Money supply, macroeconomic stability, and the implementation of interest rate targets

Journal of Macroeconomics 31 (2009) 333–344

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Money supply, macroeconomic stability, and the implementation of interest rate targets Andreas Schabert Department of Economics, University of Dortmund, Vogelpothsweg 87, 44227 Dortmund, Germany

a r t i c l e

i n f o

Article history: Received 10 June 2008 Accepted 12 August 2008 Available online 24 August 2008

JEL classification: E52 E41 E32

a b s t r a c t In this paper the relation between interest rate targets and money supply is analysed in a standard macroeconomic framework with frictionless financial markets and sticky prices. Money supplies are examined that implement equilibrium sequences satisfying forwardlooking interest rate targets. An interest rate target with a positive inflation feedback in general corresponds to an accommodating money supply, i.e., money growth rates rising with inflation. It is shown that interest rate targets (like a Taylor-rule), which are consistent with a unique equilibrium, cannot be implemented by money growth rules. Ó 2008 Elsevier Inc. All rights reserved.

Keywords: Interest rate targets Money supply Money growth rates Equilibrium determinacy Policy equivalence

1. Introduction It is now standard practice to describe the stance of monetary policy in terms of a short-run nominal interest rate. Correspondingly, it has become increasingly popular in macroeconomic theory to characterize central bank behavior by targets for the (risk-free) nominal interest rate (see Clarida et al., 1999; or Woodford, 2003). In particular, monetary policy is often summarized by forward-looking interest rate feedback rules. Nevertheless, short-run nominal interest rates actually serve as an operating target for most real world central banks, while the supply of reserves, e.g., via transfers or open market operations, acts as the policy instrument.1 Following this view, we examine money supply policies that implement interest rates targets in a standard macroeconomic model, i.e., a New Keynesian model. The aim of the paper is to disclose whether the implementation of interest rate targets/rules and thus money supply is really negligible. Put differently, it will be examined if modelling monetary policy in terms of interest rate targets or in terms of money supply rules are really equivalent, i.e., lead to identical equilibrium solution. Chowdhury and Schabert (2008) for example find that an assessment of US Federal Reserve policy based on its postwar money supply leads to different conclusions about the impact of Fed policy on macroeconomic stability than previous studies based on interest rate rules (see Clarida et al., 2000). E-mail addresses: [email protected], [email protected] ‘‘In fact, of course, any particular interest rate policy must be implemented by a specific money supply policy, and this monetary policy must be implemented by a policy of fiscal transfers, open market operations, or both.” (Lucas, 2000, p. 258). This principle for example also applies for the current conduct of US Federal Reserve policy (see Woodford, 2003, p. 26). 1

0164-0704/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jmacro.2008.08.001

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We analyze the relation between money supply and interest rates in a simple New Keynesian model, where financial markets are – like in most recent studies on optimal monetary policy – assumed to be frictionless. The model contains more than one distortion such that the central bank cannot reach the first best allocation. We then examine characteristics of interest rate targets that are implemented by lump-sum money transfers in a rational expectations equilibrium. We thereby focus on a set of forward-looking or current-looking interest rate targets that are sufficiently general to encompass widely applied monetary policy specifications (for example, a plan of an optimizing central bank acting in a discretionary way).2 Regarding the monetary policy instrument, we allow money injections to be adjusted in response to changes in the state of the economy, which departs from the common practice to restrict money growth rates to be exogenous, like in Alvarez et al. (2002) or Monnet and Weber (2001).3 This paper relates to Minford et al. (2002) and, more closely, to Auray and Feve (2008). In both studies the observational equivalence between interest rate rules and exogenous money growth rules are examined. While Minford et al. (2002) apply a sticky wage framework, Auray and Feve’s (2008) sticky price model with a cash-in-advance constraint is more similar to ours. In contrast to our study, they restrict their attention to an economy with consumption logarithmically entering the utility function and policy rules each with one parameter, which facilitates the analytical derivation of estimations for the relation between an exogenous money growth rule and a Taylor-rule (i.e., the relation between the autocorrelation coefficient of money growth shocks and the inflation feedback coefficient in a Taylor-rule).4 Our analysis complements their line of research by applying an endogenous money growth policy and by focussing on fundamental (minimum state variable) solutions, while their analysis of money growth rates under a Taylor-rule policy is conducted for an indeterminate equilibrium. Consistent with our main result, they can nevertheless not find an observational equivalence between both policies in general.5 The reason why monetary implementation matters can be summarized as follows: when financial markets are frictionless, money demand and the consumption Euler equation govern the equilibrium relation between the money growth rate and the risk-free nominal interest rate. A higher nominal (real) interest rate induces consumption and, thus, nominal (real) balances to grow, given that money demand exhibits a positive income elasticity. A nominal interest rate target that rises with inflation is therefore associated with money growth rates that also rise with inflation, which corresponds to the lack of a liquidity effect (see Christiano et al., 1997). Due to the price rigidity, the previous period price level is relevant for the equilibrium allocation. Controlling nominal money supply then induces a history dependence, since aggregate consumption is linked to real balances, which depend on the initial stock of nominal money. This history dependence vanishes when the central bank is assumed to be able to control the interest rate directly, i.e., without money supply adjustments. Hence, equilibrium sequences cannot be identical for these policy specifications.6 The policy induced history dependence under money supply policy further implies that interest rate targets satisfying the well-known Taylor-principle (i.e., nominal interest rate raised by more than one-for-one with expected inflation) cannot easily be implemented.7 Given that consumption grows with the real interest rate, these targets rule out equilibrium consumption to evolve in a history dependent way, since an explosive consumption sequence cannot be consistent with a convergent equilibrium. On the one hand, self-fulfilling expectations are excluded by this mechanism. On the other hand, it matters for the implementability of interest rate targets: when money demand links consumption expenditures to the supplied amount of money, consumption is not independent of predetermined money holdings under a money supply policy. As a consequence, money supply adjustments aimed to implement an interest rate target satisfying the Taylor-principle would lead to an unstable consumption sequence. The remainder is organized as follows. In Section 2 we describe the monetary model, where a lump-sum money transfer acts as the monetary policy instrument. Section 3 examines the equilibrium relation between money supply and interest rates. In Section 4, we apply a slightly different model and assume that real balances enter the utility function, following large parts of the literature. This analysis shows that the main result survives in the most commonly applied framework, i.e. in the standard New Keynesian model, and does not rely on particular determinacy properties of a model with a cashin-advance constraint. Section 5 concludes. 2. The model This section presents a simple infinite horizon monetary model with Lucas’ (1982) cash-in-advance specification. In contrast to common practice in New Keynesian macroeconomics, we thus explicitly consider a role for money. We apply a mon-

2 Throughout the analysis, we restrict our attention to equilibrium solutions, which satisfy common equilibrium selection criteria (see Blanchard and Kahn, 1980). 3 In both studies the equilibrium relation between money growth and interest rates is examined in a framework with segmented financial markets and flexible prices. 4 In contrast, we derive the relation between two feedback coefficients on inflation and output-gap for both types of instruments, which allows to adjust the policy instruments to a wide set of possible states. 5 This paper is also related to Vegh (2001) and Yip (2005), who examine equivalences of state contingent rules in a continuous time framework and in an endogenous growth model, respectively. 6 Correspondingly, the equilibrium determinacy requirements substantially differ between both regimes, as for example shown by Carlstrom and Fuerst (2001, 2003), Honkapohja and Evans (2003), and Schabert (2006). 7 To demonstrate the robustness of the results, we will alternatively consider the widely applied money-in-the-utility function set-up in the last part of the paper.

