for modeling the conditional probabilities. Baysia Ntworks Motivatio Th coditioal idpdc assuptio ad by aïv Bays classifirs ay s too rigid spcially for classificatio probls i which th attributs ar sowhat corrlatd. W talk today for a or flibl approach

More information

Probability & Statistics,

Probability & Statistics, Probability & Statistics, BITS Pilai K K Birla Goa Campus Dr. Jajati Kshari Sahoo Dpartmt of Mathmatics BITS Pilai, K K Birla Goa Campus Poisso Distributio Poisso Distributio: A radom variabl X is said

More information

On a problem of J. de Graaf connected with algebras of unbounded operators de Bruijn, N.G.

On a problem of J. de Graaf connected with algebras of unbounded operators de Bruijn, N.G. O a problm of J. d Graaf coctd with algbras of uboudd oprators d Bruij, N.G. Publishd: 01/01/1984 Documt Vrsio Publishr s PDF, also kow as Vrsio of Rcord (icluds fial pag, issu ad volum umbrs) Plas chck

More information

LECTURE 6 TRANSFORMATION OF RANDOM VARIABLES

LECTURE 6 TRANSFORMATION OF RANDOM VARIABLES LECTURE 6 TRANSFORMATION OF RANDOM VARIABLES TRANSFORMATION OF FUNCTION OF A RANDOM VARIABLE UNIVARIATE TRANSFORMATIONS TRANSFORMATION OF RANDOM VARIABLES If s a rv wth cdf F th Y=g s also a rv. If w wrt

More information

Testing Purchasing Power Parity between India and US

Testing Purchasing Power Parity between India and US Tstig Purchasig Powr Parity btw Idia ad US Vipul Sharma Mavidr Sigh Pahwa Gtika Sharma Abstract I today s comptitiv world th structur ad th dirctio of xtral trad ar dtrmid by th xchag rat coducts. I othr

More information

How many neutrino species?

How many neutrino species? ow may utrio scis? Two mthods for dtrmii it lium abudac i uivrs At a collidr umbr of utrio scis Exasio of th uivrs is ovrd by th Fridma quatio R R 8G tot Kc R Whr: :ubblcostat G :Gravitatioal costat 6.

More information

Exchange rates in the long run (Purchasing Power Parity: PPP)

Exchange rates in the long run (Purchasing Power Parity: PPP) Exchang rats in th long run (Purchasing Powr Parity: PPP) Jan J. Michalk JJ Michalk Th law of on pric: i for a product i; P i = E N/ * P i Or quivalntly: E N/ = P i / P i Ida: Th sam product should hav

More information

Triple Play: From De Morgan to Stirling To Euler to Maclaurin to Stirling

Triple Play: From De Morgan to Stirling To Euler to Maclaurin to Stirling Tripl Play: From D Morga to Stirlig To Eulr to Maclauri to Stirlig Augustus D Morga (186-1871) was a sigificat Victoria Mathmaticia who mad cotributios to Mathmatics History, Mathmatical Rcratios, Mathmatical

More information

Session : Plasmas in Equilibrium

Session : Plasmas in Equilibrium Sssio : Plasmas i Equilibrium Ioizatio ad Coductio i a High-prssur Plasma A ormal gas at T < 3000 K is a good lctrical isulator, bcaus thr ar almost o fr lctros i it. For prssurs > 0.1 atm, collisio amog

More information

Mount Vernon Charles Station Potential Visibility Assessment. December 4, 2017 Chesapeake Conservancy Annapolis, MD 21401

Mount Vernon Charles Station Potential Visibility Assessment. December 4, 2017 Chesapeake Conservancy Annapolis, MD 21401 Mt Vr Charls tati Pttial Vility Assssmt Dcmbr, 7 Chsapak Csrvacy Aaplis, MD Mthdlgy Bst Availabl Data Lidar pit clds rprst th bst availabl lvati data i Charls ad Pric Grg s Ctis. Ths datasts ctai millis

More information

Bipolar Junction Transistors

Bipolar Junction Transistors ipolar Juctio Trasistors ipolar juctio trasistors (JT) ar activ 3-trmial dvics with aras of applicatios: amplifirs, switch tc. high-powr circuits high-spd logic circuits for high-spd computrs. JT structur:

