Gerhard Illing (2008) Money: Theory and Practise

Transcription

1 Gerhard Illing (28 oney: Theory and racise reliminary Draf --- ommens Welcome haper 2 oney and Long Run Growh The Basic icroeconomic Framework Dynamic macro in a wo period model odern acro is abou dynamic sochasic general equilibrium in a se up wih represenaive, opimising agens in an economy wih imperfec compeiion and possibly incomplee asse markes. In order o gain he basic inuiion, we reduce he problem o he simples case: Le us consider a wo period endowmen economy wih one consumpion good per period and no uncerainy. Households can choose heir consumpion pah across ime: How much will hey consume in he firs period (he presen, how much in he second (he fuure? This seing may look a bi arificial, since (hopefully he fuure lass much longer han he presen. Bu i is sraighforward o exend i o muliple periods (or even o infinie horizon i will be jus more cumbersome. The wo-period seing allows us o characerise some key feaures of he iner-emporal allocaion in a moneary economy in he simples way. Via iner-emporal rade, households can smooh consumpion across ime. New neoclassical macroeconomics ries o capure all relevan choices on he macro level by analysing he decision of one represenaive agen. The represenaive agen framework absracs from all issues caused by heerogeneiy or disribuional issues. See discussion beween hari/kehoe and Solow (Journal of Economic erspecives 28. We allow for iner-emporal rade on perfec capial markes. Firs, we analyse real bonds paying a real rae of ineres r. Laer, we exend our basic model by inroducing governmen aciviy and money. In a moneary economy, nominal bonds pay a nominal ineres rae i.

2 The wo period real economy references of a represenaive consumer are given by he ime-separable uiliy funcion: = ( +ß [( + ]. sually, people are impaien if hey have he opion o consume a given quaniy eiher oday or o wai for one year, hey prefer o consume i righ now. This impaience is capured by he discoun facor ß<. The degree of impaience can also be characerised by he discoun rae ρ. ß and ρ are relaed by he equaion ß=/(+ρ. Impaien people need o be rewarded for posponing consumpion: fuure consumpion, measured in erms of presen value, has o be cheaper han curren consumpion. This is he case if posponing consumpion is rewarded by receiving real ineres rae paymens. In figure xx-, preferences are represened by indifference curves, characerising all consumpion bundles ( + which give he same uiliy. eople are - a leas o some exen - willing o subsiue curren for fuure consumpion. This willingness is measured by he marginal rae of subsiuion (RS - he slope of he indifference curve. ahemaically, i is he raio of marginal uiliies: d + / ( RS : = = ( + ρ d β / + ( + In he second equaion, we replace /ß by +ρ=/ß and wrie he marginal derivaives wr as ( = / The RS ells us how much fuure consumpion he consumer needs in order o be willing o give up one (marginal uni of curren consumpion. As you can see in figure xx-, along an indifference curve, he willingness o give up curren consumpion in exchange for fuure consumpion is increasing wih. To ge a beer inuiion, compare poin E and B in figure xx-. urren consumpion in E is low relaive o fuure consumpion; he oher way round in B. Bu boh consumpion bundles in figure xx- provide he same uiliy- jus as poin. In E, you would need o gain a lo addiional consumpion omorrow in order o accep giving up some consumpion oday. onversely, you would be willing o give up a lo from fuure consumpion o gain a lile bi more oday. The slope is very seep; he shadow price (he consumer s evaluaion of curren relaive o fuure consumpion is high. Bu he more consumpion becomes available oday, he more eager you are o rade i in for fuure consumpion he slope ges flaer and flaer along he indifference curve. Figure xx Ineremporal onsumpion: Indifference urves 2

3 onsumer s relaive evaluaion varies along he indifference curve. To find a reliable measure for he degree of impaience, we look a he slope (RS for he case ha consumpion is equal in boh periods as in poin. All bundles wih he same consumpion level in boh periods ( = 2 lie on he 45 o line. Since ( ( = + for = + he slope along his line is always /ß=+ρ. Saring from a posiion of equal consumpion in boh periods, you would need o gain a leas +ρ unis in he fuure in order o accep giving up one uni oday. The Wealh onsrain How o deermine he opimal consumpion pah? In sandard micro courses, consumers choose heir opimal bundle, given he budge consrain. Jus in he same way, in an ineremporal marke economy wih perfec capial markes, consumers can rade across he wo periods. Since here is jus one good in his economy we can normalise he price of his good o one. Bu consumpion of he same good nex period is quie a differen commodiy, so i has a differen price. The price of fuure consumpion in erms of consumpion oday is capured by he real rae of ineres. onsumers can save from curren income and inves savings in real bonds: S = Y - ; Savings plus he real rae of ineres r + earned adds o fuure income nex period and so allows for higher consumpion nex period. + = Y + +S (+r +. Alernaively, if consumers borrow from fuure income sreams by issuing real bonds, S <. Borrowers can consume more in he curren period. Bu nex period, hey have o pay back he bond and, in addiion, pay ineres paymens. Wih S <, fuure consumpion has o be lower han income. We can merge he per-period budge consrains ino one wealh consrain. Wealh is he presen discouned value of income in boh periods: So he wealh consrain is: + + = Y + Y + + r + + r + As in sandard micro courses, we can derive he opimal iner-emporal choice for he consumer using he Lagrangian muliplier echnique: The household maximises uiliy, subjec o he wealh consrain. This gives he firs order condiion (frequenly, his condiion is called Euler equaion: c ( Euler equaion ( + ρ = + r + c ( + (The Euler equaion is usually derived using complex dynamic echniques, such as dynamic programming, calculus of variaion or he maximum principle. For our purposes, however, i is sufficien o use he Lagrangian see appendix. The Euler equaion E* is he key equaion characerising iner-emporal demand. Le us give an inuiive inerpreaion of he Euler equaion by rewriing he condiion slighly in a more convenien way: + r + c ( = c ( + + ρ arginal uiliy ou of consumpion, adjused for he relevan rae of reurn, has o be equal across ime. Because of concaviy (u (c<, agens prefer consumpion smoohing. In figure xx2, E characerises he endowmen poin. In a growing economy wih g >, Y + > Y. If r=ρ, here are srong incenives for consumpion smoohing. onsumers would like o consume he same amoun in boh periods (poin along he wealh consrain. Wih r=ρ, marginal uiliy of consumpion across ime has o be equal, and his means ha consumpion has o be equal in boh periods. 3