A. Schabert / Journal of Macroeconomics 31 (2009) 333–344

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etary framework rather than a cashless model to identify the monetary instrument, namely, lump-sum cash injections in the asset market (or open market operations). We assume that prices are imperfectly flexible due to a staggered price setting and we introduce cost-push shocks. Thus, the central bank can raise welfare by reducing macroeconomic fluctuations, but it cannot implement the first best allocation. Throughout the paper, nominal variables are denoted by upper-case letters, while real variables are denoted by lower-case letters. There is a continuum of households indexed with j 2 ½0; 1. Households have identical asset endowments and identical preferences. Household j maximizes the expected sum of a discounted stream of instantaneous utilities U:

E0

1 X

bt Uðcjt ; ljt Þ;

ð1Þ

t¼0

where E0 is the expectation operator conditional on the time 0 information set, and b 2 ð0; 1Þ is the subjective discount factor. The instantaneous utility U is increasing in consumption c, decreasing in working time l, and satisfies the following parar  ð1 þ #Þ1 l1þ# , where r > 0 and # > 0. metric form, Uðcjt ; ljt Þ ¼ ð1  rÞ1 c1 jt jt At the beginning of period t household j is endowed with holdings of money M jt1 and a portfolio of state contingent claims on other households yielding a (random) payment Z jt . Before the goods market opens, households enter the asset market, where they can adjust their portfolio and receive money transfers. Let qt;tþ1 denote the period t price of one unit of currency in a particular state of period t þ 1 normalized by the probability of occurrence of that state, conditional on the information available in period t. Then, the price of a random payoff Z jtþ1 in period t þ 1 is given by Et ½qt;tþ1 Z jtþ1 . The households further receive wage payments and dividends Dit from monopolistically competitive firms indexed by i 2 ½0; 1. The budget constraint of household j is

Mjt þ Pt cjt 6 M jt1 þ Z jt  Et ½qt;tþ1 Z jtþ1  þ Pt wjt ljt þ Pt sjt þ

Z

1

Dj;it di;

ð2Þ

0

where Pt denotes the aggregate price level, wjt the (individual) real wage rate, and sjt a lump-sum money transfer. We further assume that households have to fulfill a no-Ponzi-game condition, limi!1 Et qt;tþi ðMjtþi þ Z jtþ1þi Þ P 0. After they leave the asset markets, households enter the goods market. In both market they rely on cash for transactions. Thus, consumption expenditures are restricted by the following cash-in-advance constraint (see Lucas, 1982):

Pt cjt 6 M jt1 þ Z jt  Et ½qt;tþ1 Z jtþ1  þ P t sjt :

ð3Þ

According to (3), the payoff from state contingent claims net of investments in a new portfolio can be used for consumption purchases. Moreover, lump-sum money transfers P t st which households receive in the asset market raise the amount of liquid funds and, thus, alleviate the goods market restriction. These cash transfers serve as the central bank’s instrument.8 We assume that households monopolistically supply differentiated labor services lj , which are transformed into aggregate R 1 11=gt 11=gt ¼ 0 ljt dj. The elasticity of substitution between differentiated labor services gt > 1 varies labor input lt where lt exogenously over time, leading to distortionary (cost-push) shocks. Cost minimization then leads to the following demand R 1 1g 1g for differentiated labor services ljt , ljt ¼ ðwjt =wt Þgt lt , with wt t ¼ 0 wjt t dj, where wt denotes the aggregate real wage rate. Maximizing the objective (1), subject to the budget constraint (2), the cash-in-advance constraint (3), the labor demand condition, and the no-Ponzi-game condition, for given initial values Z j0 and M j;1 leads to the following first order conditions:

uc ðcjt Þ ¼ kjt þ wjt ; vl ðljt Þ ¼ n1 t wjt kjt ; kjtþ1 þ wjtþ1 b kjtþ1 þ wjtþ1 ; qt;tþ1 ¼ ; kjt ¼ bEt ptþ1 ptþ1 kjt þ wjt

ð4Þ

and the goods market constraint (3), wt ðMt1 þ Z t  Et ½qt;tþ1 Z tþ1  þ P t st  Pt ct Þ ¼ 0 and wt P 0, where p denotes the inflation rate (pt ¼ Pt =P t1 ), k the multiplier on the budget constraint (2), and w the multiplier on the cash constraint (3). Further, nt ¼ gt =ðgt  1Þ denotes the stochastic wage mark-up, which will be discussed below in more detail. Furthermore, the budget constraint (2) holds with equality and the transversality condition, limi!1 btþi Et ½kjtþi ðMjtþi þ Z jtþ1þi Þ=Ptþi  ¼ 0, must be satisfied. The one-period gross interest rate on a nominally risk-free portfolio is defined as follows

Rt ¼ ½Et qt;tþ1 1 :

ð5Þ

In accordance with many recent contributions to the monetary policy literature (see, e.g., Woodford, 2003), we will consider the risk-free interest rate Rt as the central bank’s operating target. The final consumption good is an aggregate of differentiated goods produced by monopolistically competitive firms in1 R 1 1 dexed with i 2 ½0; 1. The CES aggregator of differentiated goods is defined as yt  ¼ 0 yit di, with  > 1, where yt is the number of units of the final good, yit the amount produced by firm i, and  the constant elasticity of substitution between these differentiated goods. Let P it and P t denote the price of good i set by firm i and the price index for the final good. The demand R1   ¼ 0 P1 for each differentiated good is yit ¼ ðP it =Pt Þ yt , with P 1 t it di. A firm i produces good yi employing a technology

8

As will be demonstrated below, this is equivalent to a specification where the supply of money is adjusted via asset exchanges, i.e., open market operations.

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R1 which is linear in the labor input: yit ¼ lit , where lt ¼ 0 lit di. Hence, labor demand satisfies: mcit ¼ wt , where mcit ¼ mct denotes real marginal costs. We allow for a nominal rigidity in form of staggered price setting as developed by Calvo (1983). Each period firms may reset their prices with the probability 1  / independently of the time elapsed since the last price setting. The fraction  Pit1 , where p  de/ 2 ð0; 1Þ of firms are assumed to adjust their previous period’s prices according to the simple rule Pit ¼ p notes the average inflation rate. Firms maximize their market value, which equals the expected sum of discounted dividends P e it as Dit  ðP it  P t mct Þyit .9 In each period a measure 1  / of randomly selected firms set new prices P Et 1 s¼0 qt;tþs Ditþs , whereP s se s e     / q ð p y  P mc y Þ, s.t. y ¼ ð p Þ P y . The first order condition for the price of the solution to maxe Et 1 P P tþs tþs it it t;tþs itþs itþs itþs tþs tþs s¼0 P it re-optimizing producers is given by

e it ¼ P

 1

Et

P1

s s¼0 /

Et

P1

h

þ1 s qt;tþs ytþs Ptþs p mctþs

  ð1Þs s  s¼0 / ½qt;tþs ytþs P tþs

p

i :

ð6Þ

R1    e  .  P t1 Þ þ ð1  /Þ P Aggregate output is given by yt ¼ ðP t =P t Þ lt , where ðP t Þ ¼ 0 P  ¼ /ðp it di and thus ðP t Þ t The central bank is assumed to trade with households in the asset markets. There, it also transfers money in a lump-sum R1 way, P t st ¼ 0 Pt sjt dj. Its budget constraint is given by

Pt st ¼ ðlt  1ÞMt1 ;

