IS-LM Stability Revisited: Samuelson was Right, Modigliani was Wrong

Transcription

1 Economía Vol. XXXVIII, N 75, semestre enero-juno 215, pp / ISSN IS-LM Stablty Revste: Samuelson was Rght, Moglan was Wrong Walo Menoza Bello * Pontfca Unversa Católca el Perú RESUMEN En el moelo IS-LM e Hcks, en el que se asume que la prouccón se etermna en el mercao e benes y la tasa e nterés en el mercao e nero, cuano la propensón margnal a gastar es mayor que uno, la IS tene penente postva. Moglan (1944), Varan (1977) y Sargent (1987), etermnaron que en este caso especal el moelo IS-LM es estable cuano la penente e la LM es mayor que la e la IS. En línea con Samuelson (1941), este artículo muestra que en este caso especal el moelo es estable cuano la penente e la IS es mayor que la e la LM. Sn embargo, en este caso estable, el moelo no tene un sgnfcao económco útl. Una solucón a este problema teórco es abanonar el mecansmo e ajuste keynesano y reemplazarlo por el mecansmo clásco, one la tasa e nterés se etermna en el mercao e benes y la prouccón en el mercao e nero. En este caso el moelo IS-LM es estable cuano la LM es más empnaa que la IS. Palabras clave: Hstora el pensamento económco, teoría macroeconómca, moelo IS-LM. Clasfcacón JEL: E11, B22. La establa e la IS-LM revstaa: Samuelson estaba en lo certo, Molgan equvocao ABSTRACT In Hcks s IS-LM moel, where t s assume that proucton s etermne n the goos market an the nterest rate s etermne n the money market, when the margnal propensty to spen s greater than one, the IS has a postve slope. Moglan (1944), Varan (1977) an Sargent (1987) etermne that n ths specal case the IS-LM moel s stable when the LM slope s greater than the IS. * Professor an researcher, Department of Economcs, Pontfca Unversa Católca el Perú (PUCP), 181 Unverstara Ave., Lma 32, Lma. Telephone: (213). E-mal aress: walo.menoza@pucp.pe I wsh to thank the comments by Juan Antono Morales an Oscar Dancourt an the two anonymous referees of the journal for ther valuable comments. I am solely responsble for any remanng errors.

2 124 Economía Vol. XXXVIII, N 75, 215 / ISSN In lne wth Samuelson (1941), ths artcle shows that n ths case the moel s stable when the IS slope s greater than the LM slope. However, n ths stable case the moel oes not have a useful economc meanng. One soluton to ths theoretcal problem s to abanon the Keynesan ajustment mechansm an replace t wth the Classcal mechansm where the nterest rate s etermne n the goos market an proucton s etermne n the money market. In ths case, the IS-LM moel s stable when the LM s steeper than the IS. Keywors: Economc thnkng, macroeconomc theory, IS-LM moel. JEL classfcaton: E11, B INTRODUCTION In the tratonal IS-LM moel evse by Hcks (1937), when the margnal propensty to spen s greater than one, the Keynesan multpler s negatve an the IS has a postve slope. Hcks (1937) warne that n ths specal case, only uner certan contons can the IS-LM moel be stable, an Keynes, n a letter publshe n Glboy (1939), eeme ths occurrence completely unstable. Later evelopments startng wth Moglan (1944) an nclung artcles n specalze journals, as well as sectons on macroeconomcs an mathematcs n economcs textbooks agree that, n ths atypcal case, the IS-LM moel s stable when the LM slopes more steeply than IS. Ths artcle shows that: ) The case the lterature pronounces as stable s n fact unstable, because the same stablty contons that apply to the stanar case are extene to the specal case. The moel s stable when the IS steeper than the LM. Ths result s consstent wth the vews expresse by Samuelson (1941). ) However, n ths specal stable case, the moel s lnear verson proves results wthout useful economc sgnfcance. Frstly, the equlbrum values of proucton an the nterest rate are negatve. Seconly, snce the Keynesan multpler s negatve, expanng eman results n shrnkng output. ) One soluton to ths theoretcal problem s to abanon the Keynesan ajustment mechansm an replace t wth the Classcal mechansm where the nterest rate s etermne n the goos market an proucton s etermne n the money market. In ths case the moel s stable when the LM s steeper than the IS. The followng secton revews the lterature on the specal case. Secton 3 scusses the stablty contons of the stanar IS-LM moel. Secton 4 repeats the exercse for the specal case an argues that the tratonal treatment s ncorrect. Secton 5 proposes a complementary argument to show the nstablty of the case the lterature eems stable, by smulatng the ynamc effects of an expansve monetary polcy. Secton 6 shows that