More information

Lectures 9 IIR Systems: First Order System

Lectures 9 IIR Systems: First Order System EE3054 Sigals ad Systms Lcturs 9 IIR Systms: First Ordr Systm Yao Wag Polytchic Uivrsity Som slids icludd ar xtractd from lctur prstatios prpard by McCllla ad Schafr Lics Ifo for SPFirst Slids This work

More information

The Ramsey Model. Reading: Firms. Households. Behavior of Households and Firms. Romer, Chapter 2-A;

The Ramsey Model. Reading: Firms. Households. Behavior of Households and Firms. Romer, Chapter 2-A; Th Ramsy Modl Rading: Romr, Chaptr 2-A; Dvlopd by Ramsy (1928), latr dvlopd furthr by Cass (1965) and Koopmans (1965). Similar to th Solow modl: labor and knowldg grow at xognous rats. Important diffrnc:

More information

(Reference: sections in Silberberg 5 th ed.)

(Reference: sections in Silberberg 5 th ed.) ALE. Atomic Structur Nam HEM K. Marr Tam No. Sctio What is a atom? What is th structur of a atom? Th Modl th structur of a atom (Rfrc: sctios.4 -. i Silbrbrg 5 th d.) Th subatomic articls that chmists

More information

Chapter Five. More Dimensions. is simply the set of all ordered n-tuples of real numbers x = ( x 1

Chapter Five. More Dimensions. is simply the set of all ordered n-tuples of real numbers x = ( x 1 Chatr Fiv Mor Dimsios 51 Th Sac R W ar ow rard to mov o to sacs of dimsio gratr tha thr Ths sacs ar a straightforward gralizatio of our Euclida sac of thr dimsios Lt b a ositiv itgr Th -dimsioal Euclida

More information

orbiting electron turns out to be wrong even though it Unfortunately, the classical visualization of the

orbiting electron turns out to be wrong even though it Unfortunately, the classical visualization of the Lctur 22-1 Byond Bohr Modl Unfortunatly, th classical visualization of th orbiting lctron turns out to b wrong vn though it still givs us a simpl way to think of th atom. Quantum Mchanics is ndd to truly

More information

European Business Confidence Survey December 2012 Positive expectations for 2013

European Business Confidence Survey December 2012 Positive expectations for 2013 Dcmbr 2012 Erpa Bsiss Cfic rv Dcmbr 2012 Psitiv xpctatis fr 2013 Lasrp a Ivigrs EMEA hav rctl cmplt thir latst Erpa Bsiss Cfic rv. Th fiigs sggst a psitiv start t 2013 a a mr ptimistic tlk cmpar t that

More information

Option 3. b) xe dx = and therefore the series is convergent. 12 a) Divergent b) Convergent Proof 15 For. p = 1 1so the series diverges.

Option 3. b) xe dx = and therefore the series is convergent. 12 a) Divergent b) Convergent Proof 15 For. p = 1 1so the series diverges. Optio Chaptr Ercis. Covrgs to Covrgs to Covrgs to Divrgs Covrgs to Covrgs to Divrgs 8 Divrgs Covrgs to Covrgs to Divrgs Covrgs to Covrgs to Covrgs to Covrgs to 8 Proof Covrgs to π l 8 l a b Divrgt π Divrgt

More information

LECTURE 13 Filling the bands. Occupancy of Available Energy Levels

LECTURE 13 Filling the bands. Occupancy of Available Energy Levels LUR 3 illig th bads Occupacy o Availabl rgy Lvls W hav dtrmid ad a dsity o stats. W also d a way o dtrmiig i a stat is illd or ot at a giv tmpratur. h distributio o th rgis o a larg umbr o particls ad

More information

PURE MATHEMATICS A-LEVEL PAPER 1

PURE MATHEMATICS A-LEVEL PAPER 1 -AL P MATH PAPER HONG KONG EXAMINATIONS AUTHORITY HONG KONG ADVANCED LEVEL EXAMINATION PURE MATHEMATICS A-LEVEL PAPER 8 am am ( hours) This papr must b aswrd i Eglish This papr cosists of Sctio A ad Sctio

