10. The two-period economy with sticky prices

Transcription

1 0. The two-period ecoomy with sticky prices Idex: 0. The two-period ecoomy with sticky prices Itroductio Basic model Mai assumptios Equilibrium The well fuctioig versus the Keyesia cases The classical case The Keyesia case The Keyesia cross The savig ivestmet locus What do we mea by a exogeous iterest rate? Fiscal policy Icrease i govermet expeditures today Aticipated icrease i govermet expeditures Permaet icrease i govermet expeditures Productivity slowdow Well fuctioig case Keyesia case Aalysis usig the IS curve Aimal spirits Policy implicatios Coclusios What ext?...8 Further readig...8 Review questios ad exercises...9 Review questios...9 Exercises...9 7/0/07

2 9. Itroductio A key isight of the Keyesia doctrie is that the price mechaism is too slow to esure the equilibrium i the moey market each momet i time. Whe prices are sticky, as it happes i the short term, the variable that clears the moey market is the omial iterest rate, ot the price level. The, because the iterest rate is determied i the moey market, a alterative variable is required to esure the equality betwee plaed savigs ad plaed ivestmet: this variable is real icome. I this ote, we review this argumet, i the cotext of a two-period closed ecoomy where the iterest rate is exogeously give. Aalytically, the approach is similar to that of a small ope ecoomy, the differece beig that the exogeous iterest rate is determied i the (omitted) moey market, rather tha i the iteratioal ecoomy. Sice the ecoomy is closed to fiacial trade, there ca be o mismatches betwee savigs ad ivestmet: i the cotext of the closed ecoomy, if the price (real iterest rate) fails to clear the market for goods ad services, the quatity (icome) has to do the job. This ote proceeds as follows. I Sectio 0., we preset the basic model. I Sectio 0.3, we show how the model solves whe prices are flexible ad whe prices are sticky. I Sectios 0.4 ad 0.5, we discuss how the equilibrium chages i respose to chages i the exogeous fiscal ad productivity parameters. 9. Basic model 9.. Mai assumptios Cosider a two-period closed ecoomy, where output i the curret period is predetermied, ad output i the secod period depeds o curret ivestmet oly (the capital stock depreciates fully each year). I this ecoomy, there is a govermet that purchases goods from firms, fiacig these purchases with lump taxes. The prefereces of the represetative household s are as follows l C U lc (0.) I this ecoomy, the curret (year ) atural output is pre-determied, 7/0/07

3 Q Q (0.) Future output depeds o curret ivestmet, accordig to: zf K Q (0.3). Capital is assumed to depreciate fully each year. Also for simplicity, we rule out fiacial market frictios: all domestic agets ca borrow or led ay amout of output at the prevailig iterest rate. We retai however the assumptio that the ecoomy is closed to capital flows. Firms maximize the et preset value of ivestmet: V z F K K (0.4) r The optimal ivestmet I K is implicitly defied by the first order coditio of that problem: K zfk r (0.5) Implyig: I K z,r (0.5a) The govermet budget costrait is give by: T T G G (0.6) r r The household maximizes utility (0.) subject to the budget costrait: C C (0.7) r Although the assumptio of a closed ecoomy might look restrictive, oe should remember that ecoomies are composed by tradable ad o-tradable good sectors. Hece, the aalysis below may be thought as applyig for at least some fractio of ay ope ecoomy. 7/0/07 3

4 Where life-time wealth is give by: Q T V (0.8) r T The first order coditios of the household problem deliver the Euler equatio C C r (0.9) This coditio, together with (0.7) ad (0.8) determie the household optimal cosumptio pla for the curret period: C Q T V T r (0.0) Equatio (0.0) implies that curret cosumptio depeds positively o curret output ad o future productivity, ad egatively o the iterest rate ad taxes. From (0.0), takig the partially derivative i respect to Q, we obtai the margial propesity to cosume: C Q (0.) Combiig (0.0) with (0.6) we see that i this framework, the Ricardia equivalece holds: what matters for cosumptio is curret ad future govermet spedig, ot the timig of tax collectio. This is obviously a limitatio of the model, arisig from the fact that we are postulatig lump sum taxatio. Usig (0.5.a) ad (0.6), total absorptio (or aggregate demad) i this ecoomy becomes: A C I G Q z F K G r I z, r G (0.) This expressio stresses the impact of future evets o curret spedig decisios. 9.. Equilibrium 7/0/07 4