3 Walo Menoza Bello IS-LM Stablty Revste: Samuelson was Rght, Moglan was Wrong 125 n the lnear verson of the stable moel, the equlbrum values of the nterest rate an output are negatve, an that the moel forecasts a reucton n output as a consequence of expanng eman. Secton 7 shows that only n the case n whch the Keynesan ajustment mechansm s replace by the Classcal mechansm s the stablty conton postulate by Moglan true. Fnally, the conclusons are set out n Secton BACKGROUND In the IS-LM moel, when the margnal propensty to spen (propensty to consume plus propensty to nvest) s greater than one, the Keynesan multpler s negatve an the IS curve has a postve slope. In Hcks s presentaton (1937), equlbrum n a goos market requres ex ante equalty between savng (S ) an nvestment (I ) 1. Thus, IY (, ) = S( Y, ) The slope of the IS curve, graphe on the (Y, ) space, s gven by: Y IS SY I = S I Y 1 CY I = S I Y Where Y x = Y / x s the generc form representng the partal ervatve of Y wth respect to X. The enomnator s unoubtely postve, whle the numerator can be postve, n the typcal case where the propensty to spen s less than one (C Y + I Y < 1), or negatve, when the margnal propensty to spen s greater than one (C Y + I Y > 1). One of the nnovatons n Keynes s General Theory was the ntroucton of the concepts of margnal propensty to consume an the multpler. In a letter publshe n Glboy (1939), Keynes warne that f the margnal propensty to consume s greater than one, the moel woul be unstable: My theory tself oes not requre my so-calle psychologcal law as a premse. What the theory shows s that f the psychologcal law s not fulflle, then we have a conton of complete nstablty. If, when ncomes ncrease, expenture ncreases by more than the whole of the ncrease n ncome, there s no pont of equlbrum. (Glboy, 1939, p. 634). All the subsequent lterature ealng explctly wth ths specal case (negatve Keynesan multpler an postvely slopng IS) that frst appeare n specalze journals an was then aopte by macroeconomcs an mathematcs for ncluson n 1 Hcks s text has been aapte to current termnology.

4 126 Economía Vol. XXXVIII, N 75, 215 / ISSN economsts textbooks has establshe that for the moel to be stable, the LM curve must be steeper than the IS curve. Moglan (1944) was the frst to aress ths specal case n terms of the IS an LM slopes. For the case n whch IS slopes postvely, he states that: Stablty s also possble when the IS curve rses n the neghborhoo of the equlbrum ponts as long as t cuts the LL curve from ts concave towar ts convex se (Moglan 1944, p. 64). That s, for the moel to be stable n the neghborhoo of equlbrum, the LM slope must be greater than that of the IS. Ths argument appears n Fgure 1. The LM an IS ntercept each other at ponts A an B. For Moglan, the equlbrum pont woul be stable at B, where the IS cuts the LM from ts concave to ts convex se. At ths pont, the LM slope s greater than that of the IS. At pont A, equlbrum woul be unstable. Fgure 1 Later, Huson (1957) presente the IS-LM moel s stablty conton for the specal case an also erroneously consere the unstable case to be stable. Snce Huson s IS s non-lnear an snce t shows a negatvely slopng stretch an another wth a postve slope, Huson s IS-LM moel s one of multple equlbrum, lke the one shown n Fgure 2. In ths moel:

5 Walo Menoza Bello IS-LM Stablty Revste: Samuelson was Rght, Moglan was Wrong 127 Stablty of equlbrum then requres that the IS scheule slopes upwars less steeply than the LM scheule. Consequently, B s a poston of unstable equlbrum, whle C an C' are stable. (Huson, 1957, p. 382). Fgure 2 Accorng to Huson then, the atypcal moel s stable when the LM slope s greater than that of the IS. Later, Dernburg an Dernburg (1969), Smth (197), an Varan (1977) reache the same nepenent concluson wthout referrng to Moglan s (1944) or to Huson s work (1957). Varan (1977) buls, as Huson, a moel wth multple equlbrum an arrves at the same result: when the IS slopes postvely an there are multple equlbra, the stable regon s that where the LM s steeper than the IS. Fgure 3 whch reprouces Varan s Fgure 3 (1977, p. 268) shows, as Huson, that the regon s unstable (.e. confgures a sale pont) at the ntersecton ponts between the IS an LM, when the IS s steeper than the LM, as n pont B of Fgure 3. The ntersecton ponts are stable f the LM slope s greater than that of the IS (when the IS slopes negatvely, as n pont A, or when the IS slope s postve, but less than the LM, at pont C), as shown by the recton arrows of the phase agram.