More information

8(4 m0) ( θ ) ( ) Solutions for HW 8. Chapter 25. Conceptual Questions

8(4 m0) ( θ ) ( ) Solutions for HW 8. Chapter 25. Conceptual Questions Solutios for HW 8 Captr 5 Cocptual Qustios 5.. θ dcrass. As t crystal is coprssd, t spacig d btw t plas of atos dcrass. For t first ordr diffractio =. T Bragg coditio is = d so as d dcrass, ust icras for

More information

MONTGOMERY COLLEGE Department of Mathematics Rockville Campus. 6x dx a. b. cos 2x dx ( ) 7. arctan x dx e. cos 2x dx. 2 cos3x dx

MONTGOMERY COLLEGE Department of Mathematics Rockville Campus. 6x dx a. b. cos 2x dx ( ) 7. arctan x dx e. cos 2x dx. 2 cos3x dx MONTGOMERY COLLEGE Dpartmt of Mathmatics Rockvill Campus MATH 8 - REVIEW PROBLEMS. Stat whthr ach of th followig ca b itgratd by partial fractios (PF), itgratio by parts (PI), u-substitutio (U), or o of

More information

Washington State University

Washington State University he 3 Ktics ad Ractor Dsig Sprg, 00 Washgto Stat Uivrsity Dpartmt of hmical Egrg Richard L. Zollars Exam # You will hav o hour (60 muts) to complt this xam which cosists of four (4) problms. You may us

More information

KISS: A Bit Too Simple. Greg Rose

KISS: A Bit Too Simple. Greg Rose KI: A Bit Too impl Grg Ros ggr@qualcomm.com Outli KI radom umbr grator ubgrators Efficit attack N KI ad attack oclusio PAGE 2 O approach to PRNG scurity "A radom umbr grator is lik sx: Wh it's good, its

More information

7. Differentiation of Trigonometric Function

7. Differentiation of Trigonometric Function 7. Diffrtiatio of Trigootric Fctio RADIAN MEASURE. Lt s ot th lgth of arc AB itrcpt y th ctral agl AOB o a circl of rais r a lt S ot th ara of th sctor AOB. (If s is /60 of th circfrc, AOB = 0 ; if s =

More information

CDS 101: Lecture 5.1 Reachability and State Space Feedback

CDS 101: Lecture 5.1 Reachability and State Space Feedback CDS, Lctur 5. CDS : Lctur 5. Rachability ad Stat Spac Fdback Richard M. Murray ad Hido Mabuchi 5 Octobr 4 Goals: Di rachability o a cotrol systm Giv tsts or rachability o liar systms ad apply to ampls

More information

ln x = n e = 20 (nearest integer)

ln x = n e = 20 (nearest integer) H JC Prlim Solutios 6 a + b y a + b / / dy a b 3/ d dy a b at, d Giv quatio of ormal at is y dy ad y wh. d a b () (,) is o th curv a+ b () y.9958 Qustio Solvig () ad (), w hav a, b. Qustio d.77 d d d.77

More information

Chapter 4: Exponential and Logarithmic Functions

Chapter 4: Exponential and Logarithmic Functions Erciss. 7 EXERCISES. Chatr : Eotial a Logarithmic Fuctios. a. c.. a. c. 7.89. a. 0.5 /.69 c. 5 5 + ( ). a. 5 5 5 5 5 + ( ) 5 c. + 0.086 0.050 /.96 5 ( ) + 5 + 6 + ( ) 5. 6. 7. 8. 9. 5.697 0. 0.9. a.. a.

More information

Time : 1 hr. Test Paper 08 Date 04/01/15 Batch - R Marks : 120

Time : 1 hr. Test Paper 08 Date 04/01/15 Batch - R Marks : 120 Tim : hr. Tst Papr 8 D 4//5 Bch - R Marks : SINGLE CORRECT CHOICE TYPE [4, ]. If th compl umbr z sisfis th coditio z 3, th th last valu of z is qual to : z (A) 5/3 (B) 8/3 (C) /3 (D) o of ths 5 4. Th itgral,

More information

A Review of Complex Arithmetic

A Review of Complex Arithmetic /0/005 Rviw of omplx Arithmti.do /9 A Rviw of omplx Arithmti A omplx valu has both a ral ad imagiary ompot: { } ad Im{ } a R b so that w a xprss this omplx valu as: whr. a + b Just as a ral valu a b xprssd

More information

Principles of Humidity Dalton s law

Principles of Humidity Dalton s law Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid

More information

CDS 101: Lecture 5.1 Reachability and State Space Feedback

CDS 101: Lecture 5.1 Reachability and State Space Feedback CDS, Lctur 5. CDS : Lctur 5. Rachability ad Stat Spac Fdback Richard M. Murray 7 Octobr 3 Goals: Di rachability o a cotrol systm Giv tsts or rachability o liar systms ad apply to ampls Dscrib th dsig o

More information

Figure 2-18 Thevenin Equivalent Circuit of a Noisy Resistor

Figure 2-18 Thevenin Equivalent Circuit of a Noisy Resistor .8 NOISE.8. Th Nyquist Nois Thorm W ow wat to tur our atttio to ois. W will start with th basic dfiitio of ois as usd i radar thory ad th discuss ois figur. Th typ of ois of itrst i radar thory is trmd

More information

What are those βs anyway? Understanding Design Matrix & Odds ratios

What are those βs anyway? Understanding Design Matrix & Odds ratios Ral paramtr stimat WILD 750 - Wildlif Population Analysis of 6 What ar thos βs anyway? Undrsting Dsign Matrix & Odds ratios Rfrncs Hosmr D.W.. Lmshow. 000. Applid logistic rgrssion. John Wily & ons Inc.

More information

First order differential equation Linear equation; Method of integrating factors

First order differential equation Linear equation; Method of integrating factors First orr iffrntial quation Linar quation; Mtho of intgrating factors Exampl 1: Rwrit th lft han si as th rivativ of th prouct of y an som function by prouct rul irctly. Solving th first orr iffrntial

More information

Allowing for Household Preferences in Emission Trading - A Contribution to the Climate Policy Debate -

Allowing for Household Preferences in Emission Trading - A Contribution to the Climate Policy Debate - Allowig for Hosold Prfrcs i Emissio Tradig - A Cotribtio to t Climat Policy Dbat - by Mical Alim *) ad Fridric Scidr **) Abstract: I t cott of missio tradig it sms to b tak as giv tat popl's prfrcs ca

More information

a 1and x is any real number.

a 1and x is any real number. Calcls Nots Eponnts an Logarithms Eponntial Fnction: Has th form y a, whr a 0, a an is any ral nmbr. Graph y, Graph y ln y y Th Natral Bas (Elr s nmbr): An irrational nmbr, symboliz by th lttr, appars

More information

Higher order derivatives

Higher order derivatives Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of

More information

ONLINE SUPPLEMENT Optimal Markdown Pricing and Inventory Allocation for Retail Chains with Inventory Dependent Demand

ONLINE SUPPLEMENT Optimal Markdown Pricing and Inventory Allocation for Retail Chains with Inventory Dependent Demand Submittd to Maufacturig & Srvic Opratios Maagmt mauscript MSOM 5-4R2 ONLINE SUPPLEMENT Optimal Markdow Pricig ad Ivtory Allocatio for Rtail Chais with Ivtory Dpdt Dmad Stph A Smith Dpartmt of Opratios

More information

Outline. Ionizing Radiation. Introduction. Ionizing radiation

Outline. Ionizing Radiation. Introduction. Ionizing radiation Outli Ioizig Radiatio Chaptr F.A. Attix, Itroductio to Radiological Physics ad Radiatio Dosimtry Radiological physics ad radiatio dosimtry Typs ad sourcs of ioizig radiatio Dscriptio of ioizig radiatio

More information

A Prey-Predator Model with an Alternative Food for the Predator, Harvesting of Both the Species and with A Gestation Period for Interaction

A Prey-Predator Model with an Alternative Food for the Predator, Harvesting of Both the Species and with A Gestation Period for Interaction Int. J. Opn Problms Compt. Math., Vol., o., Jun 008 A Pry-Prdator Modl with an Altrnativ Food for th Prdator, Harvsting of Both th Spcis and with A Gstation Priod for Intraction K. L. arayan and. CH. P.

More information

Where k is either given or determined from the data and c is an arbitrary constant.

Where k is either given or determined from the data and c is an arbitrary constant. Exponntial growth and dcay applications W wish to solv an quation that has a drivativ. dy ky k > dx This quation says that th rat of chang of th function is proportional to th function. Th solution is

More information

Partial Derivatives: Suppose that z = f(x, y) is a function of two variables.