5 Equilibrium i the output market requires the equality betwee icome Q ad absorptio, that is, Q A G, G, r,, z (0.3) Usig (0.), the defiitio of equilibrium (0.3) implies: Q z F K G (0.4), G I z r r This equatio summarizes the egative relatioship betwee icome ad the iterest rate, holdig fixed all other variables, such as govermet expeditures, ivestmet ad productivity. This relatioship is kow as the IS curve, ad is depicted i Figure. It is egatively sloped, because whe the iterest rate icreases, both cosumer ad ivestmet spedig declie, determiig a lower level of icome. Figure : The IS curve This curve is labelled IS because it implies the equality betwee a ecoomy total savigs ad ivestmet. Ideed, aother way of writig (0.3) is: S Q G r,, T,.. S I z,, (0.5) P, r Where 7/0/07 5

6 6 7/0/07 r T V T Q C T Q S P (0.6) G T S G (0.7) A key feature i (0.5) is that private savigs deped positively o curret output. This opes a chael through which aggregate savigs may adjust edogeously to equal ivestmet, wheever the iterest rate fails to do the job. 9.3 The well fuctioig versus the Keyesia cases The model above ca be solved i two differet maers. I the classical case, curret output is at its atural level, ad the edogeous variable is the iterest rate. I the Keyesia case, the iterest rate is exogeous, ad curret output is exogeous The classical case I the well fuctioig case, the real iterest rate adjusts so as to assure that output is always at its atural level. Defiig the atural iterest rate as r, the equilibrium is such that:,,,, z r G G A Q (0.3a) I alterative, oe ca look at the same equilibrium, usig the savigs ad ivestmet relatioship:,,,.., G P r z I S r Q S (0.5a) I ay case, the solutio for the edogeous iterest rate is: C C G I Q G K F z r (0.4a) This is o more tha the Euler equatio: remember that we are i a closed ecoomy, so the real iterest rate shall adjust to the level determied by the slope of the cosumer idifferece curve.

7 The equilibrium i the well fuctioig case is described i Figure by poit. I this poit, give the level of output (described by the vertical aggregate supply LR), the real iterest rate is determied so as to equal output ad spedig. The iterest rate i poit is labelled the atural real iterest rate, because it is the oe that equals savigs ad ivestmet whe output is at the atural level. Figure : Equilibrium: short ru ad log ru 9.3. The Keyesia case Istead of assumig that the iterest rate adjusts to clear the market of goods ad services, assume istead that prices are sticky. Whe prices are sticky, imbalaces i the moey market caot be cleared by the price level. I that case, the variable that adjusts to clear the moey market is the iterest rate: for istace, whe moey supply exceeds moey demad, this gives rise to a excess demad for bods, drivig the price of bods upwards ad by the the iterest rates dow. Sice with sticky prices, the iterest is determied i the moey market, the free variable to clears the market for goods ad services is curret output. This is illustrated i Figure by poit. This equilibrium correspods to the itersectio of the exogeous iterest rate (defied by the horizotal schedule MP, that stads for moetary policy ) with the IS curve. Because i this case the the actual iterest rate is higher tha the atural iterest rate ( r r ),equilibrium output falls short the level of atural output, Q Q, givig rise to a egative output gap. that: Aalytically, settig r r i (0.4), the equilibrium level of output, Q, is such 7/0/07 7