6 128 Economía Vol. XXXVIII, N 75, 215 / ISSN Fgure 3 Subsequent publcatons n specalze journals ealng wth ths specal case leave these authors proclame stablty contons unchange, as shown n works by Slver (1971), Chang an Smyth (1972), Puckett (1973), Burrows (1974), Cebula (198), Wang (198), Patnkn (1987) an Ros (24). The specal case an ts stablty contons are also ncorporate nto the teachng of macroeconomcs n both ol (Smth, 197, p. 313; Dernburg an McDougall, 1976, pp ; Branson, 1972, pp ), an more recent textbooks (Sargent, 1987, p. 59; McCafferty, 199, p. 149). When referrng to the specal case, Sargent states that: Ths conton s automatcally satsfe when the LM curve s upwar slopng an the IS curve s ownwar slopng. It can stll be satsfe f the IS curve s upwar slopng, prove that the LM curve s more steeply slope. (Sargent, 1987, p. 59). The same argument appears n text books on mathematcs for economcs, nclung Ferguson an Lm (1998, p. 129), Shone (22, p. 449), an Zhang (25, p. 243). In short, all the lterature startng wth Moglan (1944) states that n the case n whch the IS has a postve slope, for the IS-LM to be stable, the LM curve must be steeper than the IS curve. The followng sectons argue that the IS-LM moel s stable n the specal case where the IS slope s greater than the LM slope.

7 Walo Menoza Bello IS-LM Stablty Revste: Samuelson was Rght, Moglan was Wrong IS-LM STABILITY: THE STANDARD CASE Ths secton presents Hcks s moel (1937), n ts ynamc verson 2. In the goos market, t s assume that ajustment s by quanttes an that output ncreases ( Y > ) when there s an excess of eman n that market (EDB). In the money market, the ajustment varable s the nterest rate, whch rses ( > ) when there s excess eman (EDM ). Y = ε( EDB); ε > (1) = η( EDM ); η > (2) Where e an η are, respectvely, the proucton ajustment veloctes an the nterest rate vs-à-vs excess eman n the goos market an the money market. If Y represents output an Y stans for the eman for goos, n a close economy wthout government, the eman for goos comes from consumpton an prvate nvestment. Consumpton an nvestment are postve functons of ncome an negatve functons of the nterest rate. Consequently, eman for goos s gven by: Y + + = C(, Y) + IY (, ); < C Y < 1 (3) Excess eman n the goos market therefore equals: EDB = Y - Y = EDB(Y, ); EDB Y = C Y + I Y - 1 < ; EDB = C + I < (4) In the money market, f M s s the money supply an P s the prce level, the real money supply equals: s s M m = (5) P If m s the real eman for money, whch s rectly relate to output an nversely relate to the nterest rate: + m = m ( Y, ) (6) Consequently, the excess eman n a money market equals: + s EDM = m m = EDM ( Y, ); EDM = m > ; EDM = m <. (7) Y Y 2 See Ganolfo (1996), Ferguson an Lm (1998) an Zhang (25).

8 13 Economía Vol. XXXVIII, N 75, 215 / ISSN Replacng equatons (7) an (4) n equatons (1) an (2), respectvely, we obtan a fferental equaton system to scuss the stablty contons of the tratonal IS-LM moel. Y = EDB Y ε (, ); ε > (8) = + EDM Y η (, ); η > (9) Where e[.] an η[.] are rsng monotone functons whch are fferentable an satsfy the e[] = η[] = conton. To evaluate whether a moel s stable or not, t s worth takng nto account Ganolfo s warnng (1996) about the approprate formulaton of ajustment ynamcs n markets: The ynamc formalzaton of the Walrasan assumpton s the followng p = f D ( p ) S ( p ) [ ] Where sgn f [...] = sgn[...], f [] =, f ' () > The notaton sgn f [...] = sgn[...] means that f s a sgn-preservng functon,.e., the epenent varable has the same sgn as the nepenent varable (whch n ths case s excess eman): therefore, f excess eman s postve (negatve) the tme ervatve of p s postve (negatve),.e. p s ncreasng (ecreasng). (Ganolfo, 1996, p.172). Whch, when apple to the IS, mples: Y = ϕ I( Y, ) SY (,), sgnϕ 1 [...] = sgn[...] (Ganolfo, 1996, p. 328) 3. 1 [ ] Takng the case shown by Ganolfo as an example, snce ϕ 1 >, the nfluence of the nvestment-savng (I - S ) gap on proucton, whch equals the effect of the excess eman, also must be postve. In our presentaton, snce e >, the nfluence of the hgh excess eman on the prouct must be postve. The system mae up by equatons (8) an (9) s non-lnear an ts stuy mples analytcal ffcultes, whch we wll avo. To ths en, we wll scuss some val propertes n a local equlbrum context, through Taylor s expanson. 3 We have aapte Ganolfo s termnology to ours.

