# An inquiry into the multiplier process in IS-LM model

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1 An inquiry into te multiplier process in IS-LM model Autor: Li ziran Address: Li ziran, Room 409, Building 38#, Peing University, Beijing 00.87,PRC. Pone: (86) Internet Address: Abstract Te multiplier teory is still an important analytical tool in many macroeconomic textboos. For example, a number of textboo autors use te teory to explain te process of growt in goods maret by expanding te multiplier process into a geometric series, and tus obtain te route of economic growt. Ten some students raise an interesting question: Can we induce te dynamics of te monetary and fiscal transmission mecanism of IS-LM model troug te multiplier process? (None of te textboos involve tis problem; alternative solutions are available in economics journals, but go beyond te scope of our students nowledge.) If not, wat is te problem in te analysis of te multiplier process? Here I first sow some professor s deduction of te "monetary transmission mecanism", and ten analyze te main problems and discuss te multiplier teory. Finally, I propose a generalization of te monetary transmission mecanism approac. I trace te cange of demand and output in te process of increase respectively, and use a stocastic series of variables to reflect regularity in te teir relationsip, and obtain anoter two curves in IS-LM model representing teir relationsip In conclusion, I demonstrate teoretically tat te economy will ultimately reac its equilibrium point, following te route of LM curve between one static equilibrium point to anoter. Deduction

2 For te sae of numerical calculation, we analyze te multiplier process in a tree-department economy. We ave two important curves in IS-LM model representing te equilibrium contrail of te goods maret and money maret respectively: IS curve: C 0 + ctr 0 + I 0 + G () Y= - c ( - t) LM curve: 0 - br = A 0 - br - c ( - t) (A 0 = C 0 + ctr 0 +I 0 +G 0 ) () M (2) =Y-r, (2) P Te intersection of te two curves determines te equilibrium output of our economy: (3) Y 0 = A 0 - c ( + bm P - Were Y is output or national income; C 0 is autonomous consumption; I represents investment spending r is te interest rate and b measures interest response of investment, I 0 is autonomous investment spending; G is government purcase of goods and services; TR 0 is transfer to te private sector; t is tax rate; c measures marginal propensity to consume out of disposable income; is te interest elasticity of money demand; is te output elasticity of money supply. Some professors old te following deduction of te monetary transmission mecanism in IS-LM model: [I] Initiative effects: ()Te government increases real money stoc by M/p. (2)Good marets adjust relatively slow and Y will not cange in a sort period, wile asset maret adjusts quicly and tus interest rates will be down by M/(p) (M/p=Y-r, M/p=Y- r.oterwise te equation M/p=Y-r will not old after an increase on its left side),. (3) 2

3 (3)Lower interest rate will stimulate te investment demand and consequently increase te aggregate demand. And, te output and national income begin to rise. Te above stages can be demonstrated by te following grap: M/p R= M/P I I=b M/P = AD AD AD= Y Y [II] Induced effects: Round R= Y/ Y R A I D I Y Notes: M/p=Y-r, M/p will remain stable after te monetary policy is applied, and terefore, in order to old te equation, r will rise as a result of te increase in Y. And tis increase can be denoted as Y i Y d = (-t) Y I= - Y/ Y d = (-t) Y Y d C AD Y (Tis increase in Y is induced by increased consumption demand, and tus can be denoted as Y c ) In round, te combination of effects induced by te initiative increase in Y is Y c +Y i, we may denote Y c +Y i by Y, ten Y =[c(-t)-/] Y.Similarly, In round 2, te combination of effects induced by te cange of Y in round (denoted by Y ) is [c(-t)-/]y,or to be more exact, [c(-t)- /] 2 Y(denoted by Y 2 ).And in round 3, [c(-t)-/] 3 Y.. C=c If c(-t) -/ <, te successive terms in te series become progressively smaller, we can write out te successive rounds of increased output, starting wit te initial increase in output, we may obtain a geometric series: Y, [c(-t)-/] Y, [c(-t)-/] 2 Y.. Moreover, te total cange can be obtained by adding tem up: Y d Y Y I = - AD= C Y/ Y i = Y {+[c (-t)-/] +[c(-t)-/] 2 + }= Y/[-c(-t)+/]= Problems: b M p c( 3

