Lecture 6: Money Market and the LM Curve

Transcription

1 EC01 Itemediate Macoecoomics EC01 Itemediate Macoecoomics Lectue 6: Moe Maket ad the LM Cuve Lectue Outlie: - the LM cuve, ad its elatio to the liquidit pefeece theo Essetial eadig: Makiw: Ch. 11. The Theo of Liquidit efeece I discussig the moe maket i the log-u we showed how pices adjust to maitai the moe maket equilibium. I the shot-u, whe pices ae stick, aothe vaiable must do the job. Kees agued that i the shot-u the iteest ate adjusts to balace the suppl ad demad fo the ecoom s most liquid asset, moe. How the iteest ate adjusts i the moe maket? We have see how the demad fo moe looks like i pevious otes: M d = L( Y, ) whee hee we used the fact that i the shot-u pices ae fixed, theefoe iflatio is zeo ad i =. We still assume that: L a) > 0 Demad fo eal moe balaces iceases with total icome (moe icome = moe tasactios = moe moe eeded) L b) < 0 Demad fo eal moe balaces deceases with the iteest ate sice epesets the cost oppotuit of holdig moe (highe is the iteest ate ad highe is the etu people ca obtai b puttig thei moe ito less liquid assets like bods). We assume that the moe suppl is exogeous (it does ot deped o a othe vaiable) sice it is cotolled b the Cetal Bak (this assumptio is owadas ot 1

2 ealistic sice cetal baks ow cotol the iteest ate ad we will elax it late). uthemoe, sice i the shot u pices ae fixed, the suppl of eal moe balaces s M is completel exogeous i the shot u: M s = M I equilibium we must have that suppl =demad: M = L( Y, ) The theo of liquidit pefeece explais how the iteest ate that will adjust to keep the moe maket i equilibium i the shot-u. o the momet assume that the eal icome (Y) is fixed, so that we focus ol o the iteest ate as the mai detemiat of the moe demad. Accodig to the theo of liquidit pefeece people keeps some potfolios with ma assets i them. Some ae eall liquid ad do ot pa a iteest ate (like the moe i ou pocket o i ou cuet accout) some ae less liquid ad pa ou some iteest ate (like bods o othe secuities). Now suppose that thee is a excess of moe suppl espect to the demad. o example, the cetal bak has ijected ito the ecoomic sstem moe moe espect to the amout people ae willig to hold. Now people have a excess of moe the do ot wat to hold so the covet this excess of moe ito assets that pa a iteest ate. Suppose fo simplicit that thee is ol oe asset the pas ou a iteest ate, ad it is a pepetual bod, that is a bod the pas ou a amout A (fixed) i each ea foeve fom ow (this is just a simplificatio. Addig moe complicated assets will ot chage the mai esults). The pice of this bod toda (the peset value) is give b the followig fomula: A A A = ( 1+ ) B = That is the pice ou eed to pa to bu the bod toda. I that equatio we assume that the iteest ate is costat ove the etie peiod (this is cleal a stog assumptio, but it is made to simplif thigs. uthemoe, eve if we assume that the iteest ate is ot costat ove time, the mai esult about the egative elatio betwee value ad etu will still hold). This meas that the pice ad the etu fo this asset ae egativel elated (this is a basic esult i fiace). The esult comes fom the fact that B is followig a geometic seies (see the Appedix). Give that people have excess moe that the wat to covet ito A

3 bods, the demad fo bods will icease. B simple maket ules, if the demad of a good (i this case a bod) iceases ad the suppl is give, the pice of that good must icease (otice that the pice of bods ca chage i the shot-u while it is the geeal pice level of goods ad sevices that is stick i the shot-u). I ou case, the ol wa the pice of the bods ca icease is b a decease i the iteest ate. Theefoe, the iitial excess of moe suppl will esult i a decease of the iteest ate because of the adjustmet i the potfolios of the agets. This movemet i the iteest ate will icease moe demad (if deceases moe demad iceases) util the equilibium betwee moe suppl ad moe demad is estoed. A simila easoig applies whe thee is a excess of moe demad compaed to moe suppl. I that case people wat to hold moe moe, so the covet thei assets ito liquid moe ad so the demad fo bods will decease ad so the pice of bods. o the pice of bods to decease, iteest ate must icease. I the gaph we have the suppl of eal balaces that is a vetical lie because it is exogeous, while the demad is deceasig i the iteest ate (Y is fixed fo the momet). The equilibium iteest ate is 1 ad the equilibium eal moe balaces is simpl the amout that is supplied b defiitio. At the iteest ate thee is a excess of suppl of moe. As we have said the theo of liquidit pefeece sas that i this case the iteest must decease util the equilibium will be estoed (at 1 ). 3