8 Q G I r z F K G (0.4b) I the savigs-ivestmet framework, the equivalet coditio is: S Q r,.. S I z, P G, r, (0.5b) The Keyesia cross Figure 3 offers a alterative represetatio of the equilibrium i the goods market, based o equatios (0.) ad (0.3). Equatio (0.) implies a slope equal to i the Q, A space. Equatio (0.) also implies a positive relatioship betwee aggregate demad ad curret icome, but the slope (the margial propesity to cosume, 7.), is less tha. The well fuctioig case is described by poit. At poit, output level is at the atural level, Q Q, determied by the supply side of the ecoomy. The key feature of the model with flexible prices is that the iterest rate (price of curret output) adjusts istataeously to esure that aggregate demad equals the existig aggregate supply (ote that chages i the real iterest rate make the aggregate demad schedule i Figure 3 shift up ad dow). Because the real iterest rate moves so that aggregate demad equals full employmet, i this model the Say s Law holds: supply determies its ow demad. The Keyesia equilibrium is described by poit. I that case, the real iterest rate is too high, ad it fails to adjust to clear the goods market. Hece, aggregate demad will be too low. Sice firms will ot produce uless there is demad, output will be oly Thus, output reaches a equilibrium that falls short the full employmet level. Q Q. I the Keyesia versio of the model, curret productio respods passively to the level of aggregate demad. Now, the Say s law holds i reverse: it is the demad that determies output, ot the other way aroud. Sice the price (iterest rates) fails to adjust, the equilibrium is met through chages i quatity (icome). 7/0/07 8

9 Figure 3: The Keyesia cross The savig ivestmet locus The distictio betwee the well fuctioig case ad the case with sticky prices ca also be described with the help of Figure 4. Suppose that iitially the ecoomy is at poit 0. I poit 0, give the iterest rate, plaed savigs (at the atural output level) exceed plaed ivestmet by the distace 0. I a classical world, the excess savigs over ivestmet would cause the real iterest rate to adjust dowwards, util the equilibrium betwee savigs ad ivestmet was met, i poit. Now suppose that the real iterest rate remais uchaged, at r r. I that case, the oly way of achievig the balace betwee savigs ad ivestmet is through a fall i curret output. Graphically, the fall i output implies a shift of the savigs curve to the left. The Keyesia equilibrium is described by poit. I this poit, the iterest rate is too high, givig rise to a output gap equal to Q Q 0. Also ote that, i poit total ivestmet is less tha i poit : the price stickiess geeratig a recessio today has implicatios for future output as well. 7/0/07 9

10 Figure 4: Savigs ad ivestmet i the classical ad i the Keyesia models What do we mea by a exogeous iterest rate? I the aalysis above, we abstract from chages i moetary policy, assumig that the real iterest rate remais costat. This assumptio must be clarified. I fact, a uchaged moetary policy ca have differet meaigs. This icludes, a uchaged moey supply, a uchaged iterest rate, or a uchaged iflatio target, for istace. I our discussio, we are ot assumig that the moey supply remais costat. Istead, we assume that the cetral bak is followig some iterest rate rule, ad it will do whatever it takes to keep the real iterest rate o track. This icludes chagig the iterbak ledig rates, usig ope market operatios or whatever policy the cetral bak may fid appropriate to achieve its target for the real iterest rate. The assumptio that the cetral bak targets the moey supply uderlies the traditioal IS-LM model. I the IS-LM model, the LM curve describes the equilibrium i the moey market ad is positively sloped i the output-iterest rate locus, because with a costat moey supply, a icrease i output traslates ito a higher demad for moey for trasactio purposes, creatig a excess demad for moey ad by the a icrease i the iterest rate. 7/0/07 0