ð7Þ

where lt denotes the gross money growth rate lt ¼ mt pt =mt1 (where m denotes real balances mt ¼ Mt =P t ). We assume that the central bank transfers money, i.e., controls the money growth rate lt , contingent on current period information. It should be noted that we can, alternatively, assume that money and riskless government bonds are exclusively traded in open market operations, where their supply is characterized by ‘‘holding fiscal policy constant in the face of a government asset exchange”, Mt  M t1 ¼ ðBt  Rt1 Bt1 Þ, with Bt denoting government bonds (see Sargent and Smith, 1987). Together with a consistent goods market constraint, which then reads Pt cjt 6 Mjt1 þ Z jt  Et ½qt;tþ1 Z jtþ1  þ Rbt1 Bjt1  Bjt , this specification is equivalent, i.e., leads to an identical set of equilibrium conditions.10 Like in Alvarez et al. (2001), an open market bond purchase is then measured by lt > 1. The wage mark-up is assumed to follow the stochastic process nt ¼ gt =ðgt  1Þ ¼ n1q nqt1 expðet Þ, where q 2 ½0; 1Þ and nt1 þ ð1  qÞ n þ et , where  n > 0 and ~ xt denotes the log of a generic variable n > 1. Taking logs, one therefore obtains ~ nt ¼ q~ xt ¼ logðxt Þ, and x ¼ E0 ~ xt , ~ xt . The innovations et are assumed to be normally distributed with mean zero and a constant varnt1 þ et , where ^ nt ¼ q^ nt ¼ ~ nt   n. Throughout the paper, iance, et  Nð0; var e Þ. The stochastic process can also be written as ^ we further assume that the bounds on the mark-up fluctuations are sufficiently tight, such that the central bank can ensure the nominal interest rate to be larger than one, Rt > 1, such that the cash-in-advance constraint is binding.11 e t ; mct ; wt ; mt ; Rt ; l g1 satisfying the firms’ first order A rational expectations equilibrium is a set of sequences fyt ; lt ; P t ; P t ; P t t¼0 1 e it ¼ P e t , and P1 ¼ /ðp e 1 , the households’ first order conditions  P Þ þ ð1  /Þ P conditions mct ¼ wt , (6) with P t1 t t # r r r y t wt ¼ Rt lt nt , yt =P t ¼ bRt Et ½ytþ1 =P tþ1 , P t yt ¼ lt mt1 P t1 for Rt > 1 and P t yt 6 lt mt1 P t1 for Rt ¼ 1, lt ¼ mt pt =mt1 , the e  , the transversality condition, and a  Pt1 Þ þ ð1  /Þ P aggregate resource constraint yt ¼ ðPt =Pt Þ lt , where ðP t Þ ¼ /ðp t  , and initial values P > 0, P > 0, and m P1 ¼ M1 > 0. monetary policy, for a sequence fnt g1 1 1 1 t¼0 3. Results In this section, we examine the equilibrium relation between of the risk-free nominal interest rate Rt and state contingent money transfers lt in the benchmark model.12 In particular we assess the implementation of forward-looking or current-looking interest rate targets, which are often applied to summarize short-run monetary policy behavior (see e.g. Taylor, 1993; Alvarez et al., 2001; Woodford, 2001; Honkapohja and Evans, 2003; or Schmitt-Grohé and Uribe, 2007, to name a few). These interest rate targets for example encompass interest rate reaction functions consistent with the plan of a welfare-maximizing central bank acting in a discretionary way (see below). Following common practice, we conduct a local analysis based on a linear approximation of the model at the steady state. Applying a first-order Taylor-expansion, ^ xt will denote the percent deviation of a generic variable xt from its (deterministic) steady state value x : ^ xt ¼ log xt =x. The local approximation is conducted at the model’s steady state with a target inflation rate p > b. Different monetary policy specifications are assumed to be consistent with the same steady state, bR ¼ l ¼ p.

9 It should be noted that the application of the households’ stochastic discount factor qt;tþs in principle implies that dividends also deliver a liquidity value w, i.e., can be used for purchases of consumption goods. We neglected this property to specify the cash constraint in a conventional way. The inclusion of dividends on the right hand side of (3) would not (qualitatively) change any result in this paper, since dividends are either constant (for / ¼ 0) or – in equilibrium – solely a function of current output (for / > 0). R1 10 The households’ budget constraint would then be given by Bjt þ M jt 6 Rbt1 Bjt1 þ M jt1 þ Z jt  Et ½qt;tþ1 Z jtþ1  þ Pt wjt ljt  Pt cjt þ 0 Dj;it di, and arbitrage b pricing would imply Rt ¼ Rt . A consistent initial value for total government liabilities would be equal to zero, B1 þ M 1 ¼ 0, which ensures government solvency in any stable equilibrium with open market asset exchanges. 11 In equilibrium, households’ net wealth solely consists of money balances, Z t ¼ 0, such that the cash-in-advance constraint (3) reduces to Pt ct 6 M t1 þ Pt st . Using (7) gives P t ct 6 lt M t1 , and for Rt > 1 the equilibrium relation P t ct ¼ lt M t1 . 12 In the subsequent section, we demonstrate the robustness of the main results for an alternative money demand, i.e., for a money-in-the-utility function specification.

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Assuming that the bounds on shocks are sufficiently tight, we focus on the case where the cash-in-advance constraint (3) is always binding. In a neighborhood of the steady state the equilibrium sequences are approximated by the solutions to the linearized equilibrium conditions. The rational expectations equilibrium REE of the log-linear model is then defined as follows. ^t, m ^ t, Definition 1. A REE of the log-linear approximation to the model at the steady state with Rt > 1 is a set of sequences fp b t g1 converging to the steady state in a non-oscillatory way and satisfying ^t , R y t¼0

ry^t ¼ rEt y^tþ1  Rb t þ Et p^ tþ1 ; p^ t ¼ bEt p^ tþ1 þ xy^t þ v Rb t þ v^nt ;

ð8Þ ð9Þ

^t ¼ m ^ t; y

ð10Þ

where x ¼ vð# þ rÞ and v ¼ ð1  /Þð1  b/Þ=/, and the transversality condition, for a sequence f^ nt g1 t¼0 , initial values for nominal balances M 1 and the price level P1 , and a monetary policy. Throughout the analysis we focus on the fundamental solution, which satisfies commonly used equilibrium selection devices. Definition 2. A fundamental solution of the REE exhibits a minimum set of state variables. A REE is uniquely determined (equilibrium determinacy), if the fundamental solution is the only solution to the REE. b t g1 satisfy the ^t; m ^ t; y ^t ; R It should be noted that we disregard uninteresting and unrealistic cases where sequences fp t¼0 equilibrium conditions but exhibit oscillatory dynamics, which should be avoided by a welfare-maximizing central bank.13 It should be noted that the equilibrium condition (9) differs slightly from the standard version of the New Keynesian Phillips curve by the nominal interest rate entering the aggregate supply constraint (9) due to the presence of a monetary friction, i.e. the cash constraint (3). However, the main results are qualitatively unchanged if the aggregate supply constraint takes a ^ tþ1 þ xy ^ t ¼ bp ^t þ v^nt (see Section 4). more conventional form, p 3.1. Interest rate targets For the subsequent analysis, we consider the following form for an interest rate target that is either current- or forwardlooking, which is sufficiently general to encompass a wide variety of interest rate rules applied in the literature:14

b t ¼ q Et p ^tþi þ qn ^nt ; R p;i ^ tþi þ qy;i Et y

i 2 f0; 1g

ð11Þ

Following large parts of the literature, we first assume that the central bank is able to ensure the equilibrium interest rate sequence to satisfy certain targets (like 11), while abstracting from the particular implementation via transfers or open market operations. Under this assumption the dynamics of real and nominal balances can in principle be disregarded, since one b t g1 and subsequently for the end-of-period real balances ^ t; y ^t ; R ^t; m can separately solve for a set of equilibrium sequences fp t¼0 sequence by (10). The following proposition presents a necessary condition for equilibrium determinacy for the case where the interest rate is set according to (11). Proposition 1. Consider the case where the central bank guarantees an interest rate target (11) to be satisfied in equilibrium without controlling the money growth rate lt . Then, a necessary condition for equilibrium determinacy is

qp;i  qy;i ,1 > 1; i 2 f0; 1g;

