INSTRUMENTS OF MONETARY POLICY*

Aricles INSTRUMENTS OF MONETARY POLICY* Bernardino Adão** Isabel Correia** Pedro Teles**. INTRODUCTION A classic quesion in moneary economics is wheher he ineres rae or he money supply is he beer insrumen of moneary policy. Unil recenly pracice and heory seemed o be in disagreemen. Mos will agree ha moneary policy decision making has focused on seing a arge for he shor-erm ineres rae. However, mos heoreical work has considered he moneary policy as being a choice abou he rajecory of he money supply. One hing ha is frequen in all he lieraure is ha he moneary policy is no specified in sufficien deail. If he ineres rae is he chosen insrumen i is no described how he associaed money supply is deermined or vice versa; if he money supply is he insrumen i is no explained how he ineres rae is deermined. I is confirmed boh heoreically and empirically ha he demand for real money depends on he nominal ineres rae and on he real oupu level. Thus, unless boh he real oupu level as well as he price level are fixed, seing he nominal ineres rae is no equivalen o argeing a moneary aggregae. And vice-versa, fixing money is no equivalen o fixing he nominal ineres rae. * The views expressed in his aricle are hose of he auhors and no necessarily hose of he Banco de Porugal. This paper is based on our recen research, he main references being Adão, Correia and Teles, (2003) and (2004). This paper benefied from commens by Mara Abreu, José Brandão de Brio, José Anónio Machado, Maximiano Pinheiro and Carlos Robalo. ** Economic Research Deparmen. There are ad-hoc models where here is jus one moneary insrumen. For insance, he obsolee saic IS-LM model wih fixed prices has only one insrumen. The IS curve is he se of nominal ineres raes and oupu levels for which he good marke is in equilibrium when he supply of he good is demand deermined. The LM curve is he se of nominal ineres raes and oupu levels for which he money marke is in equilibrium. Thus, given he money supply he inersecion of he IS and he LM deermine he oupu and he nominal ineres rae. And insead, given he nominal ineres rae he IS deermines he real oupu, and given he nominal ineres rae and he oupu he LM deermines he money supply. By conras, his paper considers a sandard dynamic macroeconomic model wih microeconomic foundaions. The main resul is ha in order o obain a unique equilibrium, ha is, well defined rajecories for variables like inflaion and oupu, he cenral bank should use boh he money supply and he ineres rae as insrumens. This is a sufficiency resul as i is known ha in some paricular non-robus frameworks (), uniqueness may be obained wih less insrumens. The res of he paper is se ou as follows: secion 2 describes he lieraure. Secion 3 porrays he model. Secion 4 shows ha he Taylor principle guaranees local deerminacy bu no unique- () For insance endowmen economies wih separable logarihmic uiliy funcions in consumpion and real balances, wih converibiliy of money and no public deb. See Obsfeld and Rogoff (983). Banco de Porugal / Economic bullein / June 2004 29

Aricles ness of he equilibrium in he deerminisic version of he model. Secion 5 reveals which policy variables need o be used as insrumens in order o have uniqueness of he equilibrium in he sochasic version of he model. Secion 6 concludes. The appendix exends he resuls of secion 4 o he sochasic framework. 2. THE LITERATURE This secion provides a brief descripion of he main conribuions o he lieraure on he moneary insrumen choice problem. The earlies noorious effor was by Friedman (968), who argues agains he use of he ineres rae as an insrumen. His concern was ha if agens have irraional expecaions abou inflaion, he economy would no converge o he raional expecaions equilibrium. No maer wha nominal ineres rae he cenral bank would choose, if people expeced inflaion above he raional expecaions equilibrium, ha would resul in lower perceived real ineres rae, which would generae a higher demand for curren goods, leading o an even higher inflaion, which in urn would lead o an even lower real ineres rae, simulaing more he economy, and so on wihou bound. Unlike Friedman (968), in he recen lieraure agens are aken as being raional. The deparing poin has been ha he insrumen mus be able o generae local deerminacy of he equilibrium. Local deerminacy means ha in he neighbourhood of an equilibrium here are no oher equilibria. However, in general besides his locally