4 Figure xx2 onsumpion Smoohing for r=ρ In conras, if he ineres rae exceeds he discoun rae (r>ρ, i is opimal o consume more in he second period. Insead of consumpion smoohing, we hen ge consumpion iling owards fuure consumpion (see figure xx3! The oher way round for r<ρ. Bu he more concave he payoff funcion, he sronger he incenives for consumpion smoohing! The degree of concaviy of he uiliy funcion is capured by he elasiciy of marginal uiliy. I ells us: How does he slope of he indifference curve change wih a change in he consumpion raio c / c +? Frequenly, we consider preferences wih a consan elasiciy of subsiuion. They are represened by he ES payoff funcion: = c / ; > For = : = ln c / is he elasiciy of iner-emporal subsiuion: he willingness o give up x% of consumpion oday for % more consumpion omorrow. ES preferences allow us o solve he Euler equaion explicily: + + r + E*: = or + ρ + r + ρ + = ; + onsumpion is increasing [decreasing] over ime for seeper he pah. r > ρ [ r < ρ ]. The higher, he + + r + Define + = +, so ρ = + = + sing he Log-linear approximaion ln( + x ~ x, we ge: / = [ r ρ]. Alernaively, using c + for he log of consumpion + + c = ln, we can wrie: c = E( c+ ( r + ρ This is he basic relaion which provides a micro-foundaion for iner-emporal aggregae demand! If we urgenly wan o consume in a fixed proporion in boh periods, here is no scope for iner-emporal subsiuion. In ha exreme case, preferences are kinked he elasiciy of subsiuion is zero. Wih =, he household would like o consume he same amoun in each period independen of he ineres rae. He would choose poin K in figure xx3. In conras, wih infiniely high elasiciy of subsiuion (, indifference curves are linear wih he 4

5 slope + ρ: = + 2 /(+ρ (compare he dashed lines in he figure. In ha case, agens are indifferen beween consumpion of one uni oday or +ρ unis omorrow consumpion in boh periods is perfecly subsiuable a he rae +ρ.as long as r>ρ, consumers would hen be beer off posponing consumpion owards he final period. Figure xx3 onsumpion Tiling p o now we characerised he opimal choice of a represenaive consumer, given some arbirary level of he real ineres rae. Our key challenge, however, is o find ou he driving forces deermining he equilibrium ineres rae. In our seing, he answer is simple enough: he real rae of ineres has o be such ha he represenaive consumer is happy o consume exacly wha is available in each period. Nohing more, nohing less! On he aggregae level, he ne wealh in a (closed economy has o be zero. So raher han aking he ineres rae as given, we now jus reverse causaliy: The ineres rae has o adjus in such a way ha he represenaive consumer is happy o consume exacly wha is available for each period. Knowing his, we can use he Euler equaion joinly wih our resource consrains o deermine he marke equilibrium real rae. We call his he naural rae of ineres. The naural rae of ineres: The equilibrium real ineres rae is deermined by he equilibrium condiions: ( The Euler equaion mus hold for he represenaive consumer. (2 Resource consrains: = Y (aggregae consumpion = endowmen. Define g + as he growh rae in he economy: Y + = ( + g+ Y. If we inser he resource consrains in he Euler equaion E*, we ge as equilibrium condiion: n + r ( + + g = + + ρ This equaion looks complicaed, bu again using he log-linear approximaion ln( + x ~ x we can linearise i in he following way: gives: r n + = + g + ρ. n g = ( r + ρ Solving for he real ineres rae, his 5