ð12Þ

where ,1 ¼ ðb þ v  1Þ=x. Proof. See appendix A.1. h Condition (12), which is necessary for a unique solution to a REE, slightly differs from the well-known determinacy condition for the well-known cashless case, qp;i þ qy;i ð1  bÞ=x > 1 (see Woodford, 2003) due to the existence of the monetary friction (see for example Brueckner and Schabert, 2003). Given that ,1 > 0 holds for standard values for / (where v > 1  b), equilibrium determinacy requires the interest rate target to rise by more than one for one with current or expected future inflation, qp;i > 1 if qy;i > 0. Though, interest rate targets that are consistent with a time consistent plan do not necessarily satisfy the condition (12), the central bank can easily design a consistent interest rate target associated with equilibrium determinacy.15

13

In the working paper version of this paper the case of oscillatory dynamics explicitly considered in the analysis (see Schabert, 2005). Note that (11) encompasses targets consistent with an optimal discretionary policy plan: As shown by Ravenna and Walsh (2006) for a model that is isomorphic to the one in Definition 1, applying a second-order approximation of household welfare (1) at an undistorted steady state leads to a loss function P1 t 2 ^ t þ ðx=Þy ^2t , where y ^t serves as a measure for the output-gap. They show that a welfare maximizing plan under discretion is associated with the E0 t¼0 b ½p # ^t where a ¼  rþ# ^t ¼ y . Using the latter, an interest rate target (11) is known to be consistent its plan (see Clarida et al., 1999). first order condition ap 15 Some more details can be found in the working paper version of this paper (see Schabert, 2005). 14

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3.2. Implementation of interest rate targets We now consider the more realistic case where the central bank cannot directly set the market interest rate Rt . In order to implement an interest rate target of the form (11), the central bank instead adjusts the stock of money in a state contingent way. To be more precise, the central bank can set its instrument, i.e., lump-sum cash transfers contingent on changes in macroeconomic variables and the exogenous state. Thus, we consider a reaction function, which allows the central bank to adjust the money growth rate in the following way (i.e., in response to all relevant macroeconomic information in period t):

l^ t ¼ lp p^ t þ ly y^t þ ln ^nt ;

ð13Þ

^t ¼ m ^t þp ^t  m ^ t1 . Of course, one can alternatively assume that the central bank controls end-of-period real balwhere l ances M t =P t instead of the growth rate by (13). Yet, we refrain from modelling policy as a rule for real balances, since it would just describe a target (like the interest rate) instead of the policy instrument. b t  Et p ^ tþ1  rm ^t ¼ R ^ tþ1 , which can be Using the money demand condition (10) to eliminate output in (8) leads to rEt m b t þ ðr  1ÞEt p ^ tþ1 ¼ R ^ tþ1 . Under a money supply policy (13), rewritten in terms of the expected money growth rate rEt l the equilibrium interest rate sequence thus satisfies the following relation  b t ¼ q  Et p ^tþ1 þ qn ^nt ; R p;1 ^ tþ1 þ qy;1 Et y

where qp;1  1 ¼ rðlp  1Þ; qy;1 ¼ rly ; and qn ¼ rln q:

ð14Þ

The equilibrium relation (14) reveals that the feedback from (expected) inflation and output to the money growth rate and the corresponding feedback coefficients for an implementable interest rate ‘‘target” are closely related. In contrast to the case, where monetary implementation is disregarded (see, e.g. Proposition 1), the fundamental solution ^ t1 as a relevant state variable. This property is due to (i) to the REE under (13) exhibits beginning-of-period real money m prices evolving in a backward-looking way, (ii) the beginning-of-period stock of nominal balances M t1 being predetermined, and due to the fact that (iii) end-of-period money holdings Mt cannot be controlled independently of M t1 . As a consequence, goods market expenditures and, thereby, the equilibrium allocation in general depend on predetermined money holdings. Since real money mt1 ¼ Mt1 =Pt1 becomes a relevant state variable, the fundamental solution to the REE takes the form b t ¼ dRm m ^t ¼ dm m ^ t ¼ dpm m ^t ¼ y ^ t1 þ dme ^ ^ t1 þ dpe ^ ^ t1 þ dRe ^ nt , p nt , and R nt , where dm is the single stable eigenvalue. The followm ing proposition summarizes the main properties of the REE under (13). Proposition 2. The fundamental solution to the REE for a money supply policy (13) exhibits history dependence, ^ t1 ¼ oy ^t n om ^ t1 ¼ dm –0 and op ^ t n om ^ t1 ¼ dpm –0 if lp –1. A REE exists if only if ^ t n om om

lp < 1 þ ,1 ly ;

ð15Þ

~ 1p ; l ~ 2p Þ or lp < minfl ~ 1p ; l ~ 2p ; l ~ 3p g for r < #, or ii.) l ~ 3p < lp < minfl ~ 1p ; l ~ 2p g and is uniquely determined if and only if i.) lp 2 ðl ~ 1p ¼ 1 þ ðb þ vÞðly  1ÞðrvÞ1 , l ~ 2p ¼ 1 þ ly ,1 , and l ~ 3p ¼ 1 þ ðly  2Þð1 þ b þ vÞ=vðr  #Þ. for r > #, where l Proof. See Appendix A.2.

h

Condition (15) is a necessary and sufficient condition for the existence of a REE: if the money growth rate increases with inflation, then any rise in the latter leads to a monetary expansion which tends to raise aggregate demand. If lp  ,1 ly > 1, this feedback can lead to a further rise in the price level such that the sequences of macroeconomic aggregates become unstable. Put in technical terms, beginning-of-period real money serves as a relevant state variable under (13) such that existence of convergent equilibrium sequences requires one stable eigenvalue, which is ensured by (15). In contrast, when monetary implementation is disregarded the fundamental solution to the REE under an interest rate target (11) does not exhibit any predetermined state variable (see proof of Proposition 1). Equilibrium determinacy then requires the absence of stable eigenvalues, which relies on the interest rate target satisfying (12). The fundamental solutions examined in Proposition 1 and 2 are evidently not identical, since real money becomes a relevant state variable only if the central bank controls money supply. Nevertheless, the central bank can implement equilibrium interest rate sequences that satisfy targets of the type (11) by supplying money in an appropriate way. The following proposition summarizes some main results based on the equilibrium relation (14). Proposition 3. Suppose that the central bank supplies money according to (13). Then, 1. interest rate targets (11) satisfying qp;0 6 1, qy;0 P 0, and qn ¼ 0 can be implemented in equilibrium for ,1 > 0. 2. interest rate targets, which ensure equilibrium determinacy when money supply is neglected, cannot be implemented in an REE. Proof. See appendix A.3. h According to the first claim in Proposition 3, a simple passive interest rate target can be implemented in equilibrium by a particular choice of ln leading to qn ¼ 0. The second and more fundamental claim summarizes that an interest rate target, which is designed to ensure equilibrium determinacy in an entirely forward-looking economy (with money supply ne-

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glected), cannot be implemented via money injections. For example, a simple active interest rate target qp;i > 1 and qy;i ¼ 0 for i ¼ 1 cannot be implemented, since qp;1  1 ¼ rðlp  1Þ, qy;1 ¼ ly , and (15) can then not simultaneously be satisfied. The simple reason for this result is that all eigenvalues are unstable for such an interest rate target, which necessarily leads to unstable dynamics when money supply causes a predetermined state variable mt1 to be relevant for the equilibrium allocation. This result holds regardless whether the interest rate target is specified in terms of current or of expected future realizations of endogenous variables, and does not rely on the cash-in-advance specification of money demand (see Section 4). Note that a central bank can, nevertheless, implement a welfare maximizing plan under discretion, (8)-(10) and ^t , by transferring money in a non-destabilizing way.16 Thus, the central bank’s inability to implement interest rate ^t ¼ y ap targets has no direct consequences for its ability to implement an REE that satisfies an optimal plan (see also Chowdhury and Schabert, 2008).