4 Usig ou simple model we ca aalse what is the effect o the equilibium iteest ate whe the cetal bak iceases o deceases moe suppl. I the gaph above, a chage i the moe suppl will simpl shift the moe suppl vetical lie to the left o to the ight. How the cetal bak aises the iteest ate i the shot-u? Lookig at the gaph, the obvious aswe is b deceasig moe suppl. A decease i the moe suppl will shift the vetical lie to the left, give the demad fo moe; the esult will be a icease i the equilibium iteest ate. Wh? The idea is give b the theo of liquidit pefeece. Whe thee is a eductio i the moe suppl, at the pevious equilibium iteest ate ( 1 ), thee is a excess of demad fo moe. eople will stat covetig thei less liquid assets ito liquid moe. Theefoe, demad fo bods will decease ad theefoe (give the suppl of bods) the pice of bods must decease ad the iteest ate must icease. As the iteest ate iceases, moe demad will be educed ad the iitial excess of demad will be elimiated ad the equilibium estoed with a iteest ate that is ow highe tha the pevious oe. Now ou ca see the diffeece about the behaviou of the same vaiable i the shot ad i the log-u. I the log-u (with flexible pices) a decease i moe suppl will decease the pice level ad theefoe will decease iflatio. A decease i iflatio will decease the omial iteest ate because of the ishe effect 4

5 e ( i = + π ). O the othe had i the shot-u, give that pices ae fixed i ou model, a decease i moe suppl will INCREASE the omial iteest ate (i this case emembe that i = ). Ae these two theoies cofimed b data? Coside the example of the US moeta polic i late 70 s. I Octobe 1979, the chaima of the ED aul Volcke aouced that moeta polic would aim to decease iflatio. I that peiod the iflatio ate i the US (aual ate) was aoud 10%. Betwee August 1979 ad Apil 1980 the ED educed the moe suppl i such a wa that 8% duig that peiod. I August 1979 the omial iteest ate was 10.4%. M deceased b I Apil 1980 the omial iteest ate was 15.8%. As pedicted b the theo of liquidit pefeece the omial iteest ate iceased i the shot-u (i a peiod of 8 moths). I Jaua 1983, that compaed to August 1979 ca be cosideed i the log u, the omial iteest ate was 8.% while iflatio was 3.7%. Theefoe, as pedicted b the quatit theo, a eductio i the moe suppl will decease iflatio ad the omial iteest ate i the log-u. Notice howeve, that the iflatio ad the omial iteest ate did ot fall b equal amouts. This does ot cotadict the ishe Effect, though, as othe ecoomic chages ma have caused movemets i the eal iteest ate. This example seems to cofim that the two theoies fit well the data. Howeve we eed to kow that the peiod cosideed was a peiod whee moeta polic was coducted usig moe tagetig. The edeal Reseve used to chage moe suppl to chage the iteest ate. This is a moeta polic that ca be descibed well b ou moe maket model with exogeous moe suppl. Nowadas, moeta polic i ma couties is the evese of what is descibed above. The cetal bak sets the iteest ate ad the adjusts the moe suppl accodigl. I this case moe suppl is edogeous sice it depeds o the what the maket is askig. Now we eed to move fom the moe maket to a schedule simila to the IS, a schedule that gives us all the combiatios of Y ad that esue the equilibium i the moe maket. Such a schedule is called the LM cuve. Coside the moe maket that we have just descibed ad ow assume that Y is ot fixed but it ca chage. How to deive the LM cuve fom the moe maket equilibium: 5

6 (a) The maket fo eal moe balaces (b) The LM cuve LM 1 M 1 Suppose a icease i eal icome Y, fom Y 1 to Y. Give this icease i icome, the moe demad iceases as well. Give that the moe suppl is fixed, thee is ow a excess of moe demad at the iitial iteest ate. As we kow this excess of moe demad will esult i a icease i the iteest ate. So the iteest ate must ise to estoe equilibium i the moe maket. Notice that the LM schedule is deived fo a fixed level of the moe suppl. Theefoe, thee is a positive elatioship betwee ad Y comig fom the moe maket. This meas that the LM cuve is positivel sloped i the (,Y)-space. The LM cuve equatio: M = L( Y, ) 1) that is exactl the equilibium coditio i the moe maket. The LM cuve gives ou all the possible combiatio of Y ad that makes equatio 1) tue. Notice that the LM equatio gives ou a elatio betwee ad Y ol IMLICITLY. To see that the elatioship betwee ad Y is positive we should be able to obtai the sig of implied b equatio 1). Whe we have a implicit fuctio, we ca use the Implicit Diffeetiatio to obtai that deivative (see the Appedix). Rewite equatio 1) as: L (, Y ) L (, Y 1 ) M/ 1 Y 1 Y Y 6

