BASIC IS-LM by John Eckalbar

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1 BASIC IS-LM by John Eckalba Th ida h is to gt som flavo fo th way M woks in an IS-LM modl. W a going to look at th simplst possibl cas: Th a 3 itms gtting tadd: mony, final goods, and bonds. Th a 3 sctos: housholds, fims, govnmnt. Th is no foign scto, no labo makt, pics a xognous at P, al taxs a fixd at t. Goods makt. Housholds dmand al consumption goods, c, accoding to th lina quation c = a + b(y - t), wh y is al incom and a and b a fixd positiv constants, with b < 2. Fims dmand invstmnt goods accoding to: i = d - f, wh i is plannd al invstmnt, is th nominal intst at, and d and f a fixd positiv constants. 3. Govnmnt buys al output of g and collcts al taxs of t. 4. Th goods makt is in quilibium whn y = c + i + g, o y = a + b(y - t) + d - f + g, o y( - b) = a - b t + d - f + g, o y = (a - b t + d - f + g)/( - b), o = (a - b t + d + g)/f -y( - b)/f Dfin th IS cuv to b th st of all (y, ) points at which = (a - b t + d + g)/f - y( - b)/f...that is, th st of all and y points at which th goods makt clas. A plot of IS with lvant intcpts and slops is shown h. Can you list all soucs of IS-cuv shift? Can you show how IS would shift with changs in g, tc.? If w a at a point lik A, a goods in xcss dmand o xcss supply? (a - b t +d + g)/f A Slop IS = -( - b)/f IS LM Lt nominal mony dmand b givn by Md = k P y + h - j, wh k, h, j > 0 and P is th xognous pic lvl. Lt Ms b th mony supply. Mony dmand quals mony supply whn: y = (Ms - h + j )/ (k P) (a - b t + d + g)/( - b) y

2 = (h - Ms)/j + (k P y)/j Dfin LM to b th st of (y, ) points wh Ms = Md. Thn LM is as shown blow: Again: idntify all th ways this can shift. If w a at a point lik B, is Md > Ms? B LM Slop LM = k P/j 0 (Ms - h)/k P y (h - Ms)/j

3 Th whol stoy Th quilibium is at y and. B su you can xplain all th factos that would mak y and chang. Wh would b mov if w happnd to b at point X? (Ans.: Assum that if c + i LM + g > y, thn y incass, and if Md > Ms, thn incass. That will lt you plot out th vcto fild...mo on this in class.) Th is also a spcial fil on th dynamics, calld ISLM dynamics. You don t hav to know this it is a Mathmatica notbook, but if you know mathmatica, you might look at it. X Evyon should know what I do in class. IS y y Fo what it s woth, y = = fh h+ ML+jHa +d +g btl j bj+fkp H + blh+ M b M +kpha+d +g btl H + blj +fkp though you don t hav to b abl to div this. Could you fill this in? Evnt y g t M P k h..

4 Th big fatu of IS-LM is that y may not b at full mploymnt. If agggat dmand, c + i + g, is low, thn y may wll b lss than full mploymnt output, yf. Som hav agud that if y < yf, thn wags will fall...pics will fall...lm will shift ight (chck this with th quations)...and y will is, BUT:. Wh is th labo makt in this modl? And how do w tat this analytically? Always b suspct of discussing how a modl acts whn somthing changs that is not xplicitly in th modl. 2. If P is falling, won t that lad to xpctations of futh pic ductions, and won t this pompt tads to dlay spnding on c and i, thus causing IS to fall? 3. Popl with nominally fixd dbts may go bankupt...what dos that man? In any cas, th stoy won t unfold (i.., falling pics won t lad to full mploymnt) if:. Wags a igid. 2. Md has an intst inlastic sgmnt (liquidity tap). H s th dal on that: I ll show why in class, but th punch lin is that if Ms incass o P falls, LM will slid ight, If Md looks lik this Thn LM looks lik this. Md(y0) LM Md(y) y > y0 Md but if IS is down low on th lft hand/flat pat of LM, thn y will not incas, so montay policy won t do much. y

5 3. If invstmnt dmand is intst inlastic, IS will b vtical, and again falling P o ising M won t incas y, as you s blow. If invstmnt dmand looks lik this Thn IS looks lik this. Invstmnt dmand IS Md y Ths a th so-calld Kynsian cass, though Kyns himslf would not b happy sing his nam attachd to th tm.