4 . Tis deduction sees to use te multiplier teory to explain te mecanism. It implies tat an increase in demand will immediately induce a same amount of increase in Y. If te economy is in recession ow can te output gain suc a sarp increase? 2. Te linear discrete dynamical systems can beave in some surprising ways. Wen c(-t) -/ <,a sequence obtained from a linear discrete dynamical system bounces around te equilibrium point and te bounces get smaller so tat te sequence approaces te equilibrium point; but wen c(-t) - / >, it bounces around te equilibrium point but te bounces get larger so tat te sequence does not approac te equilibrium point. If c(-t) -/ > te initial effect is b M/(P), wile te predicted equilibrium output is C b( M + M) + ctr0 + I p, and te predicted final total c( 0 G b M b M p increase of outcome is c(.obviously, p c( < b M/(P). Tat means after te initial increase output exceeds tat of te equilibrium. We can also figure out tat in te second period, te output is below tat of te equilibrium, and in te tird period, te output exceeds tat of te equilibrium again. Just lie a cobweb grap. If te economy is in recession and output is below tat of te sufficient employment, ow can te output gain suc a sarp increase immediately after te increase in money stoc? A number of macroeconomic text-boo autors examine te government purcase multiplier as follows : Suppose te government increases purcase of goods by G, te multiplier process G/(- c), were c is te marginal propensity to consume, can be expanded into G + c G + c 2 G + + for 0< c <, and tus we obtain te route of economic growt. Neverteless, tis applies only for an economy wit a sufficiently large excess capacity to produce, for instance, an economy wit 4

5 only one customer and dozens of barbers sitting around. If we tin about output being say Boeing 747s, te analysis is more complex, and te geometric series will fail, simply because it taes time to build a plane. Te multiplier teory as its strengt, but its assumptions are very strict: () sufficient large excess capacity to produce (2) output sould exactly meet te surplus demand in eac round. (3) One maret. Instructors usually teac te teory (or tey temselves accept it literally) as axioms and leave out tese assumptions. Tus, it is reluctant for tem to use te teory to explain te case were tese assumptions are sligtly violated. Ten ow can we analyze te multiplier process? Te Generalization of te Multiplier Process Approac Wen we explain te multiplier process, te geometric series only give us a logical deduction.in fact; te number value of te increase of eac round is indefinite. We can deduce te multiplier process in anoter way. Here I trace te cange of demand and output in te process of increase respectively: Te government increases purcases of goods by G, wic increases te aggregate demand in our economy by G. In te first round, we suppose te goods maret does not adjust quicly enoug to meet te excessive demand, and income (or output) only increases by Y (<< G). Ten consumption demand will increase by c Y. Since demand still exceeds output, output will continue to increase. Ten we obtain an increase of Y 2 in te second round, and a relevant increase of c Y 2 in consumption demand. Tis cycle repeats until output meet demand. (Note tat in eac round output increases more tan demand: Y i >c Y i, and tus output will ultimately meet demand.)tus we can + obtain te equation on equilibrium point: + Yi = G+ c Y +c Y 2 +c Y 3 + = G+ c Yi, i= i= + and derive te number value of te total income increase: Yi = G /(-c). i= 5

6 Te above analysis sows an alternative metod of explaining te multiplier process, and te result is te same as we use te traditional metod of geometric series. However, tere is great difference if we analyze complicated maret. Now let us loo at te multiplier process in IS-LM model: Application in IS-LM Model If we expand te monetary multiplier process G c( (were G is te increased government purcase of goods) into a geometric series and specify a definite cange of eac round: G, [c(-t)-/] G, [c(-t)-/] 2 G.,we cannot explain te case of c(-t)-/ (-, -). Wen we apply te multiplier teory to explain IS-LM model, wic is a model involving bot te money maret and goods maret, result is unsatisfactory. Tis is partly because te multiplier G incorporates te process of cange on two separate marets into a single course, in c( wic individuals beave on money maret and goods maret separately. Tis time te multiplier process fails again. In fact, te aggregate output and aggregate demand of te nation, not te multiplier process, determine te equilibrium point in an economy. If AD>AS, output will increase; if AD<AS, output will decrease; if AD=AS, te economy reaces its equilibrium point. Many macroeconomics textboo autors do not empasize te distinction between demand and output in IS-LM model, and literally tae te equations as axioms. In fact, IS curve represents combination of interest rates and income at wic te goods maret clears. Tus te mat symbols Y, G, I, C etc. represent bot te demand and output, since tey are equal on te equilibrium point. Wen we trace te process of growt, wic is an out-of -equilibrium course, we sould consider te two concepts separately. Now we use te generalization of te multiplier process approac to deduce te dynamics of te government purcase multiplier: 6