9 Walo Menoza Bello IS-LM Stablty Revste: Samuelson was Rght, Moglan was Wrong 131 If Y e an e represent statonary (local) equlbrum values of proucton an the nterest rate n the IS-LM moel, equatons (8) an (9) can be presente n a matrx as follows: Y e EDBY EDB Y Y = ε ε e EDMY EDM η η Or ts equvalent, e Y CY IY 1 C I Y Y e = ε( + ) ε( + ) my m η η (I) Or n ts compact form: e Υ = ΩΥ Where: ε( CY + IY 1) ε( C + I) Ω= ηmy ηm The necessary an suffcent contons for ths system to be asymptotcally stable, that s, a system where all the movements flow cyclcally or non-cyclcally towars a statonary equlbrum pont, are that the Ω Jacoban matrx trace be negatve an ts etermnant postve. ) TrΩ < ) DetΩ > In ths verson of the IS-LM, the assumpton that the propensty to spen s less than one (C Y + I Y < 1) ensures that the two stablty contons are met wthout restrctons. ) TrΩ = e(c Y + I Y - 1) + ηm < ; ) DetΩ = -eη[(1 - C Y - I Y )m + m y (C + I )] > Stablty can also be scusse from the qualtatve pont of vew, by relatng the escrbe contons to the graphc representaton of the IS-LM moel. For ths purpose, we use a phase agram to represent the goos market an the money market n statonary equlbrum, that s, when output an the nterest rate reach equlbrum an stay constant ( Y = = ).

10 132 Economía Vol. XXXVIII, N 75, 215 / ISSN Statonary equlbrum n the goos market s acheve when output equals eman (excess eman s null), an output remans stable ( Y = Y Y = EDB = ). From equaton (8), the IS equlbrum equaton s obtane n the goos market when Y =. Y = Y = C(, Y ) + I(, Y ) (1) In the (Y, ) space, the IS curve has the usual negatve slope, because the propensty to spen s assume to be less than one, resultng n a postve Keynesan multpler. 1 = < Y kc ( + I ) IS 1 Where k = > s the Keynesan multpler. 1 C I Y Y Let us assume that the nterest rate sles from any pont on the IS n Fgure 4, where output equals eman. At ths pont, let us say pont M uner the IS curve, a lower nterest rate ncreases consumpton an nvestment, whch generates excess eman n the goos market. The excess eman n ths moel, whose ynamcs are expresse n equaton (8), ncreases output. Ths s the meanng of the arrows pontng to the rght from pont M. The same reasonng apples to a pont such as N, where excess supply reuces output, as shown by the left-pontng arrow that starts at that pont. Fgure 4

11 Walo Menoza Bello IS-LM Stablty Revste: Samuelson was Rght, Moglan was Wrong 133 The mechansm allowng for an excess eman to ncrease proucton s seen more clearly n the classc 45º agram such as the one shown n Fgure 5. The 45º curve shows the equlbrum ponts n the goos market, when proucton (Y ) equals eman (Y ). Curve D s a lnear representaton of equaton (1) wth a postve nepenent component an a slope that s postve, but smaller than one. Ths reflects a propensty to spen that s lower than one. Curve D has the nterest rate as a parameter. Fgure 5 shows that f the nterest rate falls from o to 1, curve D shfts upwars to D 1. Excess eman appears at the ntal output level Y. Such excess eman ncreases output. The equlbrum pont moves from A to B, an proucton rases to Y 1. Fgure 5 Statonary equlbrum s reache n the money market when supply equals the eman for money,.e. excess eman s null leang to a stable nterest rate ( m s = m = EDM = ). We reach the equlbrum equaton n the money market, the LM, when = n equaton (9). s M + = m (, Y ) (11) P The LM curve slopes postvely. Y LM my = > m

12 134 Economía Vol. XXXVIII, N 75, 215 / ISSN Let us now assume that output ncreases startng from any pont of the LM n Fgure 6, where money eman equals supply. Larger output takes us to a pont such as M, to the rght of the LM where the eman for money s also greater, creatng excess eman n that market. If we know the ynamcs of the money market represente by equaton (9), excess eman wll shft the nterest rate upwars. Ths explans the upwar-pontng arrows startng from pont M. Symmetrcally, to the left of the LM at a pont such as N, there s an excess money supply, hence the nterest rate has to fall, as shown by the ownwars-pontng arrow startng at that pont. Fgure 6 In a general equlbrum conton we can see the stanar IS-LM moel s stable, an rectonal arrows show that we are n the presence of an asymptotcally stable equlbrum. At pont (Y, ) the moel reaches statonary equlbrum.