11 The assumptio that the cetral bak targets the iterest rate is more i accordace to the real world facts: i today s world, most cetral baks act to make the real iterest rate behavig i a certai way as a fuctio of key variables, such as iflatio ad output 3. I Figure, the assumptio that the cetral bak is targetig the iterest rate materializes with a horizotal curve, labelled MP (stadig for Moetary Policy) 4. This curve may shift upwards or dowwards, depedig o the cetral bak itetios 5. By settig the MP curve uchaged, we abstract from moetary policy cosideratios. 9.4 Fiscal policy 9.4. Icrease i govermet expeditures today A icrease i govermet expeditures today shifts the IS curve rightwards, o a oe-to-oe basis. As log as the iterest rate remais uchaged, the impact o output will also be oe-to-oe 6 : Q G Thus, with sticky prices, a fiscal expasio fully fiaced with (curret or future) lump-sum taxes will be able to drive the ecoomy to full employmet. Most importat, i 3 Note that the cetral bak does ot determie the real iterest rate. But as log as prices are sticky, the cetral bak ca adjust the iter-bak ledig rate so as to iduce the desired chage i the real iterest rate. This is the tactics most cetral baks follow i our days. 4 Romer, D., 000. Keyesia macroecoomics without the LM curve. Joural of Ecoomic Perspectives, 4(), I geeral, most cetral baks are cocered with iflatio, so they set the real iterest rate to icrease wheever iflatio is foresee to icrease above target. I practice, cetral baks may also react to chages i output, settig the real iterest rate to declie whe output is below potetial, ad vice-versa. 6 The particular case i which the additioal govermet spedig is fiaced with curret taxes ( G T ) is kow as the Haavelmo Theorem [T. Haavelmo: Multiplier Effects of a Balaced Budget, i: Ecoometrica 3, 3-38, 945]. 7/0/07

12 cotrast to the well fuctioig case, private cosumptio is ot crowded out ( C 0 ): oe o had, the icrease i govermet spedig, by iducig a expasio i curret output, Q, icreases the household life-time wealth (0.8); o the other had, the extra expediture will come alog with higher curret or future taxes, by the same curret value; o balace idividual wealth does ot chage at al Aticipated icrease i govermet expeditures Equatio (0.4b) also shows that a icrease i future govermet expeditures has a deflatioary impact o curret output: Q G r The reaso is that govermet expeditures impact egatively o private wealth, reducig cosumptio today. Whether cosumptio will rise more or less tha proportioally, it depeds o how the iterest rate relates to the rate of time preferece Permaet icrease i govermet expeditures Whe govermet expeditures icrease permaetly, the govermet budget costrait chages accordig to: T T G r r The multiplier becomes: Q G G G r r I this case, the IS curve may shift up or dow, depedig o whether private cosumptio falls more or less tha the icrease i govermet spedig. This, i tur, depeds o the relative size of the iterest rate versus the rate of time preferece. 7/0/07

13 9.5 Productivity slowdow I this sectio, we examie the implicatios of a fall i the productivity parameter, z. I terms of our model, the first impact will be o ivestmet: the fact that productivity declies implies that firms will fid it profitable to ivest less, for ay give level of the iterest rate. A lower ivestmet, i tur, will reduce the household life-time wealth, so curret cosumptio will declie too. Give the curret output, the households savigs schedule expads to the right Well fuctioig case The well fuctioig case is described i Figure 5, with a move from poit 0 to poit. The fall i productivity causes ivestmet to declie ad savigs to icrease, impactig egatively o the (atural) iterest rate. The fact that the atural iterest rate declies implies that curret cosumptio becomes less expesive ad creates icetives for ivestmet, despite the productivity fall. This effect is just the eough to assure that aggregate demad remais uchaged, at the level determied by the curret atural output (the Say s law holds). Figure 5: productivity slowdow: classical case Whether total ivestmet ad savigs icrease or decrease, it depeds o the utility fuctio. I the particular case we are dealig with, the move i the iterest rate exactly offsets the shifts i the curves, so the fial levels of savigs ad ivestmet are the same as before the shock. Note however that this is ot a geeral case. 7/0/07 3

14 9.5. Keyesia case Figure 6 describes the alterative case i which the real iterest rate fails from decliig. Poit caot occur, because that would imply a excess savigs over ivestmet or, what is the same, a excess supply i the goods market. I order for savigs to match ivestmet at the prevailig iterest rate, output has to declie, movig the ecoomy to poit : the savig schedule adjusts leftwards to meet the ew ivestmet schedule at the exogeous iterest rate. Figure 6: Productivity slowdow: Keyesia case Figure 6 also illustrates a importat propositio of the Keyesia thikig, labelled as the Paradox of thrift : from poit 0 to poit, the fact that the households lifetime wealth declied with the productivity shock lead agets to icrease their plaed savigs. However, because this caused a excess of savigs over ivestmet - or isufficiet demad - curret output declied, reducig the total savigs. I the ed, idividual attempts to save more failed to produce higher savigs at the aggregate level Aalysis usig the IS curve Figure 7 explais what happes i terms of the IS curve. Iitially, the ecoomy lies i poit 0, with full employmet ad the iterest rate at its atural level. Poit depicts the classical case, i which the iterest rate adjusts dowward to its ew atural level, ad full employmet is maitaied. Poit describes the Keyesia case i which the iterest rate fails from decliig. 7/0/07 4