6 Figure xx4: The naural rae of ineres Equilibrium: = Y ; + =Y + ; Y + =(+g Y ; and = g = [ r ] n r + = ρ + g + / + + ρ sing he resource consrains Y =, we see ha he Euler equaion characerises he ineremporal IS curve: y E( y ( r ρ = + So we now have a complee characerisaion of our simple real business cycle economy: Aggregae Equilibrium: For he whole economy, demand mus be equal o supply (producion; endowmen. The equilibrium pah depends on: a references (he discoun rae and he iner-emporal elasiciy of subsiuion b Technology (endowmen; produciviy growh g real iner-emporal exchange rae In principle, here is no reason why he equilibrium real rae of ineres has o be posiive. Afer all, he relaive iner-emporal price is + r (or he inverse /(+r. In a declining economy wih negaive growh raes, he equilibrium real rae is likely o be negaive. Bu, as has already poined been ou by he Ausrian economis Eugen von Böhm Bawerk in Kapial und Kapialzins (884, impaience of consumer and/or echnological progress (generaing posiive growh rae g will cause he real rae of ineres o be posiive in normal imes. The naural rae of ineres n r is higher - The higher ρ: impaience requires sronger incenives o save! - The higher he growh rae g (because of incenives for consumpion smoohing, high growh has o be balanced by sronger incenives o save! - The lower he elasiciy of subsiuion (he more concave he uiliy, he sronger are he incenives o smooh consumpion for a given growh rae! Figure xx4 illusraes how preferences (he shape of he indifference curve affec he naural ineres rae. Since in equilibrium, he represenaive household mus be willing o consume exacly he endowmen poin, he naural ineres rae (plus mus be equal o he slope of indifference curve a poin E. The more concave he preferences, he higher mus be r. As one exreme, consider wih kinked preferences (poin K in figure xx4. The consumer is no beer off wih Y > or g >, even for r! + Y A he oher exreme, wih linear preferences (. r=ρ independen of he rae of growh g in he economy see he doed line in figure xx4. 6

7 Though experimen: Wha happens o he ineres rae if shocks disurb he economy? The answer depends very much on wheher hese shocks are emporary or permanen. a permanen increase in produciviy (A raised in boh periods: no impac b emporary increase in produciviy: (A raised jus in he period ~ lower naural rae o couner incenives o pospone consumpion c Increase in he growh rae of produciviy (g goes up: naural rae increases o couner incenives o bring forward and consume already oday Noe: sound analysis would require sochasic opimisaion. Inuiion: Incenives for precauionary savings o insure agains bad shocks. On he oher hand, wih higher risk in he fuure, giving up curren safe consumpion may require a risk premium. Disinguish beween ineres rae on safe asses (allow o insure agains risks in fuure income and ineres raes on risky asses. To give incenives o hold risky asses, a risk premium has o be paid on hese asses. Idiosyncraic risks can be diversified. So crucial: orrelaion of asses wih aggregae risk (affecing he whole economy High risk premium in imes of high volailiy. Negaively correlaed asses are expensive (have low reurn, since holding hem allows o insure agains aggregae risk. A model. α; ß Exension o many (infinie periods The wo period model may seem prey arificial, since in realiy, we don have jus o choose beween oday and omorrow. Raher, he world exends ino unknown fuure wihou definie ime horizon, making our choice problem a bi more complicaed. The bes way o capure his is o consider a household opimising from oday on unil infiniy. Forunaely, our insighs can easily be exended o such an infinie horizon economy. Surprisingly, we ge exacly he same condiions. As simples case, le us consider am economy growing a a consan rae of growh g and a consan ineres rae r n. So he represenaive consumer maximises: = ( + ρ ( c Subjec o he per period budge consrain: W + = ( + rn [ W + ( Y ] wih Y + =(+g Y and W as real wealh a period. Imposing W = and he ransversaliy condiion lim = W, all per period budge + rn consrains can be merged ino one single wealh consrain saing ha presen value of consumpion has o be equal o presen value of income: Y = = + rn = + rn sing he Lagrangian parameer λ for he wealh consrain, he firs order condiion for each period is ( = λ for all. Since his holds also for +, we know ( + ρ ( + r n + rn ( + + = λ. Dividing boh equaion gives + c ( = c ( + ( + ρ ( + r + ρ n Equilibrium: sing he resource consrain Y =, and he definiion of he growh rae g= (Y + -Y /Y we again ge he condiion for he naural rae: r n = ρ + g 7