deermined equilibrium here is an infiniy of oher equilibria ha canno be ruled ou. I is very inriguing ha all he lieraure as been saisfied wih his local deerminacy propery. To us he mulipliciy of equilibria is a disurbing resul. For i implies ha he same economic fundamenals are compaible wih many values for he macroeconomic variables. Random evens compleely unrelaed o he fundamenals, sunspos, can cause large flucuaions of he oupu and inflaion. From he view poin of he cenral bank his is undesirable, since usually is objecive is o promoe oupu and inflaion sabilizaion. In his lieraure of local deerminacy here have been a few very influenial papers. Sargen and Wallace (975) shows ha ineres rae rules ha depend only on exogenous variables do no guaranee local deerminacy and defend insead he use of he money supply as he insrumen. Mc Callum (98) shows ha if insead, he cenral bank chooses ineres rae rules ha depend on endogenous variables he Sargen and Wallace resul does no apply necessarily. The classic Taylor rule, Taylor (993), is one such example, seing he ineres rae as a funcion of he curren esimaes of he oupu gap and inflaion. Recenly he mos forceful defence of he use of he ineres rae as he insrumen is Woodford s influenial book, Woodford (2003). In his paper we presen he concep of equilibrium in a sochasic environmen. We show ha in general if he moneary auhoriy uses jus one insrumen, no maer which, here will be a large mulipliciy of equilibria. As a corollary, we ge ha here is an infinie number of equilibria when he moneary auhoriy uses only one insrumen, even if i guaranees local uniqueness. 3. MODEL We consider a cash in advance economy. The economy consiss of a represenaive household, a represenaive firm behaving compeiively, and a governmen. Producion uses labour according o a linear echnology. This environmen is he simples o sudy he insrumens of moneary policy. More complex models deliver similar resuls, as long as agens ake decisions for a leas wo periods. We consider shocks o echnology A and governmen expendiures G. The period vecor of shocks is denoed as s( A, G). The se of all possible shocks in period is denoed by S, he hisory of hese shocks up o period, which we call sae a, s 0, s,..., s, is denoed by s, and he se of all possible saes in period is denoed by S. The iniial realizaion s 0 is given. To simplify he exposiion, we assume ha he hisory of shocks has a discree disribuion. The number of all possible shocks in period is # S and he number of all possible saes in period is # S. An example may help clarify he erminology. Suppose ha G is a consan, i.e. G G for all and A for all can assume only 3 values: a high, A h, a medium, A m or a low value, A l. For each period, he number of possible shocks is 30 Banco de Porugal / Economic bullein / June 2004

Aricles =0 Char STATES = =2 Consumpion mus be purchased wih money according o he cash in advance consrain PC M (3) s 0 A h A m (A h,a h ) (A h,a m ) (A h,a l ) (A m,a h ) (A m,a m ) (A m,a l ) A he end of he period, he households receive he labour income WN where NL is labour and W is he nominal wage rae and pay lump sum axes, T. Thus, he nominal wealh households bring o period is W M R B PC W N T (4) h m l 3, S A G A G A G,,,,,. Bu he number of possible saes is differen across periods. The number of possible saes in he following period is always bigger. In period 0 here is sae, he number of possible saes in period is 3, he number of possible saes in period 2 is 9 and so on. Char provides a graphical represenaion of his example. 3.. Compeiive equilibria Households The households have preferences over consumpion C, and leisure L. These wo variables as all variables in he economy, which we describe in deail below, are a funcion of s, bu o shoren he noaion insead of wriing down Cs we wrie C. The expeced uiliy funcion is: UE uc L 0,, 0, () 0 where is a discoun facor. The households sar period wih nominal wealh W. They decide o hold money, M, and o buy B nominal bonds ha pay RB one period laer. The R is he gross nominal ineres rae a dae. Thus, in he asses marke a he beginning of period hey face he consrain M A l B W (2) (A l,a h ) (A l,a m ) (A l,a l ) The households problem is o maximize expeced uiliy () subjec o he resricions (2), (3), (4), ogeher wih a no-ponzi games condiion on he holdings of asses (2). The following are firs order condiions of he households problem: u L W u P R C (5) u C uc RE (6) P P