6 IS-LM and Fidman Th figu blow shows a gaphical mthod of diving th LM cuv. Th figus blow show what might b calld th standad cas. Mony dmand is intst lastic, whil Ms is intst inlastic. Th nd sult is that LM has a positiv slop. S if you Ms LM 0 Md(yo) Md(y) M can us th gaph to show that if th Md cuvs w intst inlastic, thn LM would b vtical, o intst inlastic. Using th quations fom ali, w s that if th intst lasticity of Md is 0, thn j = 0 and th LM is vtical...assuming that Ms also has a zo intst lasticity. If so, th LM would b vtical in this cas. What would a vtical LM imply about montay vs. fiscal policy? Daw som pictus and figu it out. This is impotant stuff. yo y y

7 What dos IS-LM hav to say about vlocity? By dfinition, V = P y/ms, so anything that affcts P, y, and/o Ms should affct vlocity, unlss th is som spcial cas at wok. Fo xampl, if g incasd, IS would shift ight and y would incas, so with P and Ms fixd, V would is. In a sns, IS-LM could b thought of as vaiabl vlocity quantity thoy, though you don t gain much by thinking of it that way... In ffct, V is not paticulaly intsting in an IS-LM famwok. In cas you cuious, you can solv to gt: V = P f H h+ ML+j Ha+d +g btl j bj+fkp ì M (Which is positiv if th intcpt of th LM is ngativ and th intcpt of th IS is positiv.) Not that if a, d, o g incass, V incass. In gnal, anything that shifts IS to th ight will T-Billq V caus V to incas and to incas...so you might s V and moving in paalll whn IS movs. But if changs in th mony dmand paamts h, k, o j caus LM to shift, V and will 5000 mov in opposit dictions. That 4000 maks this pictu FIT<90 intsting M Th gaph blow shows somthing intsting. W fit an quation fo mony dmand fom th piod 959 to 990 w gt:

8 M2 FITS Th gssion quation is M2 = DPI M2 op c Pdicto Cof Stdv t-atio p Constant DPI M2 op c s = R-sq = 99.8% R-sq(adj) = 99.8% DPI is disposabl psonal incom, and M2 op c is th oppotunity cost of holding M2 givn by th diffnc btwn th 0 ya tasuy at and th amount paid on a typical itm in M2. This quation givs a good fit ov that piod. H is th sult. S how actual mony dmand is a good fit with th quation until about 989. (Though th quation itslf is vy pimitiv.) If w continud to us that quation to th piod just aft 990, it wands way off th actual tack of M. So it looks lik Md shiftd o wnt unstabl in about 990. This is about whn th Fd quit tagting M2...and this is th ason.

9 Mo cntly, th fittd and actual lins -connctd fo a whil, but th past ya o so has shown anoth bak. ( FITS is th fittd lvl of mony dmand accoding to th quation.): FITS4 M2a Th chat abov gos though Aug 20. Moal: don t tust good looking quations.