13 Walo Menoza Bello IS-LM Stablty Revste: Samuelson was Rght, Moglan was Wrong 135 Fgure 7 Fnally, snce the secon conton for stablty, that corresponng to the postve etermnant, s equvalent to my 1 > m k( C + I ) the stanar IS-LM moel s stable when the LM curve slopes more steeply than the IS. Startng wth Moglan (1944), the lterature has extene to the specal case those stablty contons that apply to the stanar case. Below, an alternatve approach s propose whch elvers the opposte result. 4. IS-LM STABILITY: THE SPECIAL CASE When the propensty to spen s greater than one (C Y + I Y > 1), the Keynesan multpler s negatve (k < ) an the IS has postve slope. 1 = > Y kc ( + I ) IS If the IS an the LM have postve slopes, we have to etermne whch slope s greater n orer to reach stablty.

14 136 Economía Vol. XXXVIII, N 75, 215 / ISSN Our revew of the lterature foun that stablty contons n the specal case are reache base on the same equaton system (8) an (9). As a result, the same necessary an suffcent contons for stablty are acheve as n the stanar case. So, even f C Y + I Y > 1, the requrement s the same as n the tratonal moel: the LM must slope more steeply than the IS: my 1 > m k( C + I ). Ths proceure assumes that the ynamcs of the goos market, as represente n equaton (8), stll hols for the specal case 4. Y = EDB Y ε (, ); ε > (8) What oes equaton (8) tell us? That excess eman n the goos market ncreases output. Therefore, parameter e s postve. If the IS has a postve slope, that s, when C Y + I Y > 1, s t stll true that excess eman n the goos market wll ncrease output? In the tratonal case, excess eman n the goos market sappears when output rses, as represente n equaton (8). Let us assume a fall n the nterest rate. A fallng nterest rate ncreases nvestment an consumpton, an generates excess eman n the goos market. Excess eman ncreases output, whch, n turn, through ts effect on prvate expenture, ncreases eman agan. Snce eman ncreases less than output because the margnal propensty to spen s less than one the excess eman shrnks. In the movement towars a statonary equlbrum, the excess eman contnues to fall untl reachng zero, thus reestablshng equlbrum n the goos market. For a postvely slope IS, the ynamcs of ajustment n the goos market s fferent. We argue that the fferental equaton reflectng the ajustment ynamcs n ths market must be reformulate to apply the usual stablty contons. Omsson of ths aspect has le to the ncorrect treatment of stablty contons n the atypcal IS-LM moel. Let us assume, as prevously, that the nterest rate falls, nvestment an consumpton rse, an excess eman s create n the goos market. What ajustment mechansm woul be requre to restore equlbrum? What must happen to output for such excess eman to be cancele out? 4 Beses, f conton ) hols n the specal case, atonal restrctons for stablty are neee because the sgn for TrΩ s unetermne. In our revew of the lterature we foun no sgns of ths ssue havng been aresse.

15 Walo Menoza Bello IS-LM Stablty Revste: Samuelson was Rght, Moglan was Wrong 137 If output ncreases as a response to excess eman, gven that the propensty to spen s greater than one, the eman woul ncrease more than proucton; thus, the excess eman n the goos market, rather than reucng, woul rse. The excess eman woul contnue to grow nefntely, preventng the system from reachng a steay state. Ths ynamc s clearly explosve. For the process to be convergent output must contract rather than ncrease. As output shrnks, eman for goos falls more than output because the propensty to spen s greater than one, thus reucng excess eman. That s, n ths specal case, excess eman n the goos market reuces output. Or, lkewse, f the Keynesan multpler s negatve, an ncrease n eman reuces, an oes not ncrease, output. We propose an alternatve ajustment ynamcs equaton n the goos market by reproucng the mechansm we have just escrbe. In the fferental equaton for the goos market, output ncreases shoul be relate to excess supply n the market (EOB = Y - Y ), not to excess eman. The equaton that aequately reflects the new ajustment ynamcs of the goos market can be wrtten as: Y = ε( EOB); ε > (12) Excess supply n the goos market s efne by: + EOB= Y Y = EOB( Y, ); EOB = 1 C I < ; EOB = ( C + I ) > (13) Y Y Y Thus, the new ajustment ynamcs n the goos market, whch replaces equaton (8), s now wrtten as: Y = + EOB Y ε (, ); ε > ; (14) Ths formulaton s consstent wth Ganolfo s mathematcal requrement (1996, p. 328) note above. Snce there s no change n the money market, the new fferental equaton system n the atypcal case s comprse of: Y = + EOB Y ε (, ); ε > ; (14) = + EDM Y η (, ); η > (9)