Condiion (5) ses he inraemporal marginal rae of subsiuion beween leisure and consumpion equal o he real wage adjused for he opporuniy cos of using money, R. Condiion (6) is an ineremporal marginal condiion necessary for he opimal choice of nominal bonds. I says ha he uiliy oday of an addiional uni of money mus be equal o he expeced uiliy omorrow of R addiional unis of money. Firms The firms are compeiive and prices are flexible. The producion funcion of he represenaive firm is linear Y A N. The equilibrium real wage is equal o he marginal produciviy of labour, (2) The implied consrain is ha he household mus hold a ne porfolio a he end of he period ha is larger in absolue value han he presen value of is fuure ne income. Banco de Porugal / Economic bullein / June 2004 3

Aricles Governmen W A. (7) P The policy variables are axes, T, ineres raes, R, money supplies, M, and public debs, B. The governmen chooses he policy, which is defined as he behaviour of some, bu no all of hese policy variables. The governmen canno choose he behaviour of all of he policy variables because, as we will see, here are equilibrium condiions ha ogeher wih he policy deermine endogenously he values for he remaining policy variables. A policy is a se of funcions, chosen by he governmen, ha map quaniies, prices and policy variables ino policy variables. One example of a policy is he Taylor rule, ha specifies he ineres rae as a funcion of inflaion and oupu. Anoher example of a policy is a consan growh money supply. The period by period governmen budge consrains are M B M R B PG PT,. (8) 0 A each sae s equaion (8) has an ineremporal counerpar ha esablishes ha he presen expeced value of he fuure seigniorage flows mus be equal o oday s governmen responsibiliies plus he presen expeced value of he fuure governmen defici flows. This sochasic ineremporal condiion can be wrien as a funcion of only he rajecories for consumpion, leisure and policy variables. Marke clearing Marke clearing in he goods and labour marke requires C G AN, L N. We have already imposed marke clearing in he money and deb markes. Equilibrium A compeiive equilibrium is a sequence of policy variables, quaniies and prices such ha he privae agens, households and firms, solve heir problem given he sequences of policy variables and prices, he budge consrain of he governmen is saisfied and markes clear. The equilibrium condiions for he 7 variables C, L, P, B, R, M, Tare 5. They include he resources consrain C G A ( L ), 0 (9) he inraemporal condiion ha is obained from subsiuing he households inraemporal condiion (5) ino he firms opimal condiion (7) u C R, u A 0 (0) L as well as he cash in advance consrain (3), he ineremporal condiion (6), and he governmen ineremporal budge consrain. These condiions define a se of equilibrium allocaions, prices and policy variables. The number of equaions a sae s is equal o 5. The number of equilibrium variables ha mus be deermined a sae s is equal o 7. If none of he policy variables is chosen exogenously, here is an infiniy of allocaions, prices and policy variables saisfying he 5 equilibrium condiions. Since here are less equilibrium equaions han equilibrium variables here are many equilibria unless he governmen chooses exogenously some of he policy variables. There can be equilibria wih high inflaion or low inflaion as here can be equilibria wih low oupu or high oupu. Anyhing is possible. On he oher hand, if all he policy variables, axes, money supplies, ineres raes and deb are chosen exogenously, here is no equilibrium. There are many ways in which he degrees of freedom can be fulfilled. As we are primarily ineresed in sudying moneary policy we assume ha he fiscal policy adjuss o saisfy he ineremporal governmen budge consrain. In oher words, we assume ha he fiscal policy is endogenous in he sense ha whaever are he choices of he moneary auhoriy, he fiscal insrumens, B andt, adjus o saisfy he ineremporal governmen budge consrain implied by (8). 32 Banco de Porugal / Economic bullein / June 2004