7 M We ma wite equatio ) as: L( Y, ) = 0 ( Y,, M, ) = 0 ) whee we use the geeic fuctio to expess M L( Y, ). Implicit diffeetiatio sas that give ( Y,, M, ) = 0, the deivative is give b: Y = whee is the patial deivative of the fuctio with espect to. Now we have that: Y L = ad L = L L Remembe that b assumptio we have > 0 ad so Y < 0, ad < 0 ad so > 0. Usig those facts ito equatio 3) we ca see that: > 0 3) Mathematical Appedix a) Geometic Seies A geometic pogessio is a sequece of umbes whee each ew umbe is geeated b multiplig the pevious umbe b a costat tem. The costat tem is called the Geometic Ratio (R) 7

8 o example: the sequece 1,, 4, 8, 16, That sequece is a geometic pogessio with a geometic atio R equal to. The geeal fom of a geometic pogessio is: 3 a, ar, ar, ar,..., ar, ar,... 1 whee a is the iitial value. A geometic seies is the sum of all the tems of a geometic pogessio: a + ar+ ar + ar ar + ar Two impotat esults: A) The sum of the fist tems of a geometic pogessio, deoted b b:, is give a(1 R ) = 1 R B) The sum of a geometic pogessio with ifiite tems ad 0 < R < 1: = 1 a R Result B) holds ol if R is positive but less tha 1. Now coside the peset value of a seies of pamets o eceipts: 1 V = A special case: suppose that all the pamets o eceipts ae equal, i the case above this meas 1 = = = Assume that the seies of pamets o eceipts is ININITE. This is the case of the bod we have cosideed i the lectue ote. The the peset value of that ifiite seies is: V = This comes fom esult B) of a geometic seies stated befoe. Result B) was: = 1 a R 8

9 Whee a is the fist elemet of a ifite geometic seies. I ou case, if all the pamets o eceipts ae the same, we have: V = +... that is a geometic seies with fist elemet ( 1+ ) ad geometic atio 1 R = Usig those facts ito esult B): = 1 1 = = 1+ 1 (1 + = ) b) Implicit Diffeetiatio Implicit Diffeetiatio is a impotat esult that is give b the implicit fuctio theoem (we ae ot goig to state o pove this theoem). A Implicit uctio is a fuctio like: ( x, ) = 0 meaig that we do ot kow how to expess EXLICITLY (o simpl we do ot wat to) oe vaiable (fo example ) as a fuctio of the othe (fo example x). A example of a EXLICIT fuctio is: + = a bx. A example of a IMLICIT fuctio is: ( x + ) ( x + ) = 0 I the secod case to expess diectl as a fuctio of x ca be paticulal difficult (i this paticula case it is impossible). Howeve, fom a implicit fuctio we ca fid some popeties of the elatioship betwee ad x eve if we do ot kow this elatioship! d I paticula we ca calculate the deivative dx fuctio that elates the two vaiables. eve if we do ot kow the exact Implicit Diffeetiatio Rule: give a implicit fuctio of vaiables x, x,..., x ), we ca calculate the deivative: ( 1 x x i j = j i 9

10 with i j ad i, j = 1,,...,, ad whee = deivative of the fuctio with espect to x i. i ( x1, x,..., xi,..., x) is the patial x i 3 o example: ( x, ) x = 0 Accodig to the implicit fuctio diffeetiatio ule we ca fid: whee: ( x, ) x = = x, x d dx = ( x, ) = = 3 x x = = 3 x 3 Implicit diffeetiatio ca be used fo a fuctio witte i a implicit fom. o example: coside the explicit fuctio om this fuctio we ca easil calculate: + d = bx dx = a bx. Now wite the same fuctio i a implicit fom: a bx = 0 ad call this fuctio (, x) a b x = 0. If ou appl the implicit diffeetiatio ule to this implicit vesio of the fuctio ou have: d dx = x = bx whee ( x, ) ( x, ) x = = bx ad = = 1. x 10