Optimal monetary policy rules under persistent shocks

Transcription

1 Economics Working Papers (00 06) Economics Optimal monetary policy rules under persistent shocks Joydeep Bhattacharya Ioa State University, Rajesh Singh Ioa State University, Follo this and additional orks at: Part of the Economics Commons Recommended Citation Bhattacharya, Joydeep and Singh, Rajesh, "Optimal monetary policy rules under persistent shocks" (008). Economics Working Papers (00 06) This Working Paper is brought to you for free and open access by the Economics at Ioa State University Digital Repository. It has been accepted for inclusion in Economics Working Papers (00 06) by an authorized administrator of Ioa State University Digital Repository. For more information, please contact

2 Optimal monetary policy rules under persistent shocks Abstract The tug-o-ar for supremacy beteen inflation targeting and monetary targeting is a classic yet timely topic in monetary economics. In this paper, e revisit this question ithin the context of a pure-exchange overlapping generations model of money here spatial separation and random relocation create an endogenous demand for money. We distinguish beteen shocks to real output and shocks to the real interest rate. Both shocks are assumed to follo AR() processes. Irrespective of the nature of shocks, the optimal inflation target is alays positive. Under monetary targeting, shocks to endoment require negative money groth rates, hile under shocks to real interest rates it may be either positive or negative depending on the elasticity of consumption substitution. Also, monetary targeting elfare-dominates inflation targeting but the gap beteen the to vanishes as the shock process approaches a random alk. In sharp contrast, for shocks to the real interest rate, e prove that monetary targeting and inflation targeting are elfare-equivalent only in the limit hen the shocks become i.i.d.! The upshot is that persistence of the underlying fundamental uncertainty matters: depending on the nature of the shock, policy responses can either be more or less aggressive as persistence increases. Keyords overlapping generations, random relocation model, monetary targeting, inflation targeting, real shocks, persistence Disciplines Economics This orking paper is available at Ioa State University Digital Repository:

3 IOWA STATE UNIVERSITY Optimal Monetary Policy Rules under Persistent Shocks Joydeep Bhattacharya, Rajesh Singh January 008 Working Paper # Department of Economics Working Papers Series Ames, Ioa 500 Ioa State University does not discriminate on the basis of race, color, age, religion, national origin, sexual orientation, gender identity, sex, marital status, disability, or status as a U.S. veteran. Inquiries can be directed to the Director of Equal Opportunity and Diversity, 3680 Beardshear Hall, (55)

4 Optimal Monetary Rules under Persistent Shocks Joydeep Bhattacharya and Rajesh Singh y October 03, 007 Abstract The tug-o-ar for supremacy beteen in ation targeting and monetary targeting is a classic yet timely topic in monetary economics. In this paper, e revisit this question ithin the context of a pure-exchange overlapping generations model of money here spatial separation and random relocation create an endogenous demand for money. We distinguish beteen shocks to real output and shocks to the real interest rate. Both shocks are assumed to follo AR() processes. Irrespective of the nature of shocks, the optimal in ation target is alays positive. Under monetary targeting, shocks to endoment require negative money groth rates, hile under shocks to real interest rates it may be either positive or negative depending on the elasticity of consumption substitution. Also, monetary targeting elfare-dominates in ation targeting but the gap beteen the to vanishes as the shock process approaches a random alk. In sharp contrast, for shocks to the real interest rate, e prove that monetary targeting and in ation targeting are elfare-equivalent only in the limit hen the shocks become i.i.d.! The upshot is that persistence of the underlying fundamental uncertainty matters: depending on the nature of the shock, policy responses can either be more or less aggressive as persistence increases. JEL Classi cation: E3, E4, E63 Keyords: real shocks, persistence, overlapping generations, random relocation model, monetary targeting, in ation targeting Bhattacharya: Associate Professor, Department of Economics, 60 Heady Hall, Ioa State University, Ames IA ; Phone ; Fax: ; joydeep@iastate.edu y Singh: (Corresponding author) Assistant Professor, Department of Economics, 60 Heady Hall, Ioa State University, Ames IA ; Phone ; Fax: ; rsingh@iastate.edu

5 Introduction The optimal conduct of monetary policy, hether to target the money groth rate or the in ation rate, has survived as one of the most contentious issues in monetary economics. Popular until recently, Milton Friedman s (960) mechanical monetarism advised central banks to stop setting interest rates and instead set the money groth rate permanently at the estimated groth rate of the real economy. Since the 80s, hoever, the dominant paradigm in the practice of monetary policy has shifted, bringing ith it a reneed dedication to price stability via the direct control of in ation via interest rate targeting. Poole (970) presented the rst formal treatment of the larger question: ho should a monetary authority decide hether to use the money stock or the interest rate as the policy instrument. The debate at the time, as summarized by Poole, took the folloing shape: hile some argued that monetary policy should set the money stock hile letting the interest rate uctuate as it ill, others believed that monetary authorities should push interest rates up in times of boom, and don in times of recession, hile the money supply is alloed to uctuate as it ill. Poole employed a stochastic IS-LM model and used volatility of aggregate output as the basis for comparison. He reached the conclusion that if disturbances originated primarily in the IS function that summarized the real sectors of the economy [...], the money stock is the proper control instrument. But if the LM function, representing the monetary sector, is the source of the disturbances, the interest rate is the proper control variable (Poole and Lieberman, 97). The bottom-line advice as clear and extremely in uential: hen the shocks are real in nature, x the money supply; if the shocks are monetary, x the interest rate. In this paper, e broaden the scope of the original instrument problem ithin a modern optimizing frameork and pose to questions of policy relevance. First, Our goal is not to update Poole s results in a optimizing frameork; rather to use his exercise as an inspiration to pose additional instrument problems.