4. Interest elastic money demand In this last part we assess the robustness of the main results for an alternative money demand specification, which is widely used in the New Keynesian literature (see McCallum and Nelson, 1999), namely, a money-in-the-utility (MIU) function specification. Consider that households have preferences which are characterized by real balances entering the utility function in a separable way (see, e.g., Woodford, 2003):

U m ðct ; lt ; M t =Pt Þ ¼ Uðct ; lt Þ þ mðMt =Pt Þ;

ð16Þ

where mðmt Þ is strictly increasing, concave, and twice continuously differentiable. Log-linearization of the households’ first order conditions at a steady state with R > 1 and a positive elasticity of intertemporal substitution of money rm ¼  mmmmmm > 0 leads to the following money demand condition

bt : ^ t ¼ r^ct  ðR  1Þ1 R rm m

ð17Þ

Compared to the benchmark specification, money demand is now interest rate elastic and exhibits a consumption (income) elasticity that might be different from one. Due to the absence of the cash constraint (3), the aggregate supply relation (9) now reads

p^ t ¼ bEt p^ tþ1 þ xy^t þ v^nt :

ð18Þ

A rational expectations equilibrium of the MIU-model can then be defined as follows: a REE of the log-linear approximab t g1 con^t, m ^ t, y ^t , R tion to the MIU model at the steady state with sticky prices ð/ > 0Þ and Rt > 1 is a set of sequences fp t¼0 verging to the steady state in a non-oscillatory way and satisfying (8), (17), and (18), and the transversality condition, for a sequence f^ nt g1 t¼0 , initial values for nominal balances M 1 and the price level P 1 , and a monetary policy. The following proposition summarizes the local dynamic properties of money-in-the-utility function model under (13). Proposition 4. There exists a REE of the MIU model for a money supply policy (13) if only if

lp þ ,4 ly < 1; where ,4 ¼ ð1  bÞ=x > 0. The REE is uniquely determined if and only if (19) and lp þ ly

ð19Þ 1þb

x

< 1 þ 2,5 , where ,5 ¼

xþrðRþ1Þðbþ1Þ . ðR1Þxrm

Proof. See appendix A.4. h Like in the benchmark version (see Proposition 2), equilibrium sequences of output and inflation evolve in a history dependent way. The existence of a REE demands the money growth rate not to increase too strongly with inflation and output (see 19).17 If however, money supply is neglected, the fundamental equilibrium solution under (11) exhibits no endogenous state variable. Uniqueness of the rational expectations equilibrium solution under (11) then requires qp;i þ ,4 qy;i > 1 where i 2 f0; 1g, which is also known as the Taylor-principle (see Woodford, 2001). These requirements again restrict the set of implementable interest rate targets. The following proposition summarizes the main results.18 Proposition 5. Consider the MIU model and suppose that the central bank supplies money according to (13). Then, 1. The equilibrium interest rate sequence satisfies (11) with

qp;i ¼ ½rm ðlp  1Þ þ 1Cdim and qy;i ¼ ly rm Cdim ; i 2 f0; 1g

ð20Þ

where C ¼ ðR  1Þ=ðR  dm Þ.

16 17 18

Details are available upon request from the author. The same condition is shown in Chowdhury and Schabert (2008) to be necessary for equilibrium determinacy under an inertial money growth rule. Like before, an optimal plan of a central bank acting under discretion can be consistent with forward-looking interest rate targets satisfying (11).

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2. Interest rate targets (11) with qp;i þ ,4 qy;i < 1 for i 2 f0; 1g and qn ¼ 0 can be implemented in an REE. 3. Interest rate targets, which ensure equilibrium determinacy when money supply is neglected, cannot be implemented in an REE. Proof. See appendix A.5. h A (non-accommodating) money supply satisfying lp þ ,4 ly < 1 leads to a stable and positive eigenvalue dm 2 ð0; 1Þ and to C < 1. According to (20), an interest rate target violating the Taylor-principle, qp;i þ ,4 qy;i < 1, can be implemented by a money supply (13). In contrast, a central bank cannot implement an interest rate target satisfying qp P 1 and qy P 0 (or the Taylor-principle) by supplying money such that lp þ ,4 ly < 1, which is necessary for the existence of an REE (see Proposition 4). The reason is – like in the benchmark case – that an interest rate target that is designed to rule out multiple equilibria when money supply is neglected, would imply unstable dynamics when the central bank controls the stock of money. Hence, the main results derived in the cash-in-advance model with sticky prices are qualitatively unchanged in a version with an interest rate elastic money supply. Like in the benchmark model, the consumption Euler equation governs the relation between money growth and interest rates even though money demand is interest rate elastic. Thus, as long as financial markets are frictionless and money demand rises with income (consumption), a higher growth rate of money tends to be associated with higher interest rates.

5. Conclusion Monetary policy is often summarized by state contingent interest rate targets (rules). In accordance with this practice, numerous recent studies on optimal monetary policy focus on the design of such rules. Nevertheless, short-run nominal interest rates mostly serve as an operating target in reality, while they are implemented by money supply. This paper takes a closer look at money supply and interest rate targets in a monetary model where prices are sticky, financial markets are frictionless, and the central bank uses lump-sum transfers to adjust money supply. The implementation of interest rate targets that rise with inflation, is shown to be associated with money growth rates that also rise with inflation. Money supply is found to destabilize macroeconomic aggregates if the central bank aims at implementing forward-looking interest rate targets satisfying the Taylor-principle. These findings do not imply that the central bank is unable to implement a particular optimizing (welfare-maximizing) plan. The analysis in this paper rather reveals that standard sticky price models seem to be suited when the central bank is able to implement monetary policy via ‘‘open mouth operations”. However, to account for the implementation of interest rate targets via lump-sum transfers or open market operations, some kind of financial market friction, like asset market segmentation (see Alvarez et al., 2002), seems to be necessary, which relates to the solution of the well-known liquidity puzzle. These types of frictions might, however, be non-negligible for the analysis of optimal monetary policy. In this case, the design of an optimal monetary target and its implementation cannot be analyzed separately. Acknowledgements The author would like to thank Klaus Adam, Roel Beetsma, Charles Carlstrom, Matt Canzoneri, Fiorella de Fiore, Ludger Linnemann, Federico Ravenna, Leopold von Thadden, and seminar participants at the European Central Bank for suggestions and comments. Parts of paper have been written while the author was visiting the ECB Directorate General Research, as part of the Research Visitor Programme. He thanks the ECB for a stimulating research environment. This research is part of the RTN project ‘‘Macroeconomic Policy Design for Monetary Unions”, funded by the European Commission (contract number HPRN-CT-2002-00237). The views expressed in this paper are those of the author and do not necessarily reflect the views of any institution. The usual disclaimer applies. Appendix A A.1. Proof of Proposition 1 To establish the claim made in the proposition, we apply the equilibrium conditions (8)-(9) together with an interest rate b t ¼ q Et p ^tþi þ qn ^ nt where i 2 f0; 1g leading to target R p;i ^ tþi þ qy;i Et y