Partial Derivatives: Suppose that z = f(x, y) is a function of two variables. Chaptr Functions o Two Variabls Applid Calculus 61 Sction : Calculus o Functions o Two Variabls Now that ou hav som amiliarit with unctions o two variabls it s tim to start appling calculus to hlp us solv

More information

COMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.

COMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH. C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH

More information

1973 AP Calculus BC: Section I

1973 AP Calculus BC: Section I 97 AP Calculus BC: Scio I 9 Mius No Calculaor No: I his amiaio, l dos h aural logarihm of (ha is, logarihm o h bas ).. If f ( ) =, h f ( ) = ( ). ( ) + d = 7 6. If f( ) = +, h h s of valus for which f

More information

An Introduction to Asymptotic Expansions

An Introduction to Asymptotic Expansions A Itroductio to Asmptotic Expasios R. Shaar Subramaia Asmptotic xpasios ar usd i aalsis to dscrib th bhavior of a fuctio i a limitig situatio. Wh a fuctio ( x, dpds o a small paramtr, ad th solutio of

More information

Statistics 3858 : Likelihood Ratio for Exponential Distribution

Statistics 3858 : Likelihood Ratio for Exponential Distribution Statistics 3858 : Liklihood Ratio for Expotial Distributio I ths two xampl th rjctio rjctio rgio is of th form {x : 2 log (Λ(x)) > c} for a appropriat costat c. For a siz α tst, usig Thorm 9.5A w obtai

More information

terms of discrete sequences can only take values that are discrete as opposed to

terms of discrete sequences can only take values that are discrete as opposed to Diol Bgyoko () OWER SERIES Diitio Sris lik ( ) r th sm o th trms o discrt sqc. Th trms o discrt sqcs c oly tk vls tht r discrt s opposd to cotios, i.., trms tht r sch tht th mric vls o two cosctivs os

More information

Digital Signal Processing, Fall 2006

Digital Signal Processing, Fall 2006 Digital Sigal Procssig, Fall 6 Lctur 9: Th Discrt Fourir Trasfor Zhg-Hua Ta Dpartt of Elctroic Systs Aalborg Uivrsity, Dar zt@o.aau.d Digital Sigal Procssig, I, Zhg-Hua Ta, 6 Cours at a glac MM Discrt-ti

More information

Sec 2.3 Modeling with First Order Equations

Sec 2.3 Modeling with First Order Equations Sc.3 Modling with First Ordr Equations Mathmatical modls charactriz physical systms, oftn using diffrntial quations. Modl Construction: Translating physical situation into mathmatical trms. Clarly stat

More information

Discrete Fourier Transform (DFT)

Discrete Fourier Transform (DFT) Discrt Fourir Trasorm DFT Major: All Egirig Majors Authors: Duc guy http://umricalmthods.g.us.du umrical Mthods or STEM udrgraduats 8/3/29 http://umricalmthods.g.us.du Discrt Fourir Trasorm Rcalld th xpotial

More information

Intermediate Macroeconomics: New Keynesian Model

Intermediate Macroeconomics: New Keynesian Model Intrmdiat Macroconomics: Nw Kynsian Modl Eric Sims Univrsity of Notr Dam Fall 23 Introduction Among mainstram acadmic conomists and policymakrs, th lading altrnativ to th ral businss cycl thory is th Nw

More information

Supplementary Materials

Supplementary Materials 6 Supplmntary Matrials APPENDIX A PHYSICAL INTERPRETATION OF FUEL-RATE-SPEED FUNCTION A truck running on a road with grad/slop θ positiv if moving up and ngativ if moving down facs thr rsistancs: arodynamic

More information

Examples and applications on SSSP and MST

Examples and applications on SSSP and MST Exampls an applications on SSSP an MST Dan (Doris) H & Junhao Gan ITEE Univrsity of Qunslan COMP3506/7505, Uni of Qunslan Exampls an applications on SSSP an MST Dijkstra s Algorithm Th algorithm solvs

More information

Page 1 BACI. Before-After-Control-Impact Power Analysis For Several Related Populations (Variance Known) October 10, Richard A.