8 Endogenous roducion: The Labour arke p o now, we considered a pure endowmen economy. Bu he model can easily reinerpreed as a producion economy wih a given endowmen of ime N which can be used as labour inpu for producion. Assume he consumpion good can be produced using labour N as inpu wih he linear echnology (consan reurns o scale y = A N. The parameer A represens produciviy. Technological progress (an increase in A improves produciviy across ime. Le g be he growh rae of echnological progress. So we can wrie: A + = ( + g A. Here we do no analyse he driving forces for produciviy growh, bu ake echnological progress as exogenous. Le us firs also assume ha labour supply (he amoun of ime available for work is exogenous and consan across ime: N = N. In his exended economy, we have now more han one good: he consumpion good and labour N. Le be he price of, and W be he wage. In general equilibrium analysis, only relaive prices are deermined, so we can normalise = wih w=w/ as he real wage measured in unis of he consumpion good. Laer on, in secion xx, when we inroduce money, will be he nominal price of in erms of money. All income generaed will in he end accrue o households eiher as wage, capial income or profis. Wih perfec compeiion, a linear producion echnology and jus one inpu (labour, all resources produced will be paid ou o workers in equilibrium, he real wage is equal o produciviy: w=a. [To simplify, we ignore capial as inpu. Nohing depends on ha simplificaion, excep ha we don analyse capial formaion and so we will no be able o alk abou convergence o some seady sae. When you have done a growh heory course, you should be able o inegrae capial in our framework.] In realiy, of course, here is no perfec compeiion: boh producers and workers have some monopoly power. So le us now inroduce imperfec compeiion. As we will see, monopoly power creaes disorions. onopolisic disorions: arke power on he produc marke Wih marke power, he naural level of oupu (he producion level realised in marke equilibrium is inefficienly low he economy is plagued by srucural inefficiency. In order o see ha le us firs endogenize labour supply. So assume now ha he amoun of ime T available can be used eiher for work N or for leisure L: T=L+N. Again, we assume ha preferences for leisure/disuiliy of work are addiively separable. For he sake of simpliciy, we jus consider he case of logarihmic preferences. Separaing iner-emporal and inraemporal choice, le us concenrae on he opimal rade off beween consumpion and hours worked wihin each period. = ln + ln ( T N subjec o he per period budge consrain: = Y = W N + Π The opimal labour supply is deermined by he condiion ha real wage w=w/ equals W marginal rae of subsiuion beween leisure and consumpion: w = =. ( T N Wih logarihmic preferences, his gives he posiively sloped labour supply curve: N = T w The higher he real wage, he higher is he willingness o work a a given consumpion level. Assume firs ha he good is produced on a perfecly compeiive produc markes. A represenaive firm maximises Π = Y W N = Y W Y A. In equilibrium, / = W A so he real wage mus be w = A. Wih consan reurns o scale here are no / 8

9 profis: Π=. sing he equilibrium condiion = Y = A N, he efficien amoun of hours worked is: N * = T / 2 Bu if firms have marke power, hey will charge a mark-up on marginal coss W/A, driving a wedge beween marginal produciviy and he real wage. onopoly power shifs labour demand downwards. Wih consan reurns o scale he price will be = W ( + µ. A Figure xx5: The rice Seing urve A Thus we now ge w =. onopolisic disorions drive up he price of he consumpion + µ good, reducing effecive real wage. This is capured in figure xx5 by he price seing curve which is below marginal produciviy of labour. Given a compeiive labour supply curve, he T equilibrium level of employmen is inefficienly low N = < N *. Noe ha his 2 + µ inefficiency is no a resul of involunary unemploymen. Because of lower real wages, workers simply are no longer willing o work as much as under compeiive condiions. Of course, monopoly power may have serious disribuional impac: The income of workers is reduced, whereas profis are posiive. Bu neverheless, aggregaing across all households, aggregae consumpion sill equals aggregae income which equals oal producion (his is jus he same condiion you have learned in basic macro: for a closed economy, GD =GN. Because in our simple economy, I=G=, we also have GD=! Absracing from disribuional issues, we ake his fac ino accoun by imposing he condiion ha he represenaive consumer earns all income. Of course, monopoly power on he worker s side will also creae disorions: I shifs effecive labour supply (he wage curve o he lef. This way, real wage may increase relaive o he compeiive level, bu in any case, again aggregae employmen is reduced in he monopolisically disored economy. The same holds for disorionary axes. As long as leisure is no axed, axes (boh income axes or indirec consumpion axes again drive a wedge beween marginal rae of subsiuion and marginal produciviy. Taxes have a similar impac on employmen and producion as monopoly power. Key message: Y n <Y*; N<N* We capure his disorion by = Y*-Y n Focus Box for Nerds: Dixi/Sigliz ~ Blanchard/Kiyoaki opular among cenral bank academics 9