ry^t ¼ rEt y^tþ1  qp;i p^ tþi  qy;i Et y^tþi  qn ^nt þ Et p^ tþ1 ; p^ t ¼ bEt p^ tþ1 þ xy^t þ vqp;i p^ tþi þ vqy;i Et y^tþi þ vðqn þ 1Þ^nt ; ^ty ^ tþ1 Et y ^tþ1 Þ0 ¼ Aðp ^t Þ0 þ e^ which can be written as ðEt p nt . This system is entirely forward-looking, such that the REE is determined if and only if all eigenvalues are unstable. In what follows we use that the 2  2-matrix A exhibits two unstable eigenvalues if and only if detðAÞ > 1, detðAÞ  traceðAÞ > 1, and detðAÞ þ traceðAÞ > 1. Firstly, consider the case where the bt ¼ q p ^tþ1 þ qn ^ nt . The matrix A then exhibits central bank reacts to current inflation and expected future output, R p;0 ^ t þ qy;1 Et y the following properties: detðAÞ ¼ ðr þ v#qp;0 Þ=ðrb  ðb þ vÞqy;1 Þ

A. Schabert / Journal of Macroeconomics 31 (2009) 333–344

341

qy  rb  x þ xqp;0 and rb  bqy;1  vqy;1 2r þ rb þ rv þ v#  qy;1  rvqp;0 þ v#qp;0 : detðAÞ þ traceðAÞ ¼ rb  bqy;1  vqy;1 detðAÞ  traceðAÞ ¼

Hence, detðAÞ > 1 if

qp;0 þ qy;1

rb  ðb þ vÞqy;1 > 0 and v#qp;0 þ ðb þ vÞqy;1 > rð1  bÞ, detðAÞ  traceðAÞ > 1 if and only if

1vb

x

> 1;

and detðAÞ þ traceðAÞ > 1 if and only if qp;0 vðr  #Þ þ qy;1 ð1 þ b þ vÞ < 2rð1 þ bÞ þ x. Secondly, consider the case where ^ tþ1 þ qy;0 y ^ t þ qn ^ nt . The matrix A then exhibthe central bank reacts to current output and expected future inflation, Rt ¼ qp;1 p its the following properties: detðAÞ ¼ ðr þ qy;0 Þ=ðrb þ rvqp;1 Þ,

rb þ rv þ v#  qy;0 þ bqy;0 þ vqy;0  v#qp;1 and ðb þ vqp;1 Þr 2r þ rb þ rv þ v# þ qy;0 þ bqy;0 þ vqy;0  v#qp;1 detðAÞ þ traceðAÞ ¼ : ðb þ vqp;1 Þr

detðAÞ  traceðAÞ ¼ 

Hence, detðAÞ > 1 if

qp;1 þ qy;0

qy;0 rb þ rvqp;1 > 0 and qp;1 < 1b v þ rv , detðAÞ  traceðAÞ ¼> 1 if and only if

1vb

x

>1

vþ1 and detðAÞ þ traceðAÞ > 1 if and only if qp;1 < 1 þ 2xr þ qy;0 bþx . Thirdly, consider the case where the central bank reacts to bt ¼ q p ^ ^ ^ þ q expected future inflation, and output R y p;1 tþ1 y;1 tþ1 þ qn nt . The matrix A then exhibits the following properties: detðAÞ ¼ r=ðbqy;1  rb þ vqy;1  rvqp;1 Þ,

qy;1  rv  v#  rb þ v#qp;1 and rb  bqy;1  vqy;1 þ rvqp;1 2r þ rb þ rv þ v#  qy  v#qp;1 : detðAÞ þ traceðAÞ ¼ rb  bqy;1  vqy;1 þ rvqp;1

detðAÞ  traceðAÞ ¼

v b Hence, detðAÞ > 1 if qp;1 > bþ rv qy;1  v, detðAÞ  traceðAÞ > 1 if and only if

qp;1 þ qy;1

1vb

x

>1

and detðAÞ þ traceðAÞ > 1 if and only if qp;1 vð#  rÞ þ qy;1 ð1 þ b þ vÞ < 2rð1 þ bÞ þ x. Fourthly, consider the case where bt ¼ q p ^t þ qn ^ nt . The matrix A then exhibits the following the central bank reacts to current inflation and output, R p;0 ^ t þ qy;0 y properties: detðAÞ ¼ ðr þ qy;0 þ v#qp;0 Þ=ðrbÞ,

qy;0  rv  v#  rb  bqy;0  vqy;0 þ rvqp;0 þ v#qp;0 and br 2r þ rb þ rv þ v# þ qy þ bqy;0 þ vqy;0  rvqp;0 þ v#qp;0 detðAÞ þ traceðAÞ ¼ : br

detðAÞ  traceðAÞ ¼

Hence, detðAÞ > 1 if qy;0 þ v#qp;0 > rð1  bÞ, detðAÞ  traceðAÞ ¼> 1 if and only if

qp;0 þ qy;0

1vb

x

>1

and detðAÞ þ traceðAÞ > 1 if and only if qp;0 vðr  #Þ  qy;0 ðb þ v þ 1Þ < 2rð1 þ bÞ þ x. Thus, for all four possible interest rate targets satisfying (11) the condition qp;i þ qy;i 1xvb > 1 for i 2 f0; 1g is necessary for equilibrium uniqueness. h A.2. Proof of Proposition 2 To establish the claims in the proposition, we eliminate the nominal interest rate in the equilibrium conditions (8) and ^ t , leading – together with the money supply rule – to the following set of equilibrium conditions ^t ¼ m (10), and use y

^ t þ vrEt m ^ tþ1 þ v^nt ; p^ t ¼ ðb þ vÞEt p^ tþ1 þ ðx  vrÞm ^ ^t m ^ t1 ¼ ðlp  1Þp ^ t þ ln nt : ð1  ly Þm

ð21Þ ð22Þ

Given that the predetermined value of beginning-of-period real balances enters the set of equilibrium conditions, the generic form for the fundamental solution reads

^ t ¼ dm m ^ t1 þ dme ^nt ; m

^ t1 þ dpe ^nt : p^ t ¼ dpm m

ð23Þ

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Inserting the latter solution in (21)-(22), leads to the following conditions for the undetermined coefficients

0 ¼ dm ðb þ vÞdpm  dpm þ rvd2m þ dm ðx  rvÞ;

ð24Þ

0 ¼ ð1  ly Þdm  1  ðlp  1Þdpm ;

ð25Þ

0 ¼ v  dpe þ rvðqdme þ dm dme Þ þ ðb þ vÞðqdpe þ dme dpm Þ þ dme ðx  rvÞ;

ð26Þ

0 ¼ ð1  ly Þdme þ ðlp  1Þdpe þ ln :

ð27Þ

Condition (25) implies dpm ¼ ½ðly  1Þdm þ 1=ð1  lp Þ if lp –1. We now use the conditions (24)-(25) to examine the eigenvalues of the model. Suppose that lp ¼ 1. Then, the eigenvalue dm is given by dm ¼ 11l . Otherwise, we have to check y the roots of the following quadratic (characteristic) polynomial

GðXÞ ¼ X 2 þ X

ðlp  1Þx  v  b  ðlp  1Þrv þ ly  1 1 þ : rvðlp  1Þ þ ðb þ vÞð1  ly Þ rvðlp  1Þ þ ðb þ vÞð1  ly Þ