Page 1 BACI. Before-After-Control-Impact Power Analysis For Several Related Populations (Variance Known) October 10, Richard A. Pag BACI Bfor-Aftr-Cotrol-Impact Powr Aalysis For Svral Rlatd Populatios (Variac Kow) Octobr, 3 Richard A. Hirichs Cavat: This study dsig tool is for a idalizd powr aalysis built upo svral simplifyig assumptios

More information

DTFT Properties. Example - Determine the DTFT Y ( e ) of n. Let. We can therefore write. From Table 3.1, the DTFT of x[n] is given by 1

DTFT Properties. Example - Determine the DTFT Y ( e ) of n. Let. We can therefore write. From Table 3.1, the DTFT of x[n] is given by 1 DTFT Proprtis Exampl - Dtrmi th DTFT Y of y α µ, α < Lt x α µ, α < W ca thrfor writ y x x From Tabl 3., th DTFT of x is giv by ω X ω α ω Copyright, S. K. Mitra Copyright, S. K. Mitra DTFT Proprtis DTFT

More information

Homework #3. 1 x. dx. It therefore follows that a sum of the

Homework #3. 1 x. dx. It therefore follows that a sum of the Danil Cannon CS 62 / Luan March 5, 2009 Homwork # 1. Th natural logarithm is dfind by ln n = n 1 dx. It thrfor follows that a sum of th 1 x sam addnd ovr th sam intrval should b both asymptotically uppr-

More information

Introduction. Question: Why do we need new forms of parametric curves? Answer: Those parametric curves discussed are not very geometric.

Introduction. Question: Why do we need new forms of parametric curves? Answer: Those parametric curves discussed are not very geometric. Itrodctio Qestio: Why do we eed ew forms of parametric crves? Aswer: Those parametric crves discssed are ot very geometric. Itrodctio Give sch a parametric form, it is difficlt to kow the derlyig geometry

More information

EEO 401 Digital Signal Processing Prof. Mark Fowler

EEO 401 Digital Signal Processing Prof. Mark Fowler EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1

More information

Blackbody Radiation. All bodies at a temperature T emit and absorb thermal electromagnetic radiation. How is blackbody radiation absorbed and emitted?

Blackbody Radiation. All bodies at a temperature T emit and absorb thermal electromagnetic radiation. How is blackbody radiation absorbed and emitted? All bodis at a tmpratur T mit ad absorb thrmal lctromagtic radiatio Blackbody radiatio I thrmal quilibrium, th powr mittd quals th powr absorbd How is blackbody radiatio absorbd ad mittd? 1 2 A blackbody

More information

22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.

22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches. Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M

More information

Note If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark.

Note If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark. . (a) Eithr y = or ( 0, ) (b) Whn =, y = ( 0 + ) = 0 = 0 ( + ) = 0 ( )( ) = 0 Eithr = (for possibly abov) or = A 3. Not If th candidat blivs that = 0 solvs to = 0 or givs an tra solution of = 0, thn withhold

More information

Ch. 24 Molecular Reaction Dynamics 1. Collision Theory

Ch. 24 Molecular Reaction Dynamics 1. Collision Theory Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic

More information

5.1 The Nuclear Atom

5.1 The Nuclear Atom Sav My Exams! Th Hom of Rvisio For mor awsom GSE ad lvl rsourcs, visit us at www.savmyxams.co.uk/ 5.1 Th Nuclar tom Qustio Papr Lvl IGSE Subjct Physics (0625) Exam oard Topic Sub Topic ooklt ambridg Itratioal

More information

Estimation of Consumer Demand Functions When the Observed Prices Are the Same for All Sample Units

Estimation of Consumer Demand Functions When the Observed Prices Are the Same for All Sample Units Dpartmt of Agricultural ad Rsourc Ecoomics Uivrsity of Califoria, Davis Estimatio of Cosumr Dmad Fuctios Wh th Obsrvd Prics Ar th Sam for All Sampl Uits by Quirio Paris Workig Papr No. 03-004 Sptmbr 2003

More information

u x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula

u x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula 7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting

More information

Law of large numbers

Law of large numbers Law of larg umbrs Saya Mukhrj W rvisit th law of larg umbrs ad study i som dtail two typs of law of larg umbrs ( 0 = lim S ) p ε ε > 0, Wak law of larrg umbrs [ ] S = ω : lim = p, Strog law of larg umbrs

More information

A Propagating Wave Packet Group Velocity Dispersion

A Propagating Wave Packet Group Velocity Dispersion Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to

More information