Modeling Monetary Policy

Transcription

1 Modeling Monetary Policy Sauel Reynard Swiss National Bank Andreas Schabert TU Dortund University May 22, 29 Abstract In an otherwise standard acroeconoic odel, we odel the central bank as providing oney only in exchange for eligible assets in open arket operations. The relationship between the policy interest rate and expected inflation and consuption growth is affected by oney arket conditions, i.e. the varying liquidity value of eligible assets, and aggregate risk. This induces a data-consistent systeatic wedge between the policy and consuption Euler rates that standard odels equate. Moreover, as the central bank distributes back only its interest earnings and not its entire wealth to households, the relative evolution of eligible assets induces persistent and hup-shaped consuption responses to shocks. JEL classification: E52; E58; E43; E32. Keywords: Monetary Policy; Open arket operations; Liquidity preiu; Money arket rate; Consuption Euler rate; Monetary policy transission. S. Reynard: Swiss National Bank, Research Unit, Boersenstrasse 5, 822 Zurich, Switzerland. Phone: Eail: sauel.reynard@snb.ch. A. Schabert: University of Dortund, Vogelpothsweg 87, Dortund, Gerany. Phone: Eail: andreas.schabert@udo.edu. The views expressed in this paper do not necessarily reflect those of the Swiss National Bank. We are grateful to John Cochrane, Matt Canzoneri, Bezhad Diba, Marty Eichenbau, Jordi Gali, Dale Henderson, Pat Kehoe, Stephanie Schitt-Grohe, Frank Sets, Pedro Teles, Cédric Tille, as well as AEA 29, Buba/CFS/ECB, Gerzensee, IHEID, SNB, and SSES 28 conference and seinar participants for useful coents.

2 Introduction In onetary policy analysis the focus has shifted away fro onetary aggregates towards short-run noinal interest rates. Consequently, the oney arket is widely neglected in the analysis of transission and optiality of onetary policy, and oney deand is treated as a redundant eleent. This link between the onetary instruent and the private sector is replaced in current onetary acro-odels by the consuption Euler equation, which is also called the new IS-curve. It relates the policy rate to expected consuption growth and inflation, and has becoe essential for onetary transission and for the ipleentation of optial onetary policy. There are however issues with the epirical reliability of this relationship. Studies in finance provide broad evidence that consuption Euler equations fail when they are applied to asset prices or the rate of returns on bonds (see Weil, 989). This should already cast doubt on the coon practice in onetary policy analysis to assue that the real central bank interest rate, which is close to the risk-free bond rate, is tightly related to consuption growth. But what is ore worrying, in our view, are recent studies unveiling substantial failures of (iplied) consuption Euler interest rates to atch oney arket rates: interest rates generated by standard consuption Euler equations are negatively related to US oney arket rates, while their spread is negatively related to the stance of onetary policy, i.e. with the level of the oney arket rates (see Canzoneri et al., 27, and Atkeson and Kehoe, 28). Thus there sees to exist a non-negligible systeatic wedge that separates interest rates, which are claied to be identical in acroeconoic theory. Put differently, observed policy rates do not see to be related to consuption growth and inflation in the way standard odels characterize. In this paper we take a closer look at the ipleentation of onetary policy and show that an explicit specification of central bank operations can contribute to the resolution of this proble. We thereby ai at explaining the puzzling relationship between the policy rate and the Euler rate, i.e. expected values of consuption growth and inflation. We develop a acro-odel with three interest rates: a discount rate for open arket operations controlled by the central bank (the repo or policy rate), an interest rate on governent bonds (the bond rate), and the Euler rate (the rate on private debt). The odel can explain systeatic oveents of the spreads with the onetary policy stance and with aggregate uncertainty. Specifically, the liquidity preiu on bonds, i.e. the spread between Euler and bond rates, varies endogenously with the expected costs of transforing bonds into eans of payent (oney). Consistent with epirical evidence, we show that the liquidity preiu and the Euler rate can be negatively related to the policy rate. Thus changes in the policy rate are linked to aggregate deand and inflation to a saller extend than iplied by a conventional fraework where the central bank sets the Euler rate.