10 Inroduce money ino he model Now le us now modify he simple real business cycle model by inroducing money and nominal conracs. Again, we sar wih a pure endowmen economy and simply inroduce (fia money as addiional good, being he uni of accoun (numeraire and he means of paymen. is he price of in erms of money, he amoun of money available a. Why should agens be ineresed in holding money? In Wal Disney's famous comics he rich uncle Scrooge Dagober Duck likes o boas his own wealh by bahing in piles of cash. Bu usually, we don enjoy soring piles of banknoes jus o show off. Raher we are willing o hold money (cash only because of is abiliy o buy goods. oney provides only indirec uiliy ou of he consumpion of real goods people can afford by paying cash. Agens need o pay wih cash for ransacions if i is oo cosly o verify he crediworhiness for anonymous counerparies. So money is a way o sore informaion abou he abiliy o pay i saves on informaion coss. The ash in advance onsrain A naural (even hough raher ad hoc way o give money a role in his economy is o impose he ash in advance consrain. We assume ha for some subse of goods X (he so called cash goods, cash is needed if you wan o buy hem. In conras, oher goods (le us call hem credi goods may be bough wihou money. So we now have o ake ino accoun he consrain X : cash goods X canno be purchased unless paid via money. In a represenaive agen economy, o impose such a condiion seems o be prey silly if he represenaive agen consumes jus wha he produces, here is definiely no need o use money. Bu he single agen model should no be aken lierally. Raher, i is a sylised way o represen a producion economy wih many heerogenous agens consuming an enormous variey of goods which are all produced by specialised firm using specialised inpus. [All goods can be produced using he same linear echnology =X =N ]. I is sraighforward o model his explicily (The mos elegan approach is he seup of Blanchard/Kiyoaki, wih Dixi/Sigliz preferences see he Focus Box o be wrien. Here, we simply use X as composie aggregae good, consising of a huge (possibly infinie variey of differen ypes of cash goods. Each single household produces jus one specific ype, bu consumes he whole specrum of all ypes. In order o buy heses ypes, he needs o pay wih cash. In each period, he represenaive agen wans o consume boh cash and credi goods according o preferences: (, X Wha is he impac of he cash in advance consrain on he economy? We have o be precise abou he exac iming convenions. We impose he following iming sequence: A he end of each period, (- agens decide abou how much wealh o hold for he nex period. Wealh can be held eiher in form of bonds or as cash. Holding cash does no earn ineres, bu i allows he consumpion of cash goods in he nex period: X. In conras, he agen earns a nominal yield i from he wealh invesed in ineres bearing bonds: (+i B. So a he beginning of period, he agen sars wih iniial wealh W = ( + i B + accumulaed during he las period. Obviously, money is dominaed as a sore of wealh by bonds as long as i >. Tha means households ry o economise on holding cash: They will no hold more han he amoun needed for he ransacions o buy he desired cash goods X. ahemaically, he inequaliy consrain will be binding: X = (, X = (, = (, m

11 Noe ha real money balances - wrien as m = / - now show up in he (indirec uiliy funcion, even hough money does no provide uiliy iself. The indirec uiliy of money holding has been derived here from consuming cash goods purchased by holding money. uing oney in he uiliy funcion, however, is a much more general approach which can be moivaed in differen ways. An alernaive sory, for insance, yielding he same formalisaion, is he following: oney saves on ransacion coss (he shopping ime required o purchase goods. In his approach, money serves as an inermediae inpu for producing consumpion, capured by a ransacion echnology. The Focus Box Shopping ime model xx shows how o derive an indirec uiliy funcion for real money balances from he shopping ime model. The higher he nominal price level, he lower are he real money balances, so he lower he benefi from holding a nominal amoun of money. In his book, we will assume ha preferences per period can be represened by an addiive separable payoff funcion:, m = ( + V ( m ( V ( m represens he indirec uiliy of money holding. Given wealh W, he agen decides how much credi goods o consume during period. A he same ime, he decides how much wealh o accumulae for he nex period eiher in form of cash or bonds. So his per period budge consrain is: + B = Y + W I is insrucive o rewrie he budge consrain in erms of wealh, eliminaing bond holding B : Since wealh accumulaed for he nex period is W + = ( + i + B + + +, solving for B + gives B + = [ W + + ]. Insering his relaion, we can formulae he per period + i + budge consrain as difference equaion: i + W = Y + W. + i + + i + Or, solving for W : W = Y + [ W + + i + + ] + i + Solving forward for he nex period in he same way, he evoluion of wealh across wo consecuive periods is characerised by he following difference equaion: W = + i Y + W + + i + Via ieraed subsiuion, we can successively merge he evoluion of wealh across periods ino one single wealh consrain. Saring in period, ieraed subsiuion gives: wih Q = j= + i j W + Q ( Y = = = Q ( + i + lim Q W and Q = as he compound nominal discoun facor for he ime beween period and period. Noe ha Q + / Q =.The erm Q looks a bi frighening, bu i is + i + jus a generalisaion of he sandard discoun facor allowing for variable ineres raes. For a consan ineres rae i, we have Q = + i wih Q = = + i

12 Nobody is allowed o accumulae so much deb during life such ha he presen value of deb sill held a he end of he world can be posiive. So le us impose he No onzi game condiion lim Q W. This condiion saes ha he presen value of he wealh held a he end of he world canno be negaive. Noe ha we ignore uncerainy in our analysis. If we allow for risk, an equivalen condiion would hold in expeced value erms! Since you are always beer off o consume a some sage raher han o leave posiive wealh a he end of he world, he No-onzi game consrain will be sricly binding when solving he opimisaion problem. Anicipaing his fac, we impose he condiion lim Q W =. This condiion is called he ransversaliy condiion in dynamic opimisaion. sing he ransversaliy condiion, he wealh consrain is capured by he following inuiive equaion: W + Q Y = Q ( + i = = Holding money affecs he agen s wealh consrain in a sraighforward way: for he amoun of wealh held in he form money, he agen foregoes nominal ineres paymens. Naurally he opporuniy cos for holding money insead of bonds is i, as long as money provides no posiive reurn. In a world wih inflaion, he real reurn on money is even negaive: i is equal o minus he rae of inflaion. m Summary: The money in he uiliy approach (I Le denoe real money balances a he beginning of period (money held before consumpion and producion. Then, he opimizaion problem is: ( ax ( c, m = ( + ρ s.. (2 W = ( + i B + (3 + B + + = Y + W +. sing ieraed subsiuion of (2 and (3, he iner-emporal budge consrains for each collapse o he single wealh consrain: W + Q ( Y = = = Q ( + i + limq W wih nominal discoun facor Q = and Q = = + i + i Because of he ransversaliy condiion (4 lim Q W =, he wealh consrain boils down o: ( + i = = (5 W + Q Y = Q aximising ( subjec o (5 gives as firs order condiions: ( + i + + i + + E Euler equaion: = = wih = + π + ( ρ ( + ρ ( + π + The Euler equaion characerises he familiar iner-emporal opimaliy condiion. m ( m oney demand equaion: = i ( c 2