6 hen the shocks are real, does it matter hether they impinge on output or the real interest rate? Or more broadly, does it matter hether they are real income or real price shocks? Second, does the persistence of these shocks matter for the conduct of monetary policy? Our goal is strictly qualitative to understand ho optimal monetary policy (money groth and in ation rate targeting) should respond to real shocks and their persistence in a exible-price microfounded model of money. To that end, e study a to-period lived to-island symmetric pure-exchange overlapping generations model in the tradition of Tonsend (987) here limited communication and stochastic relocation create an endogenous transactions role for at money. The competing asset to money is a commonly-available linear one-period storage technology. At the end of each period, a fraction of agents is relocated from one island to the other and they are alloed to bring only cash ith them. This assumption solves the return-dominance problem of money. Risk averse agents seek insurance against this relocation risk. This justi es a role for generational banks that take deposits and holds a portfolio of cash reserves and storage on behalf of their clientele. We study to variants of this model, one in hich the young-age endoment and one here the return to the storage technology is exclusively random. We model these as AR() processes because our goal here is to study ho changes in shocks persistence a ect the optimal policy responses and the relative desirability of alternative instruments. For consistency, hoever, e ensure that the unconditional distributions of these processes remain invariant to such changes. This requires that the variance of innovations decline as the persistence of shocks increases. Typically, ith either uncertain endoment or real interest rate, banks ill care not just about the current values of these variables but also their future realizations. Almost all the ork done in this area has a quantitative focus and employ models ith sticky or staggered prices, and very fe, use elfare criteria see Woodford (003). Our approach is to ask: sans the added complexity of sticky prices, hat can a reasonable neoclassical exible-price model of money have to say about optimal monetary policy design? 3

7 For example, a bank this period cares about next period s endoment realization because the latter ill potentially in uence that period s money demand and hence the price level and therefore a ect the return on money beteen this period and the next. But next period s money demand depends on the folloing period s endoment, and so on. We assume all agents kno the relevant distributions of these shocks and form expectations about the returns on money and storage. If shocks are persistent, these expectations ill be conditional on the current realizations. We focus solely on long run stationary equilibria under hich agents expectations are coordinated across time, i.e., expectations of one generation are validated by the behavior of the next and so on ad in nitum. In the case of endoment shocks, and ith logarithmic (henceforth log ) utility, e prove that irrespective of the degree of persistence, the optimal monetary target is a zero money groth rule hile the optimal in ation rate target involves some in ation. We also sho that monetary targeting elfare-dominates in ation-rate targeting but the gap beteen the to (i.e., the relative desirability of monetary targeting) reduces as the shocks become more persistent. In the limit, hen the shocks to the endoment approach a random alk, the instrument problem vanishes: the to instruments become elfare-equivalent. We demonstrate numerically that similar results hold for more general CRRA preferences: as persistence rises, smaller policy responses in terms of money groth or in ation rate changes are needed, and the elfare gap beteen the to monetary regimes shrinks. Results under real interest rate shocks are strikingly di erent. For log utility, irrespective of the persistence of shocks, both monetary and in ation targeting are equivalent and the optimal money groth rate/in ation rate in either case is zero. For more general CRRA utility, in sharp contrast to the result under endoment shocks, monetary targeting and in ation targeting are elfare-equivalent only in the limit hen the shocks become i.i.d.! The magnitude of the policy response under either regime (money groth rate or in ation rate targeting) as ell as the elfare 4

8 gap beteen the to increases ith the persistence of these shocks. The elasticity of intertemporal substitution of consumption plays a critical role here. As in the case of endoment shocks, an optimal in ation rate under real interest shocks is alays nonnegative, but optimal money groth rate is negative only hen consumption is elastic (relative to log); else, it is positive. More interestingly, in ation targeting dominates monetary targeting hen consumption is elastic (relative to log); otherise, the latter performs better. The upshot is that even in exible-price environments, the nature of real shocks hether they impinge on output or the interest rate and their persistence, matter crucially for optimal monetary policy design. An information about the nature of the shock alone is no longer su cient. The random relocation ith limited communication model as popularized by Champ, Smith, and Williamson (997) and has been used to investigate myriad monetary policy issues in Schreft and Smith (997), Smith (00), Antinol, Huybens, and Keister (00), Antinol and Keister (006), Gomis-Porqueras and Smith (003), Bhattacharya, Haslag, and Russell (005), Haslag and Martin (007), among many others. The random relocation model is attractive because it includes a genuine reason hy money is held even hen dominated in return; it also allos for easy inclusion of institutions such as Diamond-Dybvig (983) style banks. Another strong appeal of this model is its analytical tractability. To date, hoever, researchers have either orked ith deterministic versions of the model such as Gomis-Porqueras and Smith (003) or at best, ith i.i.d. shocks, as in Bhattacharya and Singh (007). 3 Independent of its aforediscussed contribution to the area of optimal monetary policy design, our paper makes strides in the analytical treatment of persistent shocks in this popular model of money. A paper close in spirit to ours is Kryvtsov et. al. (007). In an overlapping gener- 3 The current paper is in some respects a companion piece to Bhattacharya and Singh (007). Hoever, alloing for persistent AR() shocks and adding shocks to the real interest rate makes the current analysis considerably more general and formidable. 5