Given that the model exhibits one predetermined variable, existence of a REE requires at least on stable eigenvalue. For this, we examine the value of G at X ¼ 0, which is given by Gð0Þ ¼ ½rvðlp  1Þ þ ðb þ vÞð1  ly Þ1 . Consider first the case v where Gð0Þ < 0 () lp < 1 þ bþ rv ðly  1Þ. Then the model exhibits a stable root if Gð1Þ, which is given by

Gð1Þ ¼

ly ð1  b  vÞ þ xðlp  1Þ ; rvðlp  1Þ þ ðb þ vÞð1  ly Þ l ðbþv1Þ

is positive, Gð1Þ > 0. This, obviously requires lp < 1 þ y x that there is a single stable root, Gð1Þ, which is given by

Gð1Þ ¼ 

, which is assumed to be satisfied in what follows. To ensure

ð#  rÞvðlp  1Þ þ ðb þ v þ 1Þðly  2Þ ; rvðlp  1Þ þ ðb þ vÞð1  ly Þ

has to be negative, Gð1Þ < 0. To disclose the conditions for this, we have to distinguish the cases r > # and r < #. Suppose vþ1Þ r > # () 2rv  x > 0. Then, we can conclude that Gð1Þ < 0 if lp > 1 þ ðbþ ðly  2Þ. Now suppose that 2rvx ðbþvþ1Þ r < # () 2rv  x < 0. Then, Gð1Þ < 0 if lp < 1 þ 2rvx ðly  2Þ. Finally, consider the case where ly ðbþv1Þ v . In this case there canGð0Þ > 0 () lp > 1 þ bþ rv ðly  1Þ. Then, the existence of a stable root again requires lp < 1 þ x

that

not be another stable root. Hence, the REE is uniquely determined, i.e., there is a unique stable and positive (ruling out oscil~ 1p ; l ~ 2p Þ, or iiÞlp < minfl ~ 1p ; l ~ 2p ; l ~ 3p g for r < # and latory dynamics) eigenvalue dm 2 ð0; 1Þ, if and only if iÞlp 2 ðl ðbþvÞðly 1Þ ly ðbþv1Þ ðbþvþ1Þðly 2Þ ~ ~ ~ ~ ~ ~ l3p < lp < minfl1p ; l2p g for r > #, where l1p ¼ 1 þ rv , l2p ¼ 1 þ x , and l3p ¼ 1 þ 2rvx . Existence of a REE is already ensured if

lp < 1 þ ly ðbþxv1Þ. Then, dpm ¼ ½ðly  1Þdm þ 1=ð1  lp Þ–0 if lp –1. h

A.3. Proof of Proposition 3 To establish the first claim in the proposition, we assess the equilibrium behavior of the nominal interest rate under the money supply (13). For this, we rewrite (14) in terms of current realizations of the endogenous variables, using the solution ^ t1 þ dme ^ ^ tþ1 ¼ Et ðdpm m ^ t þ dpe ^ ^ t þ ððq  dm Þdpe þ dpm dme Þ^ ^ t ¼ dm m nt and that Et p ntþ1 Þ ¼ dm p nt . Collecting for real balances m terms, we end up with the following expression

bt ¼ q p ^t þ qn ^nt ; R p;0 ^ t þ qy;0 y where qp;0 ¼ ððr  1Þ þ rlp Þdm

and qy;0 ¼ ly dm ;

ð28Þ

qn ¼ ððr  1Þ þ rlp Þððq  dm Þdpe þ dpm dme Þ þ ly dme q þ ln q; ^t ¼ y ^t . To establish the first claim, we need to show that there exists an REE. where we used the money demand condition m From (28), we can immediately conclude that a REE exists under a money supply (13) if qy;0 P 0 and qp;0 < 1, since ly dm P 0 and 1 > ððr  1Þ þ rlp Þdm () ð1=dm Þ  1 > rðlp  1Þ are satisfied for lp < 1 and ly P 0, which imply dm 2 ð0; 1Þ (see bt ¼ q p ^t further requires choosing Proposition 2). The implementation of a sequence of interest rates satisfying R p;0 ^ t þ qy;0 y ln such that qn ¼ 0. To examine the existence and the uniqueness of such a value for ln we take into account that the solution coefficients are ð1l Þd 1 functions of the policy rule parameter. We know from (24) and (25) that dm and dpm ¼ l y 1m are independent of ln . In p ðl 1Þd þl contrast, dpe and dme depend on the latter. Combining (26) and (27) reveals that dpe and, therefore, dme ¼ p 1lpe n are linear y in ln :

dpe ¼ 

ln ðx  rv þ rvðq þ dm Þ þ ðb þ vÞdpm Þ þ vð1  ly Þ : ðlp  1Þðx  rv þ rvðq þ dm Þ þ ðb þ vÞdpm Þ þ ðqðb þ vÞ  1Þð1  ly Þ

A. Schabert / Journal of Macroeconomics 31 (2009) 333–344

343

As a consequence, qn is also linear in ln . Hence, for any particular equilibrium solution there is a unique value ln , such that qn is equal to zero if the central bank sets ln ¼ ln . Evidently, the central bank can implement equilibrium sequences of bt ¼ q p bt ¼ q p ^t þ q y ^t and R ^t þ q y ^t in an analogous way. interest rates satisfying R p;0

p;1

y;1

y;0

The second claim can easily be established as follows: if an interest rate target leads to eigenvalues that are all unstable (the conditions can be found in the proof of proposition 1), an REE is uniquely determined for the case where the central bank does not control money supply. If, however, the central bank adjusts money supply according to (13) to implement such an interest rate target, the existence of an REE requires at least one stable eigenvalue. Thus, interest rate targets, which ensure equilibrium determinacy when money supply is neglected, cannot be implemented by a money supply (13). h A.4. Proof of Proposition 4 To derive the conditions for stable and non-oscillatory equilibrium sequences in the money-in-the-utility function model under a state contingent money supply (13), we replace the nominal interest rate with the money demand condition, 1 b ^ t  rm m ^ t , in the consumption Euler equation to get Rry ^t ¼ rEt y ^tþ1 þ ðR  1Þrm m ^ t þ Et p ^ tþ1 . Together with the R ¼ ry R1 t aggregate supply constraint and with (13), we obtain a three-dimensional system

0

0 B ðR  1Þrm @ 1

1 0 ^t 0 m CB ^ C B 0 1 r A@ Et ptþ1 A ¼ @ ^tþ1 1 0 0 Et y b

0

10

1

x

0

Rr

lp  1

ly

10

1 0 1 ^ t1 v m CB ^ C B C A@ pt A þ @ 0 A^nt ; ^t 0 y

which exhibits the following characteristic polynomial

Q ðXÞ ¼ X 3 þ X 2

ðR  1Þbrm ly  x  rbð1 þ RÞ  r r þ x þ Rrð1 þ bÞ þ ðly þ xlp  xÞð1  RÞrm R þX  : b br br

Since the three eigenvalues cannot separately be determined, the current values of inflation and output also depend on lagged real money. Given that Q ð0Þ ¼ R=b < 1, we know that the product of the eigenvalues is strictly positive and larger than one. Thus, there exists at least one unstable root, and either three or one positive root. Further, the value of Q ð1Þ is given rm ðxð1  lp Þ  ð1  bÞly Þ. The existence of a stable root lying between zero and one, necessarily requires by Q ð1Þ ¼ ðR1Þ br Q ð1Þ > 0, and therefore

lp þ

1b

x

ly < 1:

Uniqueness additionally requires the remaining roots to lie outside the unit circle. For this, we assess Q ðXÞ at X ¼ 1, which is given by Q ð1Þ ¼ b1r ðrm ðR  1Þðlp x þ ly ð1 þ bÞÞ  ð2 þ ðR  1Þrm Þx  2rðR þ 1Þðb þ 1ÞÞ. Two further stable roots (either complex or real) cannot exist, since there must be at least on unstable root. Thus for Q ð1Þ < 0, there exists exactly xþrðRþ1Þðbþ1Þ one stable eigenvalue. This is ensured by lp þ ly 1þb x < 1 þ 2 ðR1Þxrm . Hence, the REE is uniquely determined if and only if

i.)

xþrðRþ1Þðbþ1Þ 1þb lp þ 1b x ly < 1 and lp þ ly x < 1 þ 2 ðR1Þxrm .

h

A.5. Proof of Proposition 5 To establish the first claim in the proposition, we examine the equilibrium behavior of the nominal interest rates under a state contingent money supply (13) in the money-in-the-utility function model, as in the benchmark model (see appendix 1 b ^t ¼ ^ct , we get ^ t þ R1 ^t , and y R t ¼ ry A.3). Using the consumption Euler equation, money demand rm m 1 b tþ1 þ R R b t and with the money supply reaction function rm Et l^ tþ1 ¼ ðrm  1ÞEt p^ tþ1  R1 Et R R1

R b 1 b tþ1 ¼ ðrm l  ðrm  1ÞÞEt p ^ tþ1 þ ly rm Et y ^tþ1 þ ln rm q^nt : Rt  Et R p R1 R1 b tþ1 ¼ dm R b t þ ððq  dm ÞdRe þ dRm dme Þ^ The fundamental solution under a money supply (see Proposition 4) implies Et R nt . Thus, the current nominal interest rate is characterized by the following equilibrium relation under the fundamental solution to the REE

b t ¼ C½ðrm l  ðrm  1ÞÞEt p ^ tþ1 þ ly rm Et y ^tþ1  þ C R p



ln rm q þ

1 R1

 ððq  dm ÞdRe þ dRm dme Þ ^nt ;

^ tþ1 ¼ dm p ^ t þ ððq  dm Þdpe þ dpm dme Þ^ ^tþ1 ¼ dm y ^t þ ððq  dm Þdye þ dym dme Þ^ where C ¼ Further using that Et p nt , Et y nt , and b t ¼ dm R b t1 þ dRe ^ nt þ ðdRm dme  dm dRe Þ^ nt1 , we can rewrite the interest rate relation as R R1 . Rdm

b t ¼ Cðrm l  ðrm  1ÞÞdm p ^ t þ Cly rm dm y ^t R p " # ðrm lp  ðrm  1ÞÞððq  dm Þdpe þ dpm dme Þ þ ly rm ððq  dm Þdye þ dym dme Þ ^nt þC 1 þln rm q þ R1 ððq  dm ÞdRe þ dRm dme Þ

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To establish the second claim, consider a value ln ¼ ln such that the terms in the squared brackets equal zero. Following the line of arguments for the corresponding problem in appendix A.3, we know that there exists a unique value for ln , since all solution coefficients in the squared brackets are either independent of ln or are linear in ln . For ln ¼ ln , we get an interest b t ¼ q Et p ^tþi , for i 2 f0; 1g, where qp;i ¼ ½rm ðlp  1Þ þ 1Cdim and qy;i ¼ ly rm Cdim . For qp;i þ ,4 qy;i < 1, rate target R p;i ^ tþi þ qy;i Et y it then follows immediately that (19) can be satisfied, such that C; dim 2 ð0; 1Þ and a REE exists. The third claim in the proposition immediately follows from the requirement that there has to exist a stable eigenvalue for a REE under a money supply (13). Hence, the implementation of an interest rate target, which leads to eigenvalues that are all unstable (that is required for equilibrium determinacy when money supply is neglected), would lead to unstable dynamics under (13), such that no REE exists. h References Alvarez, F., Lucas, R.E., Weber, W.E., 2001. Interest rates and inflation. American Economic Review; Papers & Proceedings 91, 219–225. Alvarez, F., Kehoe, P.J., Atkeson, A., 2002. Money, interest rates, and exchange rates with endogenously segmented markets. Journal of Political Economy 110, 73–112. Auray, S., Feve, P., 2008. On the observational (non)equivalence of money growth and interest rate rules. Journal of Macroeconomics 30, 801–816. Blanchard, O.J., Kahn, C.M., 1980. The solution of linear difference models under rational expectations. Econometrica 48, 1305–1313. Brueckner, M., Schabert, A., 2003. Supply side effects of monetary policy and equilibrium multiplicity. Economics Letters 79, 205–211. Calvo, G., 1983. Staggered prices in a utility-maximizing framework. Journal of Monetary Economics 12, 383–398. Carlstrom, C.T., Fuerst, T.S., 2001. Timing and real indeterminacy in monetary models. Journal of Monetary Economics 47, 285–298. Carlstrom, C.T., Fuerst, T.S., 2003. Money growth rules and price level determinacy. Review of Economic Dynamics 6, 263–275. Chowdhury, I., Schabert, A., 2008. Federal reserve policy through a money supply lens. Journal of Monetary Economics 55, 825–834. Christiano, J.L., Eichenbaum, M., Evans, C.L., 1997. Sticky price and limited participation models of money: A comparison. European Economic Review 41, 1201–1249. Clarida, R., Galí, J., Gertler, M., 1999. The science of monetary policy: A new Keynesian perspective. Journal of Economic Literature 37, 1661–1707. Clarida, R., Galí, J., Gertler, M., 2000. Monetary policy rules and macroeconomic stability: Evidence and some theory. Quarterly Journal of Economics 115, 147–180. Honkapohja, S., Evans, G.W., 2003. Friedman’s money supply rule vs. optimal interest rate setting. Scottish Journal of Political Economy 50, 550–566. Lucas Jr., R.E., 1982. Interest rates and currency prices in a two-country world. Journal of Monetary Economics 10, 335–359. Lucas Jr., R.E., 2000. Inflation and welfare. Econometrica 68, 247–274. McCallum, B.T., Nelson, E., 1999. An optimizing IS-LM specification for monetary policy and business cycle analysis. Journal of Money, Credit, and Banking 31, 296–316. Minford, P., Perugini, F., Srinivasan, N., 2002. Are interest rate regressions evidence for a Taylor rule? Economics Letters 76, 145–150. Monnet, C., Weber, W.E., 2001. Money and interest rates. Federal Reserve Bank of Minneapolis Quarterly Review 25, 2–13. Ravenna, F., Walsh, C.E., 2006. Optimal monetary policy with the cost channel. Journal of Monetary Economics 53, 199–216. Sargent, T.J., Smith, B.D., 1987. Irrelevance of open market operations in some economies with government currency being dominated in rate of return. American Economic Review 77, 78–92. Schabert, A., 2005. Money supply and the implementation of interest rate targets. CEPR Discussion Paper DP5094. Schabert, A., 2006. Central bank instruments, fiscal policy regimes, and the requirements for equilibrium determinacy. Review of Economic Dynamics 9, 742–762. Schmitt-Grohé, S., Uribe, M., 2007. Optimal, simple, and implementable monetary and fiscal rules. Journal of Monetary Economics 54, 1702–1725. Vegh, C.A., 2001. Monetary policy, interest rate rules, and inflation targeting: Some basic equivalences. NBER Working Paper No. 8684. Woodford, M., 2001. The Taylor rule and optimal monetary policy. American Economic Review; Papers & Proceedings 91, 232–237. Woodford, M., 2003. Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton University Press, Princeton. Yip, C.K., 2005. Monetary Policy Targeting and Economic Growth: An Equivalence Investigation. The Chinese University of Hong Kong.