15 Figure 7: productivity slowdow ad the IS curve Aimal spirits I the sectio above, we discussed the case i which future productivity is expected to slow dow. Propoets of the eo-classical school believe that agets expectatios regardig the future are i geeral well grouded o fudametals ad ecoomic sigs that people accurately iterpret. Forecastig errors are possible, but people lear with past mistakes, so systematic expectatio errors caot occur. Propoets of the Keyesia school cotet istead that forecastig errors may be large, because expectatios are ot ecessarily ratioal. Expectatios are drive by psychological factors ad chages i collective mood ( aimal spirits ). Hece, expectatios may deviate sigificatly from reality. Chages i collective mood give rise to upward ad dowward swigs i aggregate demad, which, together with price stickiess, will cause output to fluctuate aroud its atural level, eve whe o fudametal reaso uderlies the chages i expectatios. 7/0/07 5

16 Referrig to Figure 7, the ecoomy may move from full employmet to depressio oly because people became pessimistic regardig the future Policy implicatios The Keyesia school advocates that, if the market fails to do drive the ecoomy to its atural rate i a reasoable time, the govermet should help. A first aveue to achieve this would be to use moetary policy. Referrig to Figure 7, if the cetral bak, by expadig the moey supply, successfully achieved a reductio i the iterest rate to its ew atural level, the the ecoomy would approach the full employmet. I real life, however, this may ot be that simple. First, because the cetral bak may be costraied by a madate that cares oly with price stability. Secod, eve if the cetral bak wated the real iterest rate to fall, the cetral bak iflueces the short term omial iterest rates, while what matters for spedig decisios are log term real iterest rates. I some cotexts, the cetral bak may fid it hard to drive the real iterest rates dow below a give level. This is the case of the zero lower boud, i which the omial iterest rate reaches the value of zero ad iflatio expectatios are to small to allow further declies i the real iterest rate. I that case, the alterative istrumet is a fiscal expasio. I terms of our model, a icrease i govermet spedig today or a decrease i govermet spedig tomorrow will move the IS curve to the right, causig output to expad, for each level of the iterest rate. This is illustrated i Figure 8, by a move from to 8. 7 Keyes argued that ivestmet spedig collapsed durig the Great Depressio because of a exogeous declie i the etrepreeurs expectatios regardig future profitability. 8 Note that i the eoclassical model, a fiscal expasio departig from poit would result i a upward icrease i the equilibrium iterest rate, from to, without ay impact o output. 7/0/07 6

17 Figure 8: Fiscal expasio Of course, the claim that govermets should use fiscal ad moetary policies to stabilize the ecoomy is ot free of criticism. Sceptics of macroecoomic stabilizatio policies coted that private shocks to the ecoomy are hard to idetify. Idetificatio lags ad itervetio lags reduce the effectiveess of stabilizatio. O the other had, activist policies are subject to pressures ad electoral iterests, so policymakers with discretioary power may be tempted to move away from the public iterest. 9.7 Coclusios I this chapter, we departed from the well fuctioig two-period ecoomy, by assumig that the real iterest rate does ot evolve so as to clear the market for loaable fuds. The ratioale is that prices are sticky ad the omial iterest rate is determied i the moey market. If, i plus, the cetral bak is committed to some iterest rate rule, the the real iterest rate ca be see as exogeous to the goods market. A real iterest rate lyig above the level that clears savigs ad ivestmets implies a excess savigs over ivestmet. I a ope ecoomy, this gap ca be met with a surplus i the trade balace. I a closed ecoomy, the excess supply of goods caot be absorbed, so the oly solutio for firms is to produce what is actually demaded. Thus, the ecoomy will be operatig below its potetial. I cotrast to the classical model, accordig to which more savigs brigs more ivestmet ad thereby more icome, i the Keyesia model a icrease i desired savigs 7/0/07 7