9 ations economy similar in some respects to ours, they sho that increased persistence of endoment shocks requires a smaller in ationary intervention. This is primarily because ith higher persistence, the transitory component of the shock becomes less important. 4 Although our policy results for the case of endoment shocks have a similar avor, the underlying reasons are somehat di erent. In our set up sans shocks, as is ell knon, a xed money supply (or equivalently, a constant price of consumption) is the optimal policy. 5 It is only hen the shocks induce banks to deviate from the steady state allocations that a need for optimal policy to deviate from the xed money supply rule arises. In particular, the higher the impact of shocks, the higher ill be the required deviation. A higher persistence essentially reduces the (conditional) uncertainty of next period s endoment and therefore price level uncertainty is reduced. Consequently, irrespective of the monetary regime, the policy response becomes less aggressive. In contrast, hen the shocks impinge on the real interest rate, e nd that the policy response can become more aggressive ith increased persistence. Here, if the returns are i.i.d., current realizations (on investments made last period) are irrelevant for the investment choices of the current generation. Instead, if these returns are persistent, their expectational relevance for the next period s return makes current investments contingent on current realizations. As a result, investments into cash uctuate over time. Indeed, the higher the persistence, the more volatile are the cash balances. Consequently, policy responses strengthen 4 In their model, agents precautionarily carry more money than hat is optimal from a planner s perspective. Therefore, to discourage such savings, the optimal in ation rate is positive. With i.i.d. shocks, all changes are temporary and the precautionary motive is strong. The higher the current endoment, the stronger is the precautionary motive, and the higher is the optimal in ation rate. With persistence, a change in endoment has a permanent component that does not require a proportional precautionary response. The policy need not discourage savings as aggressively. As a result, the volatility of optimal in ation declines ith the persistence of shocks. 5 In equilibrium, relocated old agents use money to purchase out of the endoment deposited by the current young. From a planner s perspective, ex-ante, hat is consumed by the relocated agents can not be stored. The social (planner s) opportunity cost of movers consumption relative to that of non-movers is thus the lost return on storage. In a decentralized economy, this is implemented by keeping the money supply xed, i.e., a zero net return on money. See Bhattacharya and Singh (forthcoming) for a detailed discussion. 6

10 ith shock persistence. Gomis-Porqueras and Smith (003) also study output and real interest rate uctuations; speci cally, they consider to-period deterministic cycles of output, real interest rate, and relocation shocks. They sho that real interest rate shocks require relatively higher nominal interest rate smoothing than do endoment shocks. The result relies critically on their assumption of elastic (relative to log) utility. We sho that the ranking of interest rate targeting vis-à-vis monetary targeting crucially depends on the elasticity of intertemporal substitution. Moreover, not only do e evaluate the best rule ithin each monetary regime, e also explicitly rank the to regimes, something that they do not. The plan for the rest of the paper is as follos. In the next section, e outline the baseline model and characterize decentralized allocations. In Section 3, e study the role of endoment uncertainty in shaping the optimal choice of monetary instruments. In Section 4, e do the same ith real interest rate shocks. Both these sections also include the results from the computational experiments under CRRA utility. Section 5 concludes. Proofs of all major results are in the appendices. The Model. Preliminaries We present a model economy that is populated by a unit mass of to-period overlapping generations of agents located in to spatially separated locations. 6 Time is denoted by t = ; ; ::;. Each to-period-lived agent is endoed ith t > 0 units of this good at date t hen young and nothing hen old. We assume is stochastic and that is revealed at the start of any date; the speci cs are provided belo in Section 3. 6 Our formulation follos Tonsend (987), and Schreft and Smith (997). Only a concise description of the environment is provided here; the reader is referred to Schreft and Smith (997) for details. 7

11 Only old-age consumption is valued. Let c t denote old-age consumption of the members of the generation born at date t; their lifetime utility is given by u(c) = c =( ); ith > 0; and u(c) = ln c hen = : All agents have access to to assets, storage and money. Each unit of the consumption good put into storage at date t yields x t+ units of the consumption good at date t +. We assume x is stochastic and that it is revealed at the start of any date; the speci cs are provided belo in Section 4. The other asset is at currency (money) that agents may acquire through sale of their endoment. Let p t denote the price level at date t. Then the gross real rate of return on money (R mt ) beteen period t and t + is given by R mt = p t =p t+ = t, here t denotes the in ation rate beteen period t and t +. We ill assume that money is dominated in its return (see belo). The sequence of events is as follos. Agents receive their endoment at the start of a period. Toards the end of the period, after the savings decisions have been made, a fraction of randomly chosen agents from each location is relocated to the other location: A relocated agent cannot collect the return on any goods she has stored, or that have been stored on her behalf, since goods cannot be transported across locations. Hoever, if an agent is carrying at currency hen she is relocated, then the currency is relocated ith it. Young agents may either invest their endoment directly into one or both the assets or go through a bank that takes deposits and invests in these assets; the speci cs of the banks problem ill follo later. Under standard assumptions discussed in Schreft and Smith (997), agents ill nd it in their best interest to deposit their entire endoment into a bank before they learn their relocation status. The bank pools the goods deposited by all the young agents and uses them to acquire a portfolio of stored goods and at currency. It issues claims to agents hose nature, timing, and size are contingent on their relocation status. If an agent does not get relocated (henceforth, a non-mover), she gets a return on her deposit next period funded by the 8