16 138 Economía Vol. XXXVIII, N 75, 215 / ISSN As before, lnearzng the equaton system (14) an (9) usng Taylor s expanson yels the followng: e Y 1 CY IY C I Y Y e = ε( ) ε( + ) my m η η (II) Or n ts compact form: Υ = Ω Υ 1 Where: e ε( 1 CY IY) ε( C + I) Ω 1 = ηmy ηm The necessary an suffcent contons for ths fferental equaton system to be stable are, as before, for the matrx Ω 1 trace to be negatve an ts etermnant postve. v) TrΩ 1 = ε( 1 C I ) + ηm < ; Y Y v) DetΩ 1 = εη ( 1 CY I ) Y m + m ( Y C + I ) > The frst conton s met wth no restrctons. To meet the secon conton, we must have: 1 m > kc ( + I ) m Y, Otherwse put, when the propensty to spen s greater than one an the Keynesan multpler s negatve, for IS-LM to be stable, the IS must be steeper than the LM: Y IS > Y LM We reach the same concluson usng the phase agram system. Snce no change has occurre n the money market, let us focus our attenton on the postvely slopng IS. Let us assume a fall n nterest rates startng at some pont of the IS n Fgure 8, where output equals an eman. At pont M, uner the IS curve, the lower nterest rate ncreases nvestment an consumpton, creatng excess eman n the goos market. In the framework of ths moel, such excess eman n the goos market - the ynamcs of whch are expresse n equaton (14) - contracts, an oes not ncrease, output, as s erroneously assume. Ths explans the arrow to the left that starts from pont M.

17 Walo Menoza Bello IS-LM Stablty Revste: Samuelson was Rght, Moglan was Wrong 139 By the same reasonng, at a pont such as N there s excess supply that ncreases output, as symbolze by the rght-pontng arrow n Fgure 8. Fgure 8 Fgure 9 replcates the 45º agram for the atypcal case. As before, curve D s the lnear representaton of equaton (1). The fference s now that ts slope s greater than one because the propensty to spen s greater than one. In these contons, equlbrum can only occur n the quarant where the value of output s negatve. We wll return to ths pont n secton 5 below. Let us smulate, as before, the nterest rate falls from to 1. In Fgure 9, f the nterest rate falls, curve D moves upwars to D 1. At the ntal proucton level Y, there s an excess eman n the goos market. Ths excess eman reuces output; t oes not ncrease t. The equlbrum pont shfts from A to B, an output falls toy 1.

18 14 Economía Vol. XXXVIII, N 75, 215 / ISSN Fgure 9 In general equlbrum, Fgure 1 combnes fgures 1 an 2 n a sngle agram, showng that the IS-LM moel s stable when the IS s steeper than the LM 5. Fgure 1 5 In secton 5 we wll show that n the specal case of a lnear IS-LM moel, the equlbrum values for output an the nterest rate are negatve.

19 Walo Menoza Bello IS-LM Stablty Revste: Samuelson was Rght, Moglan was Wrong 141 In Fgure 11 the phase agram shows that when the LM s steeper than the IS, the moel s unstable, of the sale pont type. Fgure STABILITY, INSTABILITY AND EXPANSIVE MONETARY POLICY An atonal argument n scussng nstablty contons s prove by changng an exogenous varable an evaluatng how the enogenous varables wll ajust. In stable moels, after changng an exogenous varable, these varables temporarly epart from ther ntal equlbrum values, but then change untl they reach a new statonary equlbrum. When moels are unstable, enogenous varables move away from the ntal equlbrum an never reach a new statonary equlbrum. In Dernburg an McDougall (1976) the effects of an expansve monetary polcy are llustrate wthn the context of a postvely slope IS. Ther presentaton whch assumes stablty when the LM s steeper than the IS s useful to prove evence of the error of conserng that, n ths moel, excess eman n the goos market leas to ncrease output. Dernburg an McDougall (1976) repeat the exercse wth a agram such as that shown n Fgure 12, assumng that the money market s always n equlbrum an the goos market may be temporarly n sequlbrum.