7 First, te government purcase of goods increases AD by G, and te output will increase as a result of te stimulation of excessive demand. Tis time we abandon te ambition to predict te exact value of increase in eac round. Let Y denote an uncertain increase in output in te first round, tus te cange in income is equal to Y. Ten te income increase will induce an increase of c(-t) Y in consumption demand and a decrease of Y / in investment demand. Similarly, we can analyze te cange in te second round: an increase of Y 2 in output and a cange of [c(-t)- /] Y 2 in demand. And tis cycle repeats until AD=AS. Te range of c, t and is (0, ), wile te range of b and is (0,+ ).Tus c(-t)-/ (-, ).If c(-t)-/ (0, ), ten wit te increase in AS, AD will consequently increase but on a smaller scale.( Y i > [c(-t)-/] Y i ) If c(-t)-/ (-,0),wit an increase of Y in AS, AD will decrease by [/-c(-t)] Y. All in all, AS will ultimately meet AD. Table sows te detail of te transmission mecanism, were GAP i denotes te gap between AD and AS in eac round. We obtain a stocastic series of te increase in output: Y, Y 2, Y 3,.. and a relevant series of te cange in demand [c(-t) -/] Y, [c(-t) -/] Y 2, Wen GAP i =0, te economy reaces its equilibrium point. From te equation G -[-c(-t) -/] Y i=0, we derive te total increase in = income: G Yi = = c( i and total increase in interest rate: (/) Yi = = i G i c( Conclusion: te economy will ultimately reac its equilibrium point following te route of E E 2 along te LM curve. 2 (Figure sows te route of te cange in demand and output on condition tat c (-t) -/>0, wile Figure 3 sows te case of c(-t) -/<0) Te case of te monetary transmission mecanism is similar (Figure 2): 7

8 An increase of M/p in te real money stoc sifts te LM scedule to te rigt. Te money maret adjusts immediately, and interest rates decline between point E and E 3 3 by M/(P) according to te following equations: M /p=y-r M/p= - r Te lower interest rate stimulates an increase of b M/(P) in investment demand, and ten te output begins to rise. Y denotes an indefinite increase of output in te first round. Te income increase will induce a decrease of Y / in investment demand by raising te interest rate and an increase of c(-t) Y in consumption demand, according to te following equations: M /p=y-r 0= Y - r, r= Y / I=I 0 -br, I=-b r I= - Y / C=C 0 +c[(-t)y +TR 0 ] C = c(-t) Y Similarly, we can analyze te cange in te second round: an increase of Y 2 in output and a cange of [c(-t)- /] Y 2 in demand. Tis cycle repeats until AD=AS. Finally, we obtain a series of te increase in output: Y, Y 2, Y 3, and a relevant series of te cange in demand: [c(-t) -/] Y, [c(-t) -/] Y 2, [c(-t) -/] Y 3. On te equilibrium point, we ave: b M M/(p) -[-c(-t) -/] Yi=0, p Yi = i= i= c( Advantages of tis Generalization Approac First, te metod still olds if we consider oter parameters tat determine te IS-LM model. For example, in an open economy, we can simply add a equation to modify te deduction: X=g-mY-nr, were X is net export, m is te marginal propensity to export Ten te equilibrium income determined by IS-LM model is: 8