12 goods the bank has stored. If she gets relocated (henceforth, a mover), then she gets a return on her deposit in the same period in the form of a at currency payment (hose real value ill depend on the folloing period s price level) funded by the bank s holdings of at currency. The quantity of money in circulation at the end of period t, per young agent, is denoted M t. Let m t M t =p t denote real money balances at date t. The government conducts monetary policy in one of to possible ays. The rst, called monetary targeting, is one here the government changes the nominal stock of at currency at a xed non-stochastic gross rate > 0 per period, so that M t = M t for all t. The second, called in ation targeting, is one here the government changes the nominal stock of at currency in such a ay as to keep the long-run gross in ation rate xed at : All money injections are implemented through lump sum transfers () to the young agents. The period t budget constraint of the government is t = M t M t p t = m t m t R m;t () for all t.. The bank s problem The asset holdings of young agents are assumed to be costlessly intermediated by perfectly competitive banks. Under this assumption, banks equilibrium pro ts from any generation is zero. One can equivalently think of banks being created each period by the current cohort of agents ho ish to maximize the ex-ante overall return on their cash-storage portfolio hile keeping su cient cash for the movers. Speci cally, every young agent deposits her after-tax/transfer income in the bank. The banks divide their deposits beteen stored goods s t and real balances of at currency m t, so that t + t = m t + s t. () 9

13 At the time the bank solves its problem, the current endoment t and the return x t on goods stored during previous period is knon; so is the equilibrium price level p t. But t+, x t+, and p t+ has not been realized yet. The bank cares about x t+ as it directly a ects the consumption of non-movers. Additionally, as x t+ helps predict x t+, it may also in uence the next period s money demand and the price level, hich in turn a ects the rate of return on money beteen this period and the next. The indirect e ect also holds for t+ through its e ect on the next period s money demand. Evidently the bank cannot promise a xed real return to its depositors. All it can do is let the depositors kno the amount of nominal balances being kept aside for their use on the other island. The bank knos the stochastic process for t+ and x t+ and forms expectations on the return on money R mt ; in a rational expectations equilibrium, these expectations are correct. We focus solely on long-run stationary equilibria under hich expectations are coordinated across time, i.e., expectations of one generation are validated by the behavior of the next and so on ad in nitum. De ne t mt t+ t as the ratio of cash reserves to deposits. Belo, hen e specify shock processes, e also provide conditions to ensure that money is dominated in return by storage. Then the bank ill never ant to carry cash balances across time. In that case, the bank s problem is given by t ( max E t u t [0;] R t )x t+ mt ( t + t ) +( ) E t u ( ) ( t + t ) ; (3) here E t is an expectations operator conditional on date t information. Let c mt t ( t + t ) and c nt t ( t + t ) ; the consumption allocations o ered by the bank to its moving and non-moving clientele. The rst order condition for this problem is given by E t (u 0 (c mt ) R mt ) = E t (x t+ u 0 (c nt )) : (4) Eq. (4) equates the expected marginal value of a unit of endoment saved as cash ith its expected marginal value ere it instead kept in storage. 0

14 Under CRRA preferences, i.e., u (c) = c ith > 0, the rst order condition to the bank s problem reduces to t E t R mt = ( ) [( t )] hich further simpli es to t = + ( ) E t(r mt ) E t(x t+ ) E t x t+ ; (5) : (6) Monetary policy in uences the optimal since it determines the relative return on money (R m ). max t [0;] It is possible to make further analytical progress under the assumption of logarithmic utility. The bank s problem is no reritten as E t (ln R mt ) + ln t + ( ) ln t + ( ) E t (x t+ ) + ln ( t + t ) : Observe that the bank knos the current period endoment and takes the return on money and the size of the transfer as given. Therefore, the bank s choice of ill to respond only to the second and the third terms of the previous expression. Then, the choice of t is given by (7) t = for all t; (8) hich, of course, may also be obtained from (6) by substituting =. As is ell knon, ith logarithmic utility, banks allocate deposits across the to assets to provide consumption to the to types in proportion to their population shares. As a result, the choice of is not state-contingent. This ill not be the case in the more general CRRA formulation, as is evident from (6)..3 Equilibrium under monetary and in ation targeting Enroute to solving for optimal monetary rules, e describe the monetary equilibrium under the to targeting rules. Under monetary targeting, the government xes the

15 money groth rate at : Since t = Mt M t p t e have = m t and m t = t ( t + t ) ; m t = t t ; t + t = t Hence, the equilibrium return on money is given by R mt = m t+ m t = t+ t+ t t t t : (9) t : (0) t+ With logarithmic utility, t = for all t (see (8)), and the above reduces to R mt = t+ t. Given the exogenous process for fx; g and money groth rate, a stationary equilibrium of this economy consists of a time-invariant portfolio allocation function ( t ; x t ) and a return function R mt (given by (9) and (0)) such that given R mt, ( t ; x t ) solves the banks problem (3) and R mt = (x t+ ; t+ ) (x t ; t ) t+ (x t ; t ) t (x t+ ; t+ ) : () Under in ation targeting, the government xes the in ation rate at : The real return to money is R mt = : () Here, there is no uncertainty about the rate of return on money. The government conducts monetary policy via time-varying taxes and transfers to ensure money is an asset ith a xed real return. Since t = Mt M t m p t = m t t ; e get m t = m t ( t + t ) = t t + m t t implying m t = t ( t ) m t + t t t (3) Thus, real balances follo an AR() process under in ation targeting. For future reference, under log utility ith t =, (3) reduces to m t = m m t + t (4)