10 Vlocity, Mony Dmand, and Policy Sinc V is dfind as PQ/M, MV will always qual PQ. This is not a thoy, it is a dfinition. Fom tim to tim w will call PQ nominal incom o Y. If V w a constant, montay policy aimd at stting th lvl of M would (if it succdd in stting th dsid lvl of M...a big IF) compltly dtmin th valu of Y. Not that th split btwn P and Q is not dtmind, just th poduct Y = PQ. In this cicumstanc th would pobably b a thoy about how, ov th long un, with Q moving oughly along its long un gowth path, M gowth would dtmin P gowth, o inflation. (With V constant, % M = % P + % Q, so % M - % Q = % P...mony gowth minus al gowth quals inflation. Th Fd could pick a tagt long un avag inflation at, and thn st % M to fit th abov quation and mt th inflation objctiv.) Shot un ffcts on Q could also b studid, but xta quations would b ncssay. In this cicumstanc (i.., if V w constant) popl woking at th Fd may o (supisingly) may not nd to pay a lot of attntion to M. Consid this: Suppos w hav a diffnt-looking thoy of PQ. Suppos PQ is dtmind by nominal Agggat Dmand (AD), which consists of th lvls of dmand coming fom C, I, G, and NX (nt xpots). Suppos futh that majo componnts of C and I a dpndnt upon th intst at. W could wit AD() = Y = PQ and us som divativ of th C + I + G + NX and 45-dg lin modl to dtmin quilibium Y, which would dpnd upon. This is ally just an IS cuv whn dawn in (,Y)-spac Now if th Fd contolld, it would contol Y, vn if it paid no attntion to M. In thoy, nothing whatv uls out having both V constant and AD(). Both of ths modls could b tu at th sam tim. All that is ndd is a vtical LM cuv with a nomal IS. If so, whn th Fd adjusts to mov on IS and chang Y, it is focd to chang M accoding to MV = PQ = Y. Both thois a simultanously tu, and agumnts about which is coct a pointlss. Not that if V w not constant, but w highly pdictabl, ssntially th sam montay policy could b followd. That is, if V w known to b autonomously gowing (i.., gowing du to changs in th fficincy of th paymnts systm, ath than, say, to changs in intst ats) at a at of, say, 3% p ya, th Fd could us % M + % V = % P + % Q. This was Fidman s advic in th lat 60s and 70s...Fo xampl, if long un % Q = 2.5, % V = 3, and tagt inflation is, thn th Fd s montay policy ul would b to st % M = =.5. Rmmb fom th Cambidg vsion of th quantity thoy that if V is constant and mony dmand quals mony supply, thn M = kpq = ky, wh k is a constant. This mans, and this is impotant, M dmand is a fixd atio, k, of PQ o Y. And this mans that mony dmand is not a function of intst ats,. If mony dmand w a function of, Md(), thn sinc in quilibium V = PQ/Ms = PQ/Md()...V would b a function of...and not a constant. Suppos mony dmand is a stabl function of and Y in th sns that whn mony dmand quals mony supply, w can wit M = ky - j + ε, wh k and j do not vay ov tim and th o tm ε is small nough to b ignod. This would giv us th usual LM cuv. If w also had th usual IS cuv that divs fom a stabl AD() function, thn again, montay policy could focus ith on M o. If th Fd wantd Y = Y f = AD( f ), wh f is th intst at that sts AD() = Y f, th Fd could tagt M at ky f - j f. But wouldn t it b simpl fo th Fd to simply tagt at f, buying and slling bonds as ndd to kp = f. Thn makts would automatically mov M to ky f - j f. (Kyns has som thoughts on this that w will xplo lat.) With gad to vlocity, if M = ky - j, V = Y/(kY - j). Oth things (lik C, I, G, NX) bing qual, you would xpct to s V positivly colatd with. In fact this is th way things lookd fo M until about 980 and fo M2 until about 990.

11 If V w wildly atic and unpdictabl, thn th would b littl to no point in tying to contol M., sinc M would not colat with any of th Fd s ultimat objctivs. In that cas, th Fd might as wll tagt th numb of goats in Bidwll Pak. If instad, th Agggat Dmand had som pdictabl action to, th Fd would simply tagt, and adjust th tagt as ncssay to nudg Y in th dsid diction. In fact, whn it was cla that V was bcoming unpdictabl stating in th aly 80s, th Fd finally gav up on tagting M. V2 lost its histoical lation to in about 990, and th Fd quickly gav up on it as wll. Th is now som vidnc that V MZM has a stabl lationship with (actually its oppotunity cost), but th is lativly littl intst in fimly tagting MZM.

12 So if you viwd th wold though IS-LM lnss, you would bliv that th ffctivnss of montay vs. Fiscal policy dpnds citically upon th slop of LM. As w look closly at this, what w find is that LM hav it s nomal upwad slop if ith Md o Ms is intst lastic. Lt us look into ths in tun. Intst lasticity of Md. W will look at th thoy bhind two diffnt asons fo bliving that Md is intst lastic. Spculativ Mony Dmand und Ctainty Assum a consol pays $ p ya fov. It s makt valu at tim t will b /t, wh t is th intst at at tim t. Any on tad s xpctd gain fom holding a consol fo on ya would b + t+ wh is th xp ctd int stat attim t +. t+ looking atth fomula abov is th int st paymnt, and is th capitalgain oloss on th bond t+ t t A littl aithmtic shows that t+ Gain= 0 iff t = < + t t+ + t+ Gain> 0 iff t > + t+... so you would hold th consol t+ Gain< 0 iff t < + t+... so you would hold M W tak any on tad s intst xpctations as a givn and ask whth sh/h would want to hold mony o bonds at vaious valus of t. Fo xampl, suppos you xpctd th intst at t.476 to b 5% in on ya. Thn t+ t = = , and if t >.476 you want to hold bonds, whil if t <.0476 you want to hold mony fo spculativ puposs. In this cas.0476 is th citical intst at. You mony dmand cuv will look lik this (at lft). Md