9 C 0 + ctr 0 + I 0 + G 0 + g Y= (b+ n) c( m+ 4 and te series will be more complex to demonstrate te mecanism, but te result is similar. Second, it can, if not very reluctantly, explain te effect of monetary policy wit a orizontal IS curve and te effect of fiscal policy wit a vertical LM curve: ()If b, we obtain a orizontal IS curve.now we trace te monetary policy: Te lower interest rate stimulates an infinitely large increase of b M/(P) in investment demand, Y denotes an increase in output in te first round.te income increase will induce a decrease of Y / in investment demand by raising te interest rate, and an increase of c(-t) Y in consumption demand. Here we ave a flat (or almost orizontal) demand curve AE 2 wit a negative slope, and a relatively steeper supply curve. Since Y only generates infinitely large canges in investment demand, and tus avoid te embarrassment of explaining an infinitely large decrease of M/(p) in output wen applying te traditional metod of geometric series. We obtain: b M p lim Yi M = = b i= c( p (2)If 0, we obtain a vertical LM curve. Let us trace te effect of fiscal policy: First, te excessive government purcase of goods increases AD by G. Y denotes an increase of output in te first round. Ten an increase of c(-t) Y in consumption demand and a decrease of Y / in investment demand. Since an infinitely small Y can be large enoug to mae te decrease of investment demand counteract G, ere we ave a vertical supply curve E E 2 and a demand curve AE 2 wit a negative slope. Tus we avoid te problem of explaining an infinitely large decrease of G / in output. Finally we obtain: (5) G lim Yi = 0 i= c( =0 (6) 9

10 NOTES. For example, te following textboos expand te multiplier process into a geometric series: Maniw (996), Gordon (993), Hall and Taylor (993), Dornbusc and Fiscer (994) 2. It seems tat te effects on consumption and investment bot occur witin te same round, in fact, tey are two separate processes. For example, suppose te effect on investment lags m rounds beind tat on consumption demand and tus GAP equals G -[-c(-t)] Y i -/ + Yi. Wit te increase of output,gap still becomes progressively smaller.we can consider te series in anoter way: increased consumption demand c(-t) Y, c(-t) Y 2, c(-t) Y 3 cange of investment demand - / Y, -/ Y 2,-/ Y 3,increase of output Y, Y 2, Y 3 In te long run, we can still add up te aggregate effect of eac round and get te same result: Yi=-/ Yi+ c(-t) = = i= i -m i= i Yi+ G i = 3. Robert J. Gordon explained tat: Finding temselves wit more money tan tey need. Tis raises te prices of stocs and reduces te interest rate. Te initial decline in interest rate is called `liquidity effect` of a monetary expansion. Te lower interest rate raises te desired level of autonomous consumption and investment spending requiring an increase in production. Tis is te `income effect` of a monetary expansion."(gordon 990) Rudiger Dornbusc summarizes te stages in te transmission mecanism as follows:() Cange in real Money supply (2) Portfolio adjustments lead to a cange in asset prices and interest rates (3) Spending adjusts to te cange in interest rate (4) Output adjust to te cange in aggregate demand (Dornbusc and Fiscer 994.) REFERENCES () Dornbusc, R., and S. Fiscer Macroeconomics. 6t. ed. New Yor: McGraw-Hill. 0

11 (2) Froyen, R Macroeconomics: Teories and policies. 5t. ed. Upper Saddle River, N.J.: Pren-tice-Hall. (3) Gordon, R. J Macroeconomics, 5t ed. New Yor: Harper Collins College Publisers. (4) Maniw, N. G.996. Macroeconomics. 3rd. ed. New Yor: Wort. Table Round Effect on C Effect on I Combination effects GAP i of eac period G c(-t) Y (-/) Y c(-t) Y -/ Y G -[-c(-/] Y 2 c(-t) Y 2 (-/) Y 2 c(-t) Y 2 -/ Y 2 G -[-c(-/] ( Y + Y 2 ) m c(-t) Y m (-/) Y m c(-t) Y m -/ Y m m G -[-c(-/ ûyi. c(-t) Y (-/) Y c(-t) Y -/ Y G -[-c(-/] ûyi R IS 2 i= i= IS E 2 E A B G/ c( LM G G

12 R IS E LM 2 M/(p) E 2 2

13 R IS 2 o IS LM E G G c( E 2 Figure 3 Te income increase is smaller tan te government spending if c(-t)-/ <0 B A / Slope= <0 c(- t) Y(AD) 3

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