16 here m is the persistence term. A stationary equilibrium of this economy is described by a time-invariant portfolio allocation function () ; that given and the stochastic process for x solves banks problem described by equation (3) given by (6). The solution leads to a stationary process for real money demand governed by (3). As discussed in Bhattacharya and Singh (forthcoming), in steady states, a planner constrained by limited communication faces a return of x on stored goods; such a planner ho allocates beteen the movers and the non-movers ould choose an allocation (c m ; c n ) so as to maximize u(c m ) + ( ) u(c n ) subject to c m + ( ) c n =x = : The marginal condition is u 0 (c m ) = x u 0 (c n ) : This is the intratemporal e ciency or intragenerational e ciency condition connecting marginal utilities of movers and non-movers at any date. In a steady state, a government trying to replicate the planning solution ould face a static problem and hence ould need to pay attention solely to this intratemporal margin. With shocks, hoever, the government s problem does not remain static. An intertemporal (intergenerational) margin appears because shocks hit di erent generations asymmetrically. No the government pays attention to providing some amount of intergenerational insurance. To achieve this, the government may opt to trade o intratemporal for intertemporal e ciency and this causes optimal monetary policy to deviate from hatever policy achieves intratemporal e ciency alone. A road-map of hat lies ahead is in order. We start by analyzing the optimal monetary rules under monetary and in ation targeting policies in the case here endoments follo an AR() process but storage return is xed. We are able to derive clean analytical results for the case of logarithmic utility. We go on to compare stationary elfare across the to targeting policies. We then repeat these exercises in the case here the endoment is xed but the return to storage follos an AR() process. In both cases, e extend the scope of our results to the case of CRRA utility 3

17 by means of numerical computations. 3 Endoment uncertainty We start o by xing the return on storage to x for all dates and alloing the endoment process to be stochastic. Shocks to the endoment are intended to represent real output shocks. Our goal is to investigate ho optimal monetary policy responds to such shocks, and in particular, to the persistence of such shocks. We assume that t follos an AR() process of the form: t = t + " t ; (5) here " is i.i.d. ith mean ( ) and variance ( ), and here and are the unconditional mean and variance of the endoment process. This speci cation allos us to study ho optimal policies vary ith the persistence ( ) of the shocks hile keeping the unconditional mean and variance of these shocks constant. We also assume that x > + ( ) e min. This ensures that x > E t (R mt ) alays holds, and banks never carry cash for the non-movers. We adopt an ex-ante measure of elfare. This allos us to obtain the stochastic analog of the golden rule in a stochastic overlapping generations economy ith nitely lived agents. Here, a generation s elfare is de ned as its lifetime expected utility here an unconditional expectation is taken ith respect to stationary distributions of exogenous as ell as endogenous variables. This measure, by construction, treats all generations symmetrically and makes each of them representative. The unconditional expectation ensures that the derived policy rules are state-uncontingent or timeless. We rst consider the optimal money groth rate under monetary targeting, then characterize the optimal in ation rate under in ation targeting, and nally establish a ranking beteen the to. 4

18 3. Monetary targeting Monetary policy has di erent e ects on the to groups, movers and non-movers. The latter s consumption is given by x( t + t ) hile the formers by R mt ( t + t ): Using (9), it is easily veri ed that the consumption of non-movers is given by c nt = xt + and that of the movers by c mt = t+. Then, indirect utility as a function of at + t is obtained by evaluating (7) at t = and using the equilibrium return given by (0): W t () = E t fln t+ j t g + ( ) ln (x t ) ln ln : (6) Notice that has no e ect on the rst to terms on the r.h.s. of the elfare expression. Ex-ante stationary aggregate elfare is de ned as W E fw ()g here E is the unconditional expectation. Thus, e have W = ( ) ln x + E (ln ) ln ln (7) What is the best from the standpoint of stationary elfare? We de ne ~ arg max fw g. Since W is assumed to be concave in, ~ solves d d W = 0; [notice ~ maximizes the last to terms in (7)]. For future reference, e de ne ~ W max W. Proposition Under logarithmic utility, irrespective of, the optimal monetary policy is to keep the money supply xed, i.e., ~ =. Notice from equation (6) that the terms containing are independent of both the current and/or future endoment. In that case, optimal monetary policy can safely ignore the aforediscussed intergenerational margin and ork solely to reach an e cient intragenerational margin. 5

19 3. In ation targeting The stochastic process for real balances under in ation targeting as given by (4) can be reritten as m t = m m t + t = X ( m ) s t s : s=0 This, hen combined ith the AR() process for endoments, yields " m t = X # X X X s " t s m s " t s + m s " t s 3 m s 3 " t s + :::: : s=0 s= Denote the mean and variance of the stochastic process for m de ned in (4) by m and m respectively. The folloing lemma contains some pertinent information about the mean and variance of the stochastic process for m: s= s=3 Lemma For log utility, under in ation targeting, the mean and the variance of the stochastic process for real balances are given by: m = ; + m m = ( ) 4 + ( + m ) m + 4 m + m m (8) Moreover, the variance declines monotonically ith. Since the unconditional mean and variance, and, are xed, Lemma shos that the variance of real balances or equivalently net-of-transfer income (recall that + = m=) depends on the persistence of the endoment process and the persistence of monetary process. Notice hoever, that mean real balances depends solely on m and not on. The folloing lemma, hich compares ith m ; ill permit further progress m toards evaluation of the optimal in ation target and comparison across the to targeting schemes. 6