3 The odel is based on a general equilibriu fraework, where oney deand is introduced by a cash-in-advance constraint. The odel ainly differs fro standard onetary acro-odels by three assuptions: First, we assue that financial arkets are separated. The asset arket, where agents trade interest bearing assets and cash, opens at the end of each period. Before, the oney arket opens, where agents can acquire cash fro the central bank in exchange for interest bearing assets discounted with the rate set by the central bank, i.e., the repo rate. Bonds bought today can be cashed in the next period at the repo rate. The bond rate is therefore closely linked to the expected future repo rate in open arket operations, while the spread between these rates increases on average with aggregate uncertainty and investors relative risk aversion. Thus, the bond rate exhibits a risk preiu. Second, we assue that only governent bonds are eligible in open arket operations, while other assets (here, privately issued debt) cannot be cashed at the central bank. The ain property is that the aount of eligible assets is not unliited. Access to oney is thus bounded by private sector governent bond holdings and cannot be eased by holdings of other securities issued by the private sector. Due to this property, governent bonds are perceived as a closer substitute for cash, which gives rise to a liquidity preiu. Thus, in equilibriu we observe a spread between the bond rate and the interest rate on privately issued debt, which are not eligible for open arket operations. 2 The debt rate, which corresponds to the above entioned consuption Euler rate, thus differs fro the bond rate, while the spread depends on the state of the econoy. In particular, a higher repo rate raises the price of oney in ters of bonds, i.e. reduces the aount of oney per unit of bonds supplied to the central bank, and leads to a decline in the liquidity preiu. Third, we assue that the central bank transfers its revenues to the fiscal authority. Following central bank practice (see Meulendyke, 998), we assue that it reinvests payoffs fro aturing securities in new interest bearing assets. The associated interest rate earnings are then transferred to the fiscal authority, while financial wealth is held by the central bank as the counterpart of outstanding oney. 3 As a consequence, the distribution of eligible securities between the private sector and the central bank changes over tie and, in particular, varies with the onetary policy stance. This property exerts an additional effect of onetary policy on the private sector behavior. When we exaine the transission of onetary policy shocks, which will be substantially affected by these assuptions, we consider prices to be set in an iperfectly flexible way, to allow for realistic inflation dynaics. When the constraint in open arket operations ( discounted value of bonds held by the private sector new oney ) is binding, the odel s 2 Bansal and Colean (996) endogenously derive a liquidity preiu by assuing bonds reduce transactions costs. 3 This differs fro the coon assuption in general equilibriu acro-odels that the central bank transfers seigniorage (defined as the change in the onetary base) to the fiscal authority. 2

4 predictions substantially differs fro results generated by standard odels. Consider, for exaple, an unexpected increase in the repo rate, i.e. a positive innovation to a Taylor-type feedback rule for the repo rate. Since aggregate deand is constrained by the aount of short-ter bonds discounted with the repo rate (plus oney carried over fro the previous period), which represents the aount of oney the private sector can get through open arket operations, the higher repo rate has a negative effect on the level of noinal consuption. Under sticky prices, onetary policy affects real consuption. However, onetary policy rather ipacts on the level of real consuption than on its growth rate, as iplied by the consuption Euler equation in standard odels. Here onetary policy has a saller initial ipact on the level of consuption than in standard odels. When the central bank increases its policy rate, part of that increase reflects a decrease in liquidity preiu such that expected consuption growth is less affected than in a standard odel. Moreover, due to the third assuption above, the rise in the repo rate further affects consuption through its ipact on the distribution of eligible securities. If, for exaple, onetary policy is persistently tightened by a higher repo rates, the central bank deands ore bonds in exchange for oney. With reduced bond holdings, the constraint in the oney arket tends to becoe even tighter in the next period, which is responsible for a hupshaped consuption response. Hence, an inertial rise inthereporateleadstoadeclinein the consuption growth rate, which together with lower expected inflation iplies the Euler-rate to fall, consistent with epirical evidence (see Canzoneri et al., 27). Finally, the odel provides a siple explanation for the existence and at the sae tie for the lack of a liquidity effect: When the central bank controls the oney growth rate and the open (or oney) arket constraint is binding, there is negative relation between newly injected oney and the repo rate, since the stock of eligible bonds is predeterined by the last period investent decision. 4 As a consequence, a oney injection leads to an unabiguous liquidity effect, i.e. a decline in the repo rate and in the bond rate. In contrast, the debt/euler rate increases due to the well-known anticipated inflation effect. The latter typically leads to a lack of a liquidity effect in standard sticky price odels (see e.g. Christiano, et al., 997), which we also found for the version of odel where the oney arket constraint is not binding. The paper is organized as follows. Section 2 presents epirical evidence on short-ter interest rates and spreads. In section 3, the odel is developed. In section 4, we exaine the behavior of interest rates and spreads in the odel. Section 5 presents responses to interest rate and oney supply shocks, and section 6 concludes. 4 This is of course due to the firstaboveentionedassuptiononthetiingoffinancial arkets. 3