13 you Md m Md Md Md h total At th sam tim, I might xpct intst ats to b 7% in on ya. Thn my citical intst at is.07/.07 =.065. In th abov gaph, m has high intst at xpctations, h has low ons, and you is in th middl. Th makt-wid total spculativ dmand cuv on th ight is th hoizontal sum of th th on th lft. With many tads, th cuv will b smooth in appaanc. Of cous, all of this is static. W an t analyzing how intst at xpctations volv ov tim o what dtmins thm. This much should b obvious though:. Th a spculativ asons fo holding mony. 2..Th mony dmand cuv is intst lastic. 3. Whn popls intst xpctations is, th mony dmand cuv shifts up o to th ight, and this will caus LM to shift up/ight. 4. Actual ats dpnd stongly upon xpctations, and xpctations dpnd upon ats. Th intst lasticity of tansactions dmand: Baumol modl. Suppos that ov a ya-long intval you a going to spnd a total of T dollas in a stady stam. You hav th option of holding cash, which pays no intst o holding, say, a bond that tuns p dolla p ya. Evy tim you go to th bok to gt cash, you incu a tansaction cost of b dollas p visit. You could withdaw all you cash on Januay and sav a lot on tansactions costs by only having to go to th bok onc p ya, but thn you would hav to hold an avag of T/2 dollas duing th ya and you intst oppotunity cost would b high...it would b T/2 if w igno compounding. To sav intst oppotunity cost you could withdaw mony quit oftn, but thn you tansaction costs, b, would b high. How do you manag you cash withdawals? W can figu this out as follows: Lt C b th siz of ach cash withdawal and assum that all withdawals a of qual siz. It follows that you nd to mak T/C withdawals p ya, so you total tansaction cost will b bt/c. You avag mony holding will b C/2, so you annual intst oppotunity cost will b C/2. So you total cost of ady cash is bt C this total. Th tchniqu is to tak th divativ of C + 2. What you want to do is pick C so as to minimiz bt C C + 2 with spct to C, st th sult bt qual to zo, and solv fo C. Th valu of th divativ w sk is 2 + C 2 qual to zo and solving fo C w hav th optimal cash withdawal givn by. Stting this

14 C bt = 2. This is th famous squa oot ul. Sinc C/2 is in a sns on s avag mony dmand, this quation tlls us about th intst lasticity of mony dmand. Not that if incass, th optimal C dclins. Not also that if b o T incass, mony dmand iss. Sinc T will tnd to colat with a pson s incom, this sms to mak on bliv that, in ou mo common notation, Md = f(y, ).

15 Md(y0) Md(y) Ms LM 0 M yo y y Intst Elasticity of Mony Supply Evn if Md is intst inlastic, if Ms is intst lastic so is LM. I ll giv a quick gaph and w will xplo dtails in a minut. Th figu blow shows Md as compltly intst inlastic, with Ms as lastic. Not that this giv LM th usual shap. Why might Ms b intst lastic? Mayb whn incass popl switch fom dmand dposits to tim dposits, and this fs up svs, o mayb cuncy flows into th banking systm, and this givs banks svs and allows thm to incas loans and dposits. On way to xplo this is via th mony multipli. H s how:. M2 = C + D + T wh C is cuncy hld by th public, D is dmand dposits, and T is tim dposits 2. C = c M2 Wh th cuncy atio, c, is a positiv constant 3. T = t M2 Wh t, th tim dposit atio, is a positiv constant. W assum that - c - t > 0 4. H = R + C 5. R = RR + XR Wh H is high powd mony, with R bing bank vault cash o bank sv dposits at th Fd. H is ith svs o cash, which could b svs if th public dcidd to put th cash in th bank. Rsvs a ith quid svs, RR, o xcss svs, XR 6. RR = D Wh is th quid sv atio. Tim dposits don t hav quid svs. 7. XR = x D x is th banking systms xcss sv dmand atio. If you substitut into quation 4, you gt

16 M 2 = ( + x)( c t ) + c H. wh is th monymultipli o" mm." ( + x )( c t ) + c Th ida is that th Fd influncs H and, whil th banks influnc x, and th public dtmins c and t. Th sulting mony supply is nd sult of all of thi actions. To illustat: Opn makt puchass of bonds by th Fd will incas H, and this will incas M2, if vything ls mains constant. If th Fd incass, thn M2 should fall, sinc ( - c - t) > 0. If th banks want to hold mo xcss svs, i.., x incass, thn mm falls and so dos M2 with vything ls hld constant. Whn would x incas? Mayb whn intst ats a low, hnc th intst lasticity of Ms. If t incass, mm will is and so will M2. This might happn whn intst ats is hnc th intst lasticity of Ms. Excis: What happns whn c incass? All of ths factos point to th intst lasticity of Ms, and thfo a positiv slop fo LM. Qustion: How would w us IS-LM if th Fd followd a policy of adjusting Ms in whatv way ncssay to hold = 0?