20 Lemma As increases m = declines. In particular, m 9 (( )+) > for m ( ) = 0 >= (m e ) (( )+) = > for all ( ) + = m (0; ) ( e ) as! >; Thus, hen endoment shocks are highly persistent, the per-unit variance of net-oftransfer income approaches that of the endoment process. Under in ation targeting, there is no uncertainty regarding the return on money. The only remaining uncertainty is about the net-of-transfer income hich is the sum of the endoment and the transfer income. When the persistence of the endoment process increases, it gains relatively higher importance (relative to the stochastic process for money balances hich determines the transfer income) in the determination of the net-of-transfer income. In the limit, as the endoment process becomes a random alk, the signi cance of monetary process vanishes completely. As the unconditional variance of the endoment process is constant, a reduction in the relative importance of monetary process reduces the unconditional variance of the net-ofincome process. An increase in the persistence of endoment process thus makes in ation targeting more desirable, as ill become clear further belo. Evaluating (7) at t = and using () obtains indirect utility at date t as W t () = ln + ( ) ln x + ln ( t + t ) = ln + ( ) ln x + ln mt It is apparent that has to e ects on elfare, one through its e ect on the return on money (captured by the ln term above) and the other via its e ect on posttax/transfer income (captured by the ln ( t + t ) or ln mt terms above). Having characterized the rst and second moment properties of the net-of-transfer income under in ation targeting, e are no ready to study the properties of an optimal in ation target. To do so, e rst de ne the ex ante stationary aggregate elfare as W E (W ()) = ln ln + ( ) ln x + E (ln m) : (9) 7 :

21 Notice that the last term on the r.h.s. the mean value of log post tax income. of (9) corresponds to E(ln ( t + t )) or We de ne the optimal in ation rate as ~ arg max fw g and its corresponding elfare as W ~ max fw g. Henceforth, e assume that W is strictly concave in ; then ~ solves d W = 0. d Proposition Under logarithmic utility, optimal in ation targeting involves setting a positive in ation rate, or ~ > for all [0; ). Furthermore, lim! ~ =. The proof of Proposition relies on a second-order Taylor approximation of the last term on the r.h.s. of (9) around m. 7 An intuition for this result is as follos. The rst thing to note is that hile in ation targeting eliminates rate of return uncertainty, the uncertainty about the post-tax/transfer income + = m remains. A riskaverse agent ould thus prefer to have the highest expected value for m ith minimum accompanying volatility. It can be checked from Lemmas and that raising in ation rate achieves both objectives, it follos that choosing a positive in ation rate ( > ) may be desirable. Intuitively, in order to give mt the date t + young such that their storage allocation, ( m t to date t movers the government ill have to tax ) ( t+ + t+ ) ; leaves of goods for date t mover s consumption. In other ords, taxes/transfers should m be such that ( ) ( t+ + t+ ) = t t+. This is the rationale behind (4), hich is reritten belo as ( t+ + t+ ) = ( ) ( t + t ) + t+: (0) Equation (0) makes clear that the autocorrelation beteen total income at to adjacent periods is negative and the strength of this correlation becomes smaller as rises. Thus setting > may improve intertemporal e ciency; this ay shocks to income do not get transmitted over time as easily. Proposition also states that as the persistence of endoment shock increases, the optimal in ation target approaches unity. This directly follos the results stated 7 Observe that E ln m = ln m m m and this is here Lemma helps. 8

22 in Lemma. As the persistence of endoment process gets larger the per unit net of transfer income volatility under in ation targeting approaches that of the endoment process. As a result, the policy correction through in ation needs to be less aggressive. We no proceed to anser the question: hich targeting regime, set at its on optimal rate, achieves higher elfare? Proposition 3 Under logarithmic utility, optimal targeting of the money groth rate is stationary-elfare superior to optimal targeting of the in ation rate for all <. The elfare gap beteen the to regimes shrinks as increases; as!, both regimes are stationary-elfare equivalent. Recall that optimal monetary targeting involves setting = ( xing the money supply) thereby making the post-tax/transfer income exactly equal to the endoment. In this setting, as discussed earlier, both non-movers and movers consumption variability is solely due to the endoment uncertainty. On the other hand, optimal in ation targeting involves xing the in ation rate thereby eliminating any uncertainty ith respect to the return on money; the residual uncertainty, in this case, is ith regard to the post-tax/transfer income. Why is monetary targeting superior? A xed money supply rule achieves exante intratemporal e ciency; it does/can not a ect the fundamental endoment uncertainty, captured by. Compare this to a zero net in ation rate policy. Of course, as e have seen, = can achieve ex-ante intratemporal e ciency; hoever, it is associated ith a higher volatility of post-tax income. In order to reduce the per unit volatility of net-of-transfer income mm, needs to be increased. The optimal thus trades o intratemporal e ciency against the bene t received from the reduction in income volatility. Yet, as Lemma shos, for any degree of persistence of endoment shocks, the per-unit volatility of net-of-transfer income under any in ation rate is higher than the endoment volatility, hich equals the net-of-transfer income volatility under monetary targeting. Overall, relative to monetary targeting, optimal 9