5 2 Epirical behavior of interest rates This section presents the epirical behavior of the different interest rates considered in the odel and the relationships between the. The odel contains the Euler rate R d,apolicy rate R, i.e. the price of oney in ters of bonds inside open arket operations, and an interest rate R on an asset that the central bank accepts in exchange for oney in its open arket operations, i.e. the price of oney in ters of bonds outside open arket operations. Euler rate vs. policy rate This sub-section copares the epirical behavior of two interest rates that standard odels equate, i.e. the Euler rate and the policy rate. In our odel there is a third interest rate R, i.e. the interest rate on assets accepted by the central bank in exchange for oney in open arket operations. In this sub-section we focus on the spread between the fed funds rate and the Euler rate, given that epirically and in the odel both R and R ove relatively close to each other and contrast significantly with the behavior of R d. Thus for epirical coparison with the Euler rate we can interchangeably use R or R, with only negligible quantitative differences (see below). First, the epirical interest rate iplied by standard Euler equations is coputed. The ethodology is siilar to Fuhrer (2) and Canzoneri et al. (27). In a standard Euler equation, the inverse of the gross noinal interest rate Rt d canbeexpressedas µ uc,t+ P t = βe t, () u c,t P t+ R d t where β is the discount factor, u c is arginal utility of consuption, and P is the price level. With a standard CRRA utility function, leading to a arginal utility of consuption c σ t, and under conditional log-norality the Euler equation can be written as " # σ (E t log c t+ log c t ) E t log π t+ +rt d = β exp + σ2 2 var t log c t+ + 2 var, (2) t log π t+ + σcov t (log c t+, log π t+ ) where π t = P t /P t. Equation (2) is used to copute the iplied standard Euler (net) interest rate r d, where the conditional oents are estiated fro a six-variable VAR, Y t = A + A Y t + v t, assuing v i.i.d.n (, Σ), σ =2and β =.993. The variables included in Y (966Q-28Q2) are log per capita real personal consuption expenditures on nondurable goods and services, log change in the deflator of this consuption easure, log price of industrial coodities, log per capita real disposable personal incoe, federal funds rate, and log per capita real non-consuption GDP. Moreover, a segented (974Q) tie trend is included in A. Figure displays the coputed standard Euler interest rate r d and the fed funds rate r, as well as the spread between these two rates, s,t = rt d r, in percent. The Euler rate averages at.4 percent, whereas the federal funds rate averages at 6.5 percent; thus the 4

6 Standard Euler interest rate Federal funds rate Spread Figure : Euler and federal funds rates (%) average spread is about 5 percentage points. Inflation averages at 4.4 percentage points over the period considered. The federal funds rate and the Euler rate, which should be identical according to standard acroeconoic odels, display no apparent co-oveent. The fed funds rate is strongly negatively correlated with the spread, a fact that has recently been pointed out by Atkeson and Kehoe (28), while using Sets and Wouter s (27) odel. Thus, the unexplained wedge between the federal funds rate and the Euler rate are substantially related to the federal funds rate. At low frequency, the Euler and federal funds rates are positively correlated, which is ainly due to inflation trends (upward in the 97s and then downward in the 98s) that ove both rates in the sae direction. These trends evidently distort the correlation between the Euler and policy rates in coparison to a theoretical environent with constant steady-state inflation. In order to correct for these inflation trends and to isolate short-run (business cycle) interest rate dynaics fro longer ter oveents, we HP-filter (λ = 6) the interest rate series. The correlations between HP-filtered variables will be used to assess theoretical oents of our odel, which will be exained around a given steady-state inflation rate. Figure 2 displays the sae variables as in Figure but HP filtered. The bold line is inus the detrended federal funds rate. Thus, there is an apparent negative cooveent between fluctuations of the spread and of the policy rate. Also, the Euler and policy rates 5