Aricles Now, he number of relevan variables is 5 and he number of relevan equaions 4, being one of hem, (6), a sochasic dynamic equaion. By couning equaions and unknowns, i would seem enough in order o ge deerminacy ha he governmen would have jus one moneary insrumen, as ha would be equivalen o adding o he remaining equilibrium condiions anoher condiion, which would resul in a sysem wih he same number of equaions as unknowns. Tha inuiion is wrong because one of he equaions, (6), is a sochasic dynamic equaion. If he environmen was deerminisic, (6) would be a firs order difference equaion and in order o ge a unique soluion i would be enough o have an iniial or erminal condiion. Because he environmen is sochasic, he number of condiions necessary o ge uniqueness is much larger as we will see below. In secion 5 we show ha in general by seing only a funcion for one of he moneary policy variables uniqueness of he equilibrium is no achieved. As we explain in secion 4, his implies ha by simply following an ineres rae rule, even if i guaranees local deerminacy, he moneary auhoriy is allowing an infinie number of equilibria, many of which can be associaed wih very high inflaion levels. 4. LOCAL DETERMINACY AND INTEREST RATE RULES The lieraure is currenly dominaed by a rule-based approach o moneary policy. According o he lieraure local deerminacy is among he mos desirable properies ha a rule mus possess. Local deerminacy means, as we said before, ha in he neighbourhood of an equilibrium here is no oher equilibrium. In his secion we clarify wha is mean by an ineres rae feedback rule guaraneeing local deerminacy and show ha for a sandard environmen local deerminacy is achieved if he Taylor principle is followed. Roughly speaking, he Taylor principle is verified if in response o an increase in inflaion he increase in he nominal ineres rae is higher. This secion is an excepion, as here, o simplify he exposiion we consider a deerminisic environmen, i.e. A A and G G for all and uc, L C v L. In he appendix we presen he more complex sochasic counerpar. Le R be he seady sae compeiive equilibrium for he ineres rae and le be he seady sae compeiive equilibrium for he inflaion rae. Then, R =, where is he real ineres rae. Assume ha he cenral bank conducs a pure curren nonlinear Taylor rule (3) : R R, P where (he Taylor principle), and. P Afer subsiuing he Taylor rule in (6) ge z z where z. By recursive subsiuion we ge z z k, for all k and. () There is no condiion o pin down he iniial value for inflaion. Since he iniial inflaion level can be any value here is an infiniy of equilibrium rajecories for he inflaion rae. Neverheless, hey can be ypified in 3 classes. Eiher inflaion is consan,, or here is an hyperinflaion,, or inflaion is approaching zero, 0. This is easy o verify. If 0 hen () implies ha for all. If 0 hen () implies ha and, since. If 0, hen () implies ha and 0 since. Thus, when he cenral bank follows a Taylor rule ha obeys he Taylor principle i is able o ge local deerminacy. In a neighbourhood of he seady sae inflaion here is no oher equilibrium inflaion rajecory. Bu we have jus seen ha here is an infiniy of oher equilibria for inflaion which converge o zero or o infiniy. These resuls beg wo inerrelaed quesions: Why is local deerminacy such an ineresing propery? Or why has mos of he lieraure assumed ha undesirable equilibria do no happen? We do no know he answer o hese quesions. (3) Usually he Taylor rule is presened in is linearized form. As can be verified he linearized version is, R R =. Banco de Porugal / Economic bullein / June 2004 33

Aricles There may be insiuions ha we have ignored in he model, which can be used o eliminae some of hese undesirable equilibria. For insance, in some models an hyperinflaion can be eliminaed if he cenral bank has sufficien resources and can commi o buy back is currency if he price level exceeds a cerain level. We are no going o pursue his issue here. Those readers ineresed in his opic should sar by seeing he seminal paper of Obsfeld and Rogoff (983). In general, here are sill an infiniy of equilibria ha pass hese ypes of ess. I is easy o verify, using an argumen similar o he one above, ha if he Taylor rule did no obey he Taylor principle, i.e., here would be jus wo ypes of equilibrium. The seady sae and an infiniy of equilibria converging o he seady sae. A firs sigh i would seem ha i would be preferable ha a cenral bank would follow a Taylor rule ha did no saisfy he Taylor principle, as undesirable equilibria, hyperinflaions or hyperdeflaions would no be possible. This conclusion is no correc because whenever here is mulipliciy of equilibria i may be possible ha sunspos can cause large flucuaions in inflaion. Inflaion can flucuae randomly jus because agens come o believe his will happen. The ineresed readers should sar wih Farmer (993). 