23 in ation targeting distorts the intratemporal e ciency margin and additionally leaves the net-of-transfer income more volatile. This makes its less desirable overall. Proposition 3 also states that as the persistence of endoment process increases, the elfare under in ation targeting approaches that achievable under monetary targeting. Once again, the result follos from Lemma. As increases, the volatility of net-of-transfer income under in ation targeting decreases and the elfare gap accordingly shrinks. When the endoment process is a random alk, the signi cance of the monetary process in net-of-transfer income determination completely vanishes, and net-of-transfer income volatility not only equals that of endoment volatility, but is also independent of the in ation rate. Since intratemporal e ciency is ensured, the in ation rate instrument can play no additional role. As under monetary targeting, it is optimal to have ~ = ; hence, an identical elfare is obtained. 3.3 CRRA utility We no extend our analysis to incorporate the more general CRRA utility form: u(c) = c =( ) here is the coe cient of relative risk aversion. Our objective here is to verify hether the avor of the results from Section 3 continue to obtain for aay from unity. Since it is not possible to pursue this analytically, e ill resort to numerical analysis belo. By combining (5) ith (3), the expression for period t elfare is given by W t ( ) ( t + t ) ( t ) E t x t+ ( t + t ) E t (R mt ) = : t () Under monetary targeting t ( t ), t + t ; and R mt are given by (6), (9), and (0), respectively. Thus, for given probability distributions for and, the equilibrium function is obtained as a xed point of (6). Evidently, under monetary targeting, the equilibrium t [denoted t ()] is a function of, and the period t realization 0

24 of. Under in ation targeting R m =. Then is obtained from (6), and then t + t = mt is obtained from (3). Next, the optimal policies and optimal elfare levels are de ned as ~W i max i W i, ~{ arg max W i ; i f; g Finally, e represent ~ W i in terms of its consumption equivalent ~c i by using h ~W i = ~c i i = ( ) : Our choice of parametric speci cation is as follos. We x = 0: and assume that the long-run distribution of is log-normal. For the AR() speci cation, e assume ln t = ln t + " t, here " N ( ( ) ; ( )) ; = 0 obviously nests the i.i.d. speci cation. Belo e present results for = 0; 0:5, and 0.9. In particular, e compare optimal money groth and in ation rates under the to policies, along ith their respective elfare levels, for [0:5; :]. 8 A fe ords about the computational algorithm is in order. Under monetary targeting, the main step entails computing the xed point of as a function of, depending on the nature of the shock. To do so, e guess an initial function, 9 and numerically iterate on (6) to convergence. This is done for a xed. Once the function is obtained, evaluating () by averaging over a large number of simulations obtains the ex-ante elfare. By repeating this exercise for di erent values of, e easily obtain ~ and ~ W. Under in ation targeting, e rst x. This yields directly. After assuming an initial value of m 0 = m e, e let the computer simulate (3), and for each observation of or ; and m compute (). An average obtained 8 We nd that [0:5; :] is a fairly representative range, and the qualitative nature of our results continue to hold hen this range is enlarged. 9 Our initial guess is ( t ) =. The convergence to the equilibrium function at any desired accuracy is reasonably fast.

25 from the previous step yields ex-ante elfare. By repeating this exercise for di erent values of, e obtain ~ and ~ W. We have analytically established that hen = (log utility), ~ = and ~ > :

26 Figure (a) shos that ~ > for all ; and Figure (b) shos that ~ for all. 0 Figure (a): Optimal against (endoment shocks) Figure (b): Optimal against (endoment shocks) 0 The ~ curve for = 0 the i.i.d. case follos the same shape as the other to, is tangential to them at =, and lies strictly belo = 0:5 at all other. If presented together, ~ for = 0:9 becomes indistinguishable from the x-axis. 3

27 As under log utility, ~ > reduces the net-of-transfer income volatility. Figure (a) shos that for any given, ~ is increasing in. Intuitively, as gets higher i.e., the utility is more concave, the income volatility hurts more. A more aggressive policy response is needed and thus ~ is increasing in. Figure (a) also shos that, for any given, ~ is decreasing in. Recall from the discussion folloing Lemma that net-of-transfers income volatility is decreasing in the persistence of the endoment shocks. As a result, the policy is less aggressive, i.e., it gets closer to ~ = as increases. Why is ~ < for all 6=? Roughly, hen <, the main factor that dominates the policy choice is the banks disproportionately high allocation on non-movers consumption. This is because the bank perceives the consumption of the to types as gross substitutes and the return on cash that goes to movers consumption is uncertain. Then ~ <, by transferring resources to movers, aligns banks choices ith hat is ex-ante optimal. On the other hand, hen >, banks choice of deposits saved for movers is too volatile over time. Ex-ante, ~ < then provides an appropriate compensation. Irrespective of the value of, the deviation of banks choices from ex-ante optimality is primarily due to the money s rate of return uncertainty. As the unconditional distribution is preserved by assumption, an increase in the persistence of endoment shocks implies a reduced spread of its next period s conditional distribution. As a result, the banks choices are closer to hat ex-ante optimality requires. This is evident from Figure that shos that is closer to the average and less steep as increases. This leads to the result shon in Figure (b): the policy is less aggressive A planner constrained by the fact that movers must consume from current endoment ill choose allocations such that ex-ante E (u 0 (c m )) = x E (u 0 (c n )). In a decentralized equilibrium hoever the bank s choice is governed by E (u 0 (c m ) R m ) = xe (u 0 (c n )). Setting ~ < aligns the decentralized marginal condition ith that of the planner. For technical details of this argument the reader is referred to Bhattacharya and Singh (007), ho obtain similar results for i.i.d. shocks. 4

28 as increases, i.e., it gets closer to ~ =. Figure : as a function of under monetary targeting Figure 3 presents the percentage gain in steady state elfare, expressed in terms of equivalent consumption, that is obtained under monetary targeting relative to 5