20 142 Economía Vol. XXXVIII, N 75, 215 / ISSN In Fgure 12, an expansve monetary polcy (an ncrease n M s ), shfts the LM to the rght. Accorng to Dernburg an McDougall: The shft n the LM curve causes the nterest rate to fall mmeately to 1. Ths agan means that ntene nvestment excees savng an that ncome must therefore rse. But the rse n ncome stmulates further nvestment because of our assumpton that nvestment s a functon of the level of profts an ncome. Consequently, the orgnal monetary sturbance causes ncome to rse; ths cause atonal nvestment to be nuce, an ths, n turn, causes ncome to rse stll further. Wll ncome contnue to rse nefntely, or wll a new equlbrum pont be foun? In the present case the rse n ncome causes the nterest rate to rse an to ampen nvestment more raply than the rse n ncome stmulates further nvestment. In other wors the rate of nterest that keeps the money market n equlbrum rses more raply than the rate of nterest that keeps the prouct market n equlbrum. Consequently, a new stable equlbrum pont wll be reache at Y 1 an 1. The path of ajustment agan follows the arrows upwar along the LM curve (Dernburg an McDougall 1976, p. 244) Fgure 12 shows the above mentone authors graph an argument. The re arrows hghlght ther propose ajustment ynamcs. Fgure 12 What s the mstake n ths reasonng? Postng that when the nterest rate falls to pont B, the resultng excess eman ncreases output. In ths atypcal IS-LM moel excess eman reuces output, rather than ncreasng t. Therefore, snce at pont B

21 Walo Menoza Bello IS-LM Stablty Revste: Samuelson was Rght, Moglan was Wrong 143 there s excess eman n the goos market, output shoul fall an the blue arrows shoul pont to the left, not to the rght. Therefore, the moel s unstable an the system oes not reach a statonary equlbrum. Fgure 13 shows the stable case, when the IS s steeper than the LM. As before, the expansve monetary polcy shfts the LM to the rght, an short term equlbrum s reache at pont B, wth a lower nterest rate an the same proucton level. As n B the lower nterest rate has create excess eman n the goos market, output must fall to restore equlbrum n that market. As output falls, so too oes the eman for money, an the nterest rate slps as well. Downwar ajustment contnues (followng the arrows) along the new LM curve untl t reaches (Y 1, 1 ). At ths pont, a new statonary equlbrum s reache. Fgure 13 The fact that an expansve monetary polcy shoul reuce nstea of ncreasng the nterest rate, as n the unstable case shown Fgure 11, s consstent wth Samuelson s observaton seven ecaes ago. By smulatng the effects of an expansve monetary polcy n fferent scenaros, nclung a propensty to spen of greater than one, he establshe that:..the only theorem whch remans true uner all crcumstances s that an ncrease n the amount of money must lower nterest rates f equlbrum s stable. (Samuelson, 1941, p. 12). If the LM were steeper than the IS, as n Fgure 12 - whch reprouces Dernburg s an MacDougall s arguments (1976) - an expansve monetary polcy woul ncrease the nterest rate.

22 144 Economía Vol. XXXVIII, N 75, 215 / ISSN STABILITY, NEGATIVE KEYNESIAN MULTIPLIER AND THE RELEVANCE OF THE ATYPICAL CASE We have argue that n the atypcal case, IS-LM moel s stable when the IS s steeper than the LM. In ths secton we wll show that n ths specal stable case, the moel yels results of no useful economc sgnfcance. Frstly, n the moel s lnear verson, output an nterest rate equlbrum values are negatve. Seconly, snce the Keynesan multpler s negatve, expanng eman results n fallng output. We have aopte the lnear verson of the IS-LM moel to llustrate how to obtan analytcal equlbrum values for output an the nterest rate. Equlbrum n the goos an money markets s gven by: Y = Y = C + a Y - a 1 + I + a 2 Y - a 3 (15) M s - P = m s = m = b Y - b 1 (16) Where C + I = A o o s autonomous prvate expenture, M s s the nomnal money supply, P s the prce level, an all the parameters are postve. The IS an LM equatons are rawn from these formulae, respectvely: Ao 1 = a + a ka ( + a ) Y s M P bo = + b b Y 1 1 (17) (18) where k = 1 1 a a < 2 s the Keynesan multpler. Equlbrum n the goos market can also be expresse n such a way that the consequences of havng a negatve Keynesan multpler wll be more clearly apparent. In ths case, as shown n equaton (19), greater eman prompte by rsng autonomous expenture or a slng nterest rate leas to lower output. Y = k[a o - (a 1 + a 3 )] (19) The IS an LM slopes are gven by: 1 bo = > ; = > Y ka ( + a ) Y b IS 1 3 LM 1

23 Walo Menoza Bello IS-LM Stablty Revste: Samuelson was Rght, Moglan was Wrong 145 To meet the secon stablty 6 conton, the IS must be steeper than the LM: 1 b > ka ( + a ) b That s, 1 3 b 1 + k(a 1 + a 3 )b > 1 Furthermore, the statonary equlbrum values for output an the nterest rate are obtane from equatons (17) an (18). Y e e kb1 b k a a b A ka ( 1+ a3) b k a a b M s = P ( ) (2) ( ) ( ) bk b k a a b A 1 b k a a b M s = P ( ) (21) ( ) ( ) Gven the stablty conton b 1 + k(a 1 + a 3 )b >, the statonary equlbrum values of output an the nterest rate are evently negatve, whch makes no economc sense. Fgure 14, where equatons (17) an (18) are represente wth the IS slope larger than the LM slope, reprouces ths result. Statonary equlbrum s reache n the quarant where nterest rate an prouct values are negatve. Fgure 14 6 The trace conton s met wthout restrcton.