5. EXOGENOUS POLICY INSTRUMENTS We are ineresed in idenifying wha are he exogenous insrumens of policy ha guaranee ha here is a unique equilibrium for allocaions and prices. This provides a measure of degrees of freedom in conducing policy. This is a quesion of policy relevance. As menioned above, i is associaed wih he insrumen problem in moneary economics on wheher o use he ineres rae or he money supply as he moneary policy insrumen. Under very general condiions he sysem of equaions defining he equilibrium can be summarized by,, u C R L R M CR C R E u C R L R M CR where CR and, C, 0 (2) LR mean ha consumpion and leisure depend only on he level of he ineres rae. 5.. Conducing policy wih consan funcions In his subsecion, we show ha in general when policy is conduced wih consan funcions for he policy insrumens, i is necessary o deermine exogenously boh ineres raes and money supplies. Suppose he pah of money supply is se exogenously in every dae and sae. In addiion, in period zero he ineres rae, R 0, is se exogenously and, for each, for each sae s, he ineres raes are se exogenously in # S saes ha follow. In his case (2) a dae 0 would deermine he R in he remaining sae, since # S of he R s were already given. The usage of (2) for he oher daes would deermine recursively all he R s ha were no se exogenously. Thus, here is a single soluion for he allocaions and prices. Similarly, here is also a unique equilibrium if he nominal ineres rae is se exogenously in every dae and sae, and he money supply is se exogenously in period 0, as well as, for each and sae s, in he # S saes ha follow. Thus, we have he following resul when policy is conduced wih consan funcions: in general, if money supply is deermined exogenously in every dae and sae, and if ineres raes are also deermined exogenously in he iniial period, as well as in # S # S saes for each, hen he allocaions and prices can be deermined uniquely, similarly, if he exogenous policy insrumens are he ineres raes in every sae, he iniial money supply and he money supply, in # S # S saes, for, hen here is in general a unique equilibrium. Char 2 illusraes his resul for he example of secion 3. For insance, a unique equilibrium can 34 Banco de Porugal / Economic bullein / June 2004

Aricles s 0 =0 endogenous Char 2 INSTRUMENTS = =2 A h A m A l (A h,a h ) (A h,a m ) (A h,a l ) (A m,a h ) (A m,a m ) (A m,a l ) (A l,a h ) (A l,a m ) (A l,a l ) endogenous M 0. We can use he argumen used before. A any sae s, given M and R here is one equaion (2) ha relaes s wih period, and # S equaions for he subsequen R s, which are implied by he feedback rule. Thus, o obain he # S values of he R s and he # S values of he M s, he moneary auhoriy needs o se # S values for he M s. In general, a similar resul holds if he moneary policy is conduced wih money feedback rules. When he moneary policy is conduced wih a money feedback rule in order o have a unique equilibrium, i is necessary o deermine exogenously he ineres rae in # S saes, for each sae s,, as well as R 0. be guaraneed if for he saes wih a circle one of he insrumens, be i he money supply or he ineres rae, is deermined endogenously by (2) and in he remaining saes money supply and ineres rae are exogenous (4). 5.2. Conducing policy wih feedback rules I is commonly assumed ha policy is conduced wih feedback rules, in paricular, ineres rae feedback rules. In his subsecion, we argue ha he resuls of he previous secion do no change if insead he moneary policy is conduced wih feedback rules for he policy insrumens insead of consan funcions. The use of ineres rae rules ha depend on curren or pas variables (hese are he ype of rules ha guaranee local deerminacy) preserves he same degrees of freedom in he deerminaion of he equilibrium. I is sill necessary o deermine exogenously he levels of money supply in some of he saes. When he policy is conduced wih curren or backward ineres rae feedback rules in order o have a unique equilibrium, i is necessary o deermine exogenously he money supply in # S saes, for each sae s,, as well as (4) If insead, axes were exogenous, a single moneary insrumen would be enough o ge a unique equilibrium. For insance if he cenral bank se exogenously he ineres rae and he fiscal auhoriy se axes exogenously, he price level would be deermined by he governmen ineremporal budge consrain. This resul is known as he fiscal heory of he price level. See Woodford (2003). 