13 According o he money demand equaion, money will be held such ha he raio of marginal uiliies beween money and credi goods is equal o he opporuniy cos of holding money he nominal rae of ineres! The higher i (he higher he rae of inflaion, he less money is held To gain deeper inuiion, le us derive he explici soluion firs for log uiliy (=:, X = ln( + ln( m. Wih logarihmic preferences, we ge as Euler equaion ( + r + + / = and as money demand funcion: = + ρ i ore generally, for he uiliy funcion wih consan elasiciy of subsiuion: (, = α + ( α The firs order condiion give as micro-based version of he L curve: ( i = α α The Fisher equaion: There is a sraighforward relaion beween he real rae of ineres r and he nominal rae of ineres i : i = r :+ π I is called he Fisher equaion, since Irving Fisher emphasised his relaion. In our model, i is obvious ha his condiion has o hold. Assume ha he consumer can choose beween buying real or nominal bonds. Nominal bonds pay some fixed ineres rae in nominal erms. Real bonds are indexed bonds, wih paymens adjused for he rae of inflaion. Therefore, he real rae of reurn is known in advance, whereas he nominal reurn varies wih he rae of inflaion realized over he life of he bond. Hence, neiher he purchaser nor he issuer faces any risk ha an unanicipaed increase or decrease in inflaion will erode or boos he purchasing power of he bond s paymens. Indexed bonds insure agains inflaion risk. 3

14 Focus: Inflaion linked Bonds S: TIS Treasury Inflaion-roeced Securiies. These bonds are mainly issued by governmens. The world's firs known inflaion-indexed bonds were issued by he ommonwealh of assachuses in 78 during he Revoluionary War. These bonds were invened o deal wih severe warime inflaion and wih angry disconen among soldiers in he.s. In modern imes, he Briish governmen began o issue inflaion-linked Gils (gil-edged securiies in 98 afer an exended period of high inflaion. In 28, he worldwide marke for inflaion-linked bonds comprises over $.5 rillion of he inernaional deb marke. This sounds like an enormous sum, bu i is only a small share of he oal amoun of ousanding bonds. The.S. Treasury inroduced inflaion-indexed noes firs in January 997. Economiss had urged he reasury o issue his insrumen as useful financial innovaion for quie some ime. A ha ime, however, many observers have been raher skepical abou he demand for such kind of bonds. They argued ha no many invesors will be ineresed in buying such bonds, so marke will no be liquid. Some cynics even suggesed, in order o save cos, he reasury should send he ads for hese new securiies jus o he members of he American Economic Associaion. Indeed, in 999 hese bonds accouned for less han 2 percen of he markeable federal governmen deb ousanding. hp:// Economis: Long-daed inflaion-linked bonds are probably he bes mach for pension liabiliies and demand for hem is so high ha, in Briain, real yields are remarkably low. hp:// During he Weimar Republic he Reichsbank, hen Germany s cenral bank, played his game for real and sensaionally los conrol. The accepance of consanly rising prices, known as inflaionary expecaions, seized he minds of he people and he enire world of commerce. Indexing prices o inflaion only made hings worse: i reached,24% in 922, which was jus abou bearable, bu galloped o 5,7,,% during 923. aper money became he cheapes form of heaing fuel. By Ocober 923, 4.2 billion Reichsmarks could (or migh have bough you a dollar. Anyone who had suppored Germany s effor in he firs world war by buying governmen bonds go almos nohing in reurn when a new currency, he Renenmark, was inroduced in November. Bu inflaion was haled. Tha is he background which promped he 948 currency law o forbid indexaion clauses wihou he Bundesbank s permission. This permission was always refused, excep for some long-erm renal agreemens. Due o he resisance of he Bundesbank, i ook unil arch 26 o inroduce he firs indexed governmen bonds. The issue of 5.5 billion of bonds was linked o he harmonised index of consumer prices in he euro zone (excluding obacco. No many German invesors were ineresed: abou 6% of he bonds were sold naionally; mos were placed elsewhere in Europe, especially France. Bu i s a sar. Gerhard Schleif, co-head of he German Finance Agency which handles governmen borrowing, inends o add anoher 5 billion- billion o he same issue over he nex wo years. While ineres raes and inflaionary expecaions are low his is he cheapes way for governmens o borrow money. ar 4h 26 hp:// 4