17 Montay vs. Fiscal Policy On of th poblms with IS-LM as it is convntionally usd is that it allows on to ask and gt answs to nonsns qustions. Fo xampl, it is possibl to us IS-LM to answ a qustion lik, What happns if w hav an incas in g? O what happns if w hav an incas in M? (Ths may not sm lik cazy qustions, but thy a.) And IS-LM allows us to daw an atificially solid lin btwn montay and fiscal policy. Why a ths nonsns qustions? Think of [ g] as an vnt. W can ask (insid IS- LM), what happns following this vnt. But this sot of vnt can nv happn. Lik vy oth ntity, th govnmnt has a budgt constaint. Th constaint looks lik this: g = t + Pb B + M This says that th govnmnt s spnding must b financd somhow...ith by taxs, t, bond sals to th public that bing in Pb B, wh Pb is th pic of bonds and B is th quantity of bonds hld by th public, o by pinting mony, M. Givn this, w can ask about th ffcts of vnts lik: [ g, t] o [ g, Pb B] o [ g, M], but [ g] maks no sns. This is a big dal. I can t pdict what an incas in g will do unlss I know how it will b financd.. H s how all this can giv is to an atificial distinction btwn montay and fiscal policy. Consid a fw T accounts:. Th Fd buys bonds fom th public. Th vnt is [ M and B ]. This is montay policy. Sinc Dposits and Rsvs initially incas qually, th will b xcss svs, XR, which pompt banks to mak mo loans and xpand th mony supply. So w hav M ising and no chang in g...looks lik montay policy. tasuy fd banks public 8B 8R 8R 8D 9B 8D (XR > 0) 8D 8L 8L 8D 2. What if th tasuy incasd t and spnt th mony buying a 747 fom Boing (th public scto)? Th vnt is [ g and t]. H s what th accts would look lik. Th financing is abov th dashd lin and th spnding is blow. This looks lik fiscal policy, sinc w hav g and t changing, but no chang in M and not much commotion in financial tasuy fd banks public 8D at Fd 9tax civabl 9D at Fd 8B747 9R 8D of Tas 9D of T 8R 9R 8R 9D 8D 9D 9B747 8D 9tax liab. ts. mak 3. What if th tasuy incass g and sold bonds to th public to financ th puchas? Th vnt is [ g and B]. Th financing is abov th lin and th spnding is blow th lin. Not

18 that this involvs an incas in g and no chang in M, but th will b considabl montay commotion du to th bond sal. 4. What if th tasuy spnt mo on g by simply unning down its dposits at th Fd? Th t a s u y f d b a n k s p u b l i c 8 D a t F d 8 B 9 R 9 R 9 D 8 D o f t 8 B R 8 R 8 D 9 D a t F d 9 D o f t 8 B 9 D 8 D 9 B vnt is [ g and M]. Th T acct is just th itms blow th lin in th abov figu. This tim w gt an incas in g and an incas in M. Is this montay o fiscal policy? 5. Finally, what if th Fd bought a B747 o a nw comput systm? Is that montay of fiscal policy? As intst ats dop towad zo h and in Japan, popl woy about th ability of Montay policy aimd at ducing...on of th options is to hav th cntal bank buy things oth than vy shot T-bills...th futh on gos fom t-bills, th clos montay policy coms to bing fiscal policy. Do you s now why it isn t pop to ask about th ffct of an incas in g with no discussion of financing? t a s u y f d b a n k s p u b l i c 8 B R 8 R 8 D 9 B D ( X R > 0 ) 8 D 8 D 8 L 8 L

19 Th old tmplat was: fiscal chang g [ g] chang t [ t] Montay OMO Chang discount at Chang sv q...[ M] but I lik:

20 GDPal V

(, ) which is a positively sloping curve showing (Y,r) for which the money market is in equilibrium. The P = (1.4)

(, ) which is a positively sloping curve showing (Y,r) for which the money market is in equilibrium. The P = (1.4) ots lctu Th IS/LM modl fo an opn conomy is basd on a fixd pic lvl (vy sticky pics) and consists of a goods makt and a mony makt. Th goods makt is Y C+ I + G+ X εq (.) E SEK wh ε = is th al xchang at, E