18 has a deflatioary impact i the ecoomy. For istace, the aticipatio of a productivity fall will depress curret cosumptio ad demad, causig a recessio today. Because people have setimets, these swigs may happe just because of expectatios, eve if the uderlyig fudametals are soud. The govermet ca fix a low demad, either by expadig the moetary policy so as to decrease the real iterest rate, or by icreasig expeditures o a temporary basis What ext? Whe prices are sticky, the omial iterest is determied i the moey market. This gives the cetral bak the power to ifluece ecoomic activity. If, for istace, the real iterest rate icreases above its atural level, the cetral bak may adjust its policy tools to drive dow the short term iterest rates ad by the, to achieve a reductio i the log term rates closer to the atural level. Drivig the log term real iterest rate dow may ot be however that easy, especially durig fiacial crisis. First, the omial iterest rate may reach the zero lower boud, makig it hard to drive the real iterest rate further dow without iflatio. Secod, because, i a cotext of high ucertaity, real iterest rates i private cotracts may depart sigificatly from the iterest rates set by the cetral bak i the moey market. Third, i a cotext of high ucertaity, leders may refrai from ledig, regardless the level of iterest rates. I the ext hadouts, we review these argumets. Further readig Adolfatto, D., Macroecoomic Theory ad Policy, Chapter 7. Sachs, J., Larrai, F., Macroecoomics i the global ecoomy. Pretice Hall, 993, Chapter. 7/0/07 8

19 Review questios ad exercises Review questios Exercises 9.. (Edowmet ecoomy) Cosider a ecoomy where full employmet curret ad future output are iitially give Q Q 000. Further assume that i this ecoomy, the prefereces of the represetative cosumer are give by U lc lc. Iitially, there are o govermet taxes ad expeditures. a) (Natural iterest rate). From the household optimizatio problem, fid out the atural iterest rate i this ecoomy. Represet the equilibrium i the Keyesia cross ad fid the expressio of the IS curve. b) (Dark future). Suppose that people expect istead future output to be Q 800. How much will be the atural iterest rate i this case? Assumig that the iterest rate is free to move, explai what happes usig the savigs schedule, the Keyesia cross ad the IS curve. c) (Keyesia case) Now cosider the case i which the real iterest rate caot fall below zero. Explai explai the adjustmet of the ecoomy referrig to the savigs schedule, the Keyesia cross ad the IS curve. d) (Fiscal expasio) Departig from (c), examie the implicatios of a govermet itervetio amoutig to G 00. Explai how the savigs schedule, aggregate expediture ad the IS curve will shift. Will it make a differece if these expeditures are fiaced with curret taxatio of with future taxatio? 9.. (Ivestmet ecoomy) Cosider a ecoomy where atural output i the curret period is Q 00. Output i the secod period depeds o curret 0.5 ivestmet, accordig to Q zk. The prefereces of the represetative cosumer are give by U lc lc. Iitially, there are o govermet taxes ad expeditures. a) (Iitial state). Suppose that z 40. Fid out the optimal ivestmet, cosumptio, ad the atural iterest rate i this ecoomy. Depict the equilibrium i the savigs-ivestmet schedule. b) (Productivity slowdow). Examie the implicatios of a fall i expected productivity to z 0, cosiderig two cases: (b) the iterest rate is free to adjust. (b) the iterest rate caot fall below zero. I particular, show what 7/0/07 9

20 happes i the savig-ivestmet diagram, i the Keyesia cross ad to the IS curve. c) (Paradox of thrift): Comparig (a) with (b), explai the Paradox of thrift. d) Returig to the iitial case, with z 40, examie the implicatios of a positive trasitory shock o curret output, from Q 00 to Q 875. Agai, compare the well fuctioig case with the case i which the real iterest rate caot fall below zero. I particular, show what happes i the savig-ivestmet diagram, i the Keyesia cross ad to the IS curve. e) Departig from (d), explai how fiscal policy could be used to drive the ecoomy to full employmet. 7/0/07 0