15 Le us use he specificaion = ( i (wih α=/2 o ry o undersand how moneary policy affecs he equilibrium oucome. In his chaper, we assume ha all prices are perfecly flexible even in he shor run. To make i simple, consider he case ha real endowmen Y in he economy is growing a a consan rae g. If so, in a seady sae, real consumpion mus n also grow a he same rae g. So for all, / = Y / Y = g = [ r ρ]. Obviously, in his economy he naural rae of ineres is consan across ime: r n = ρ + g. oneary policy will be neural and super-neural in he sense ha i does no affec he real rae of growh of producion. I will affec, however, he well being of consumers: A higher nominal rae of ineres forces consumers o economise on money balances; so i will raise ransacion cos for shopping ha is, i shifs consumpion away from cash o credi goods. A posiive nominal rae of ineres works like a ax on money holding, inroducing inefficiencies ino he economy. The inefficiency is lower he lower he ineres rae. Since producion of fia money is nearly cosless, a policy rying o minimize he inefficiencies would ry o implemen a nominal ineres close o i would saiae consumers wih money holding. Such a policy has firs been recommended by ilon Friedman 968 as Opimal Quaniy of oney. Equivalenly, he cenral bank could pay nominal ineres on money balances a a rae equal o he marke nominal ineres rae. aying ineres on reserves held by commercial banks is nowadays sandard for many cenral banks. Bu i would be adminisraively exremely cumbersome o pay ineres on cash. In he same way, i is difficul o implemen a negaive ineres rae on cash as has been suggesed by Silvio Gesell in his proposal of Schwundgeld, aken up in he discussion of zero bound on nominal ineres by Goodfriend and Buier and anigirzoglou (2. A posiive nominal rae of ineres (or, more generally, a lower ineres rae on money compared o bonds provides seigniorage income for he governmen. Le k= /Y be he amoun of non-ineres earning money held by he privae secor as share of nominal GD. Then, i k is he seigniorage revenue for he governmen (see figure xxx6. A i, people are willing o hold k, creaing seigniorage income i k. Wih a lower ineres rae i people increase money balances up o k generaing seigniorage i k. So on he one hand, seigniorage is reduced by (i -i k - he blue shaded recangular A. On he oher hand, he increase in money holding raises seigniorage by he area E i (k -k. Since a lower ineres rae raises consumers welfare by he area A+B, oal welfare (as sum of consumers and producers (governmen s surplus is definiely higher, he lower he ineres rae i. Figure xx 6 Friedman s opimal quaniy of money 5

16 Since ax revernue is essenial for financing governmen aciviies, he deadweigh loss creaed from ineres raes has o be raded off agains he deadweigh loss caused by oher, more convenional axes. Following he heory on opimal axaion, Edmond helps argued ha i is opimal o use boh menas of axaion (see Illing 997, page 64ff. How is he equilibrium nominal rae of ineres deermined? Figure xx6 may give he impression ha his is prey easy: Since he cenral bank is he monopoly producer of money, he ask seems o be simply o fix eiher he money supply or he nominal ineres rae. Equilibrium on he money marke will hen be realised a he inersecion of supply and demand: = k( i Y Bu hings are much more ricky, since we need o deermine no jus i, bu also he iniial price level.. Wih money demand being ineres rae elasic, he iniial price level and he nominal ineres rae depend on he whole expeced fuure pah of moneary policy from now on unil infiniy. I is simply no sufficien jus o se he curren ineres rae, equilibrium is inherenly forward looking. The reason is fairly inuiive: According o he Fisher equaion, e he nominal rae of ineres is he sum of he real rae and expeced inflaion: i = r + π. So seing he nominal rae boils down o pursue a credible inflaion sraegy: The higher he expeced fuure rae of inflaion, he higher he nominal ineres rae. In a forward looking, non-sochasic raional expecaions equilibrium, he expeced rae of inflaion has o be equal o he realised rae of inflaion. In an economy wih perfecly flexible prices, he cenral bank canno affec he real rae of ineres [This is no longer rue when a leas some prices are sicky. If so, he cenral bank can seer he shor run real rae of ineres by changing he nominal rae, as long as his policy has no adverse impac on inflaionary expecaions!]. So i can only affec he nominal rae via commiing o some specific pah for he rae of inflaion by choosing some desired rae of inflaion π*. Le us discuss wo differen sraegies o implemen some arge π* as a seady sae policy Sraegy of consan money growh In he micro-based L curve derived above le us impose he equilibrium condiions: S = Y ; =. A each, equilibrium on he money marke is characerised by = Y ( i ; Wha pah of money supply is required o implemen some seady sae i + π *? How is he. iniial price level deermined by such a policy? Since expecaions abou fuure moneary policy have an impac on curren money holding via π e, we need o look a a dynamic ineremporal relaion: real money balances depend on he fuure expeced price pah. For his reason, self fulfilling bubbles for he price pah may be possible. As long as we rule ou such bubble pahs wih self fulfilling inflaionary expecaions, we will ge a unique equilibrium wih a policy of consan growh rae of he money supply. Laer, however, in chaper xx, we will see ha condiions for a unique seady sae equilibrium are quie ricky. Self fulfilling inflaionary expecaions ( speculaive hyperinflaion may make money useless as a medium of exchange wihou violaing aggregae resource consrains! oney has o be essenial o rule ou such a price pah! onsider now a policy of consan money growh + = ( + µ and assume ha i will resul in some consan seady rae of inflaion π, wih seady sae nominal ineres i = r n + π = r n 6