6. CONCLUSION Under he assumpion ha he fiscal policy was endogenous, a moneary policy ha uses jus one moneary policy insrumen, eiher he nominal ineres rae or he money supply, is no able o eliminae he mulipliciy of equilibria. In paricular, a Taylor rule ha obeys he Taylor principle generaes local deerminacy. Bu local deerminacy is sill consisen wih an infiniy of equilibria. Any level of inflaion can be an equilibrium. Since mos cenral banks have he sabilizaion of inflaion as heir main objecive i is crucial o know how a unique equilibrium for inflaion can be achieved. To obain uniqueness of he equilibria, i is sufficien for he cenral bank o use is wo insrumens simulaneously. The cenral bank mus choose ineres raes and money supplies concurrenly. REFERENCES Adão, Bernardino, Isabel Correia and Pedro Teles, 2003, Gaps and Triangles, Review of Economic Sudies, 70, p. 699-73. Adão, Bernardino, Isabel Correia and Pedro Teles, 2004, Insrumens of Moneary Policy, mimeo, Federal Reserve Bank of Chicago. Farmer, Roger, 993, The Macroeconomics of Self-Fulfilling Prophecies", MIT Press. Friedman, Milon, 968, The Role of Moneary Policy, American Economic Review, 58, -7. McCallum, Benne, 98, Price Level Deerminacy wih Ineres Rae Policy Rule and Ra- Banco de Porugal / Economic bullein / June 2004 35

Aricles ional Expecaions, Journal of Moneary Economics, 8, 39-329. Obsfeld, Maurice and Kenneh Rogoff, 983, Speculaive Hyperinflaions in Maximizing Models: Can We Rule Them Ou, Journal of Poliical Economy, 9, 675-687. Sargen, T. J. and Neil Wallace, 975, Raional Expecaions, he Opimal Moneary Insrumen, and he Opimal Money Supply Rule, Journal of Poliical Economy, 83, p. 24-254. Taylor, John B., 993, Discreion Versus Policy Rules in Pracice, Carnegie-Rocheser Conference Series on Public Policy 39, p. 95-24. Woodford, Michael, 2003, Ineres and Prices, Princeon Universiy. 36 Banco de Porugal / Economic bullein / June 2004

Aricles APPENDIX In he appendix we sudy local deerminacy in he sochasic environmen. The inroducion of he concep of he ime-invarian equilibrium is necessary o sudy local deerminacy. In order o proceed an assumpion is made, for each sae s, he shocks A, G have an idenical and independen disribuion. The ime-invarian equilibrium is a compeiive equilibrium wih he propery ha i is jus a funcion of he shock. Formally, he ime-invarian equilibrium is a uple for consumpion, leisure, ineres rae, money growh and Ms inflaion, Cs, Ls, Rs,, ha saisfies he relevan compeiive equilibrium condi- Ms ions. These condiions are given by (3), (9), (0) and (2), C( s) M( s ) C( s ) M( s ) s, C(s ) GA L( s), u u C L s s R, A u s s Eu s C C R. (3) For a given Rs he wo middle equaions deermine Cs and Ls. Given he firs equaion deermines he growh rae of money beween a sae and any of is subsequen saes. Finally (3) deermines Rs. To economize on noaion we now assume wihou loss of generaliy ha he uiliy funcion is separable and linear in consumpion. In his case (3) can be wrien as R Tha is he ime-invarian nominal ineres rae does no depend on he shocks. Suppose ha he cenral bank conducs a pure curren Taylor rule: R R (4) P where (he Taylor principle), and. P Afer subsiuing (4) in he households ineremporal condiion, (3), we ge E z z (5) where z. By recursive subsiuion we ge E E... E z k k... k,. z, for all (6) In he following paragraph we supply an heurisic proof ha he only equilibria are he ime-invarian equilibrium and an infiniy of oher equilibria which have he characerisic ha in some saes of naure eiher inflaion is going o infiniy or is going o zero. Since if z hen z wih posiive probabiliy. The proof is by conradicion. Assume i was no converging o infiniy wih posiive probabiliy, hen i would be bounded wih probabiliy one, which means ha no maer how arbirary in he fuure you ake he z s is expeced value would be bounded wih probabiliy one. Bu since he exponen is a consan smaller han one by aking s sufficienly large will ge he lef hand side of (6) smaller han he righ hand side. By a similar argumen if z, have z 0 wih posiive probabiliy. Thus, when he cenral bank follows a Taylor rule ha obeys he Taylor principle i is able o ge local deerminacy. In a neighbourhood of he ime-invarian equilibrium inflaion here is no oher equilibrium. We have jus seen ha he oher equilibria which are infinie in number are eiher associaed wih inflaion converging wih probabiliy bounded from zero o infiniy or o zero. Banco de Porugal / Economic bullein / June 2004 37