29 in ation targeting. Figure 3: % change in ~c i from folloing ~ over ~ Proposition 3 shoed that the elfare gap beteen monetary and in ation targeting is decreasing in ; the same holds for all. Figure 3 also shos that the elfare gap is increasing in for any given. Intuitively, a relatively higher income volatility under in ation targeting hurts more as increases, thus making monetary targeting even more desirable. In the next section, e study uncertainty regarding the return to the storage technology. In that case, as e demonstrate belo, the higher the persistence of the storage return shocks, the larger is the elfare gain under the elfare-superior regime. 4 Uncertain return on storage In this section, e shut o the endoment uncertainty and instead allo the return on storage to be uncertain. Speci cally, e assume that each unit of the consumption good stored at date t yields x t units at the start of date t. Further x t is assumed 6

30 to follo an AR() process: x t x e = x (x t x e ) + " t ; and " t i.i.d. here it is assumed that x e > ; " t ["; "] ith E t f" t g = 0 and represents shocks to the real interest rate: The above process implies that x t is distributed over support h i x e + " ; x e + " and its unconditional mean and variance are given by x e > x x and x ", respectively. Conditionally, hoever, E fx t jx t g = ( x x ) x e + x x t. It is further assumed that E fx t jx t g > for all x t ; hich requires that x e + x x " > or " > x x (x e ). This ensures that a strictly positive amount of storage is held by banks at all times. Our goal is to repeat the previous set of exercises, i.e., investigate ho optimal monetary policy responds to real interest rate shocks, and in particular, on the persistence ( x ) of such shocks. Since the structure of the results is roughly similar to those presented above for endoment shocks, e ill necessarily be more brief in their presentation belo. In the case of log utility, t = for all t; in particular, is independent of x: Under monetary targeting, since m t = ( + t ) and t = Mt M t p t = m t, it follos that m t = = for all t. Further, R mt = p t = m t+ = p t+ m t : Under in ation targeting, R mt = for all t. Further, m t = ( + t ) and t = M t M t p t = m t m t. In turn, m t = m t +. In a stationary equilibrium, m t = = for all t. Thus, in either case the indirect utility is given by W z = ln z+( ) E t fln x t+ jx t g+ln ln + z for z = ; : () Proposition 4 Irrespective of the speci cation of shocks to storage returns, monetary and in ation targeting are equivalent under logarithmic utility. requires xing the money supply or equivalently the price level. 7 The optimal rule

31 Under log preferences, banks alays spend fraction of deposits to acquire cash reserves. For a xed money supply, the net-of-transfers income is also simply the endoment in this case, the banks keep aside a fraction of the endoment to purchase cash. Prices are constant period after period and the rate of return on money is unity. The same equilibrium obtains if prices ere instead xed. But, hy is a constant money supply optimal? To anser this, compare () ith (6). The terms containing are independent of current and/or future return on storage. Once again, optimal monetary policy ignores the intergenerational margin and ~ = obtains an e cient intragenerational margin. Does this result continue to hold in the more general CRRA form of utility? We no sho that the equivalence of the to instruments described in Proposition 4 breaks don hen the storage shocks are persistent, i.e., x > 0. Under in ation targeting, the rate of return on money is xed;i.e., R mt = for all t. The equilibrium t [denoted (; x t )] is readily obtained from (6) as (; x t ) = + ( ) n oi he t x t+ jx t (3) Hoever, under monetary targeting, uncertain storage returns also contribute to return on money uncertainty. In this case, (; x t ) is obtained from (6) as (; x t ) = + ( ) (;x t) (;x t)( ) ( "E t x t+ (;x t+ )( ) (;x t+ ) x t )# (4) As is evident, the current does depend on the return to storage in the folloing period. If x is persistent, the distribution of x t+ is contingent on x t and in turn t is a function of x t ; otherise not (as the folloing Proposition makes clear). Proposition 5 For CRRA utility, under i:i:d: shocks to storage returns ( x = 0), monetary targeting and in ation targeting are equivalent. In either case, the best rule 8

32 involves xing the money supply or equivalently the price level. If x > 0 holds, the equivalence breaks don. When banks allocate their deposits into cash and storage, the return on either of them is not knon. While the return on storage is unknon by assumption, the return on money depends on the future price level hich in turn depends on next period s allocations. When the return to storage is i.i.d., its current realization does not help predict its future values. Then t is independent of x t. As all other exogenous variables, including the money groth rate, are constant, is time-invariant, and so are the transfers and deposits at the bank. Once again, prices gro at the money groth rate. The rate of return on money and real balances are constant, hich also means that movers consumption is constant over time. Why is ~ = then? Notice that in the decentralized equilibrium movers directly consume a constant fraction of endoment, hereas the non-movers consume only stored goods ith uncertain returns. From a planner s perspective a unit of endoment that goes to movers obtains u 0 (c m ) hereas a unit reserved for non-movers obtains E t fx t+ u 0 (c nt )g. Equating the to and then comparing ith (4) obtains ~ =. On the other hand, hen shocks are persistent x t does help in predicting x t+ and then t does depend on x t. A constant money supply does not lead to a stationary prices anymore, and the equivalence breaks don. What optimal money groth/in ation rates obtain for x > 0 (and 6= )? Which targeting regime is superior for di erent values of? As before, e resort to numerical simulations. Our choice of parametric speci cation is as follos. We x = 0: and assume that the long-run distribution of x is log-normal. For the AR() speci cation, e assume ln x = x ln x t + " t, here " N ( ( x ) ; x ( x)) ; x = 0 obviously nests the i.i.d. speci cation. Belo e present results for x = 0:5; 0:8, and We compare optimal money 9