17 + Y + We have = + µ ; = + g Y and i = = = + π + i i rn. In each period, money marke equilibrium gives: = ( ; = + ( i + i+. Dividing Y Y + Y i Y + µ = Y µ i = Y + = + g So = + π = π* = µ g + g Given some (and so π, he iniial price level. is now uniquely deermined. Wha happens if a some sage, unexpecedly, he rae of growh of money supply suddenly reduced o a lower level which is expeced o say a ha level forever? Wih a lower money growh rae, fuure inflaion will be reduced. This lowers he nominal ineres rae and makes money holding more aracive. olicy of a pure ineres rae peg (Sargen/Wallace Fix nominal ineres rae: { i = i + u} wih E ( u = oney supply is endogenous a All real variables are well specified: Real balances are uniquely deermined by: V m ( m = i b Rae of inflaion is uniquely deermined (Fisher equaion: E( p p = E( π = E( i r + No speculaive bubble for he price pah. Bu: Iniial price level is no specified - here is nominal indeerminacy: If m (wih is a soluion o V m ( m = i, hen also any λ m (wih λ. The difference equaion: [ E ( p p ] = i - r = consan is no sufficien o pin down he iniial price level: + If is a soluion (wih, hen also λ wih λ Key Lesson: ure ineres rae peg leaves price level indeerminae. In conras, feedback ineres rae rules are able o conrol price level Ineres Rae Feedback Rules Example: Inflaion Targeing/ or rice Level Targeing Announced (redible enral Bank Rule: i = i + f ( π π* + u nder wha condiions do we ge a deerminae price level? Taylor principle 7

18 *Opional aerial Focus Box for Nerds: ash goods in a ash in advance model Formally, he cash-in-advance model and he shopping ime model can be seen as special cases of he I approach. Take he IA model: ax ( x, l s.. = ( + ρ s.. ( W = ( + i B + (2. X + B = ( l + W (3 X For lim Q W = W + Q ( l = Q ( X + i = = p X for all X is a cash good which can be purchased only via cash. In conras, leisure l can be consumed wihou cash (i represens a credi good. The IA consrain drives a wedge beween consumpion of cash and credi goods: / X = + i. For i >, he consrain X becomes binding (giving a / l posiive demand for money. So in he objecive funcion, we can subsiue X = m = /. Thus, he opimizaion problem is formally equivalen o maximizing uiliy ou of real money balances and consumpion of credi goods. Similarly, he shopping ime model can be ransformed as a problem of maximizing he indirec uiliy funcion V x, m = ( f ( x, m ( Formally, as in he I approach, ( and (2 give (5, so he Lagrangian is: ( X, l + µ + = ( + ρ W + Q ( l Q ( X + i λ ( X = = = This gives he Euler equaion above and he equivalence of a money demand funcion. The demand for money is decreasing wih he ineres rae: Wih increasing i, less cash goods relaive o credi goods are consumed. The requiremen o pay via cash forces people o hold money one period in advance, so he effecive price of consumpion for cash goods increases o + i. Obviously, he cash in advance consrain inroduces a disorion beween he effecive price of consumpion and of producion for cash goods. Figure xx Disorion beween cash and credi goods 8

19 *Opional aerial Focus Box for Nerds:Shopping ime model A represenaive agen has sandard preferences defined on real consumpion: ax ( X, l = ( + ρ The basic idea of he shopping ime model is ha money reduces ransacion coss (shopping ime required for purchasing goods. In a sense, money serves as an inermediae inpu for producing consumpion. This can be capured by a ransacion echnology T, m wih T > ; < c T m (, m ( T represens he real resources used up when is consumed and m are real money balances available. The higher he sock of real money balances, he lower hese resources (money allows savings on ransacion coss. So he per period budge consrain has o be modified according o: i + l + T(, m ( Now, redefine gross consumpion as: X = + T, m. Since X is increasing in X, his ( funcion can be invered o = f ( X, m. Thus, we can reformulae he opimizaion problem via he indirec uiliy funcion: ( f ( X, m, l ( + ρ = s.. W + Q ( l = Q ( X + i = = This is formally equivalen o he I approach. In paricular, he special case wih exogenous labour supply N = gives a I funcion V X, m = ( f ( X, m excep Y ha now, he srucure of he indirec uiliy funcion ransacion echnology: X = + T, m. ( V ( ( X, m The FO for consumpion pah and real money balances are: ( c X + i + = + ( + ρ ( + π ( + c + X + ( c m m 2 : = = i ( c x X is well specified by he 9