24 146 Economía Vol. XXXVIII, N 75, 215 / ISSN Ths IS-LM moel confguraton leas us to a stuaton where exogenous shocks prouce analytcally strange outcomes. For example, n ths moel, an autonomous expanson of consumpton or nvestment or, f we ntrouce government to the moel, expanng government expentures woul lea to fallng output an nterest rates. Ths s because greater expenture generates excess eman n the goos market, thus reucng output. Fallng output reuces eman for money n the money market, thus brngng the nterest rate own. In Fgure 15, n vew of a larger autonomous expenture, the IS curve shfts upwars. In the new pont of ntersecton wth curve LM, whch has not shfte, both output an the nterest rate are smaller. Fgure A SOLUTION TO THE IMPASSE: THE CLASSICAL MECHANISM Although n strctly mathematcal terms our results are correct, n theoretcal an emprcal terms the noton that frms reuce proucton to meet a rse n eman for ther goos, or that the equlbrum values of output an the nterest rate are negatve, rener the results absur. In formal terms, the only way to justfy the poston of Moglan an later economsts woul be to abanon the Keynesan moel, whch assumes that the ajustment n the

25 Walo Menoza Bello IS-LM Stablty Revste: Samuelson was Rght, Moglan was Wrong 147 goos market occurs through changes n proucton an the ajustment n the money market through changes n the nterest rate. If, as n the Classcal case where real money eman oes not epen on the rate of nterest proucton s etermne n the money market an the nterest rate n the goos market, t can be shown, wth the same formal arguments use, that the stable case s that where the LM s steeper than the IS. Ths efnton of Classcal moel correspons to Hcks (1937). If we aopt the Classcal ajustment mechansm, the ynamcs of the etermnaton of the nterest rate an output woul be gven by: r = g + EDB Y g (, ); > ; (22) Y = h + EOM Y h (, ); > (23) Equaton (22) ncates that the nterest rate rses when there s excess eman n the goos market an equaton (24) shows us that proucton rses when there s excess supply n the money market. Lnearzng the equaton system (22) an (23) yels the followng: e r gc I g 1 CY IY e = ( + ) ( ) hm hm Y Y Y Y Or n ts compact form: Υ = Ω Υ 2 Where: e gc ( + I) g( 1 CY IY) Ω 2 = hm hm Y The necessary an suffcent contons for ths fferental equaton system to be stable are: ) TrΩ 2 = gc ( + I ) hm < ; Y ) DetΩ 2 = gh ( C + I ) my + m ( 1 CY I ) Y >

26 148 Economía Vol. XXXVIII, N 75, 215 / ISSN The frst conton s met wthout problem. To meet the secon conton, we must have, 1 m < kc ( + I ) m Y, that s, when the propensty to spen s greater than one but the ajustment mechansm s Classcal, for IS-LM to be stable, the LM must be steeper than the IS: Y LM > Y IS However, none of the stues revewe eparte from the Keynesan ajustment mechansm when scussng the stabltes of the IS-LM moel n the specal case. 8. CONCLUSIONS Ths paper argues that n the specal case where IS s postvely slope, for the IS-LM moel to be stable, the IS must be steeper than the LM. Ths challenges the lterature that, startng wth Moglan (1944), hols that stablty s reache when the LM s steeper than the IS. Only Samuelson (1941), by smulatng the effects of an expansve monetary polcy n fferent scenaros, nclung the case of a propensty to spen of greater than one, proposes a result that s consstent wth ours. However, n ths specal stable case, the moel yels results of no useful economc sgnfcance. Frstly, n the moel s lnear verson, the equlbrum values for output an the nterest rate are negatve. Seconly, snce the Keynesan multpler s negatve, expanng eman results n fallng output. In ths paper one soluton to the problem entfe was rase. That soluton s to abanon the Keynesan ajustment mechansm an replace t wth the Classcal mechansm where the nterest rate s etermne n the goos market an proucton n the money market. In ths case, the IS-LM moel s stable when the LM s steeper than the IS.

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