7 are negatively correlated at business cycle frequency Standard Euler interest rate (HP filtered) Federal funds rate (opposite, HP filtered) Spread (HP filtered) Figure 2: HP-filtered Euler and federal funds rates Table presents the (unconditional) correlations between the federal funds rate r, the Euler rate r d, and the spread s, using standard Euler equations as well as our own odel s Euler equation. 5 Table Epirical correlations Standard Euler equation Our odel s Euler equation corr(s,r).98.9 corr r d,r There is a strong (close to inus one) negative correlation between the spread and the policy rate. The Euler rate and the policy rate are negatively correlated as well, as in Canzoneri et al. (27) in the case of real rates. 6 The correlations presented in Table are relatively siilar for both Euler rates. 5 Details on this latter rate can be found derived in the appendix. The difference between the standard Euler equation and our own odel Euler equation is ainly due to a cash-in-advance constraint. Overall, these two Euler rates differ only slightly, except in accelerating inflation (late 97s) and disinflation (early 98s) episodes, as well as around 992 and 23 with the drops in the policy rate. 6 Canzoneri et al. (27) reported correlation between real rates is saller (.37) andtheyfind a positive correlation between noinal rates, which coes fro the inflation trends, as explained above. 6

8 Policy rate vs. oney arket rate In this subsection we briefly assess the epirical counterpart of the spread between the policy rate R and interest rate R, which easure the relative price of oney inside and outside open arket operations. For this we assess onthly data for the effective federal funds rate and the (overnight and 3-onth) US$-LIBOR since January 2. In general, the LIBOR lies slightly above the policy rate (see fig.?? in appendix 9). The average spread between the federal funds rate and the overnight (3-onth) LIBOR has been 7 (25) basis points, when the recent financial crisis period (back to August, 27) is oitted. 3 The odel In this section we develop a acroeconoic fraework where the asset arket and the oney arket are separated. There are four different types of agents: households, firs, the central bank and the governent. We abstract fro financial interediation and assue that households directly trade with the central bank in open arket operations. Households can invest in governent bonds and non-interest bearing oney, and they can borrow and lend aong each other using a full set of noinally state contingent clais. Their deand for oney is induced by assuing that goods arket transactions cannot be conducted by using credit. This is odelled by a cash-in-advance constraint, i.e. by assuing that households have to hold oney for goods arket purchases. They can get oney fro the central bank only in exchange for securities in open arket operations. To give a preview, financial arkets separation will lead to a spread between the governent bond rate and the policy (repo) rate, i.e. a risk preiu, whereas the spread between the Euler and governent bond rates, i.e. a liquidity preiu, will be due to the special role of governent bonds in open arket operation. Throughout the paper, upper case letters denote noinal variables, lower case letters real variables, and variables without an index (i or j) aggregate variables. 3. Tiing of events The tiing of arkets and the specification of open arket operations will be iportant for our results. We will focus on the case where only governent bonds are eligible in open arket operations (like in Lacker, 997, or Schabert, 24). The tiing of events in each period is as follows: There is a continuu of infinitely lived households indexed with i [, ]. A household i enters a period t with noinal assets carried over fro the previous period t : Mi,t H + B i,t + D i,t, where M H denotes holdings of oney, B governent bonds, and D private debt. 7

9 . Aggregate shocks aterialize, labor is supplied by households, and goods are produced by firs. 2. Households enter the oney arket, where they can engage in open arket operations with the central bank. There, oney can be traded only in exchange for eligible securities, and is supplied via outright sales/purchases and via repurchase agreeents. The relative price of oney Rt (for both types of trades) is controlled by the central bank and will be called repo rate: Bi,t/R c t = I i,t, where I i,t is the aount of oney received by household i and Bi,t c the aount of bonds the CB gets. We assue that only governent bonds are eligible B c i,t B i,t. (3) When household i leaves the oney arket its bonds holdings equal B i,t B c i,t. 3. Households enter the (final) goods arket, where oney is assued to be the only accepted eans of payent. Thus goods arket expenditures are restricted by oney carried over fro the previous period plus additional oney acquired fro the central bank via current period open arket operations: P t c i,t I i,t + M H i,t, (4) where c i denotes purchases of the final consuption good and P its price level. When household i leaves the goods arket, its oney stock equals I i,t + Mi,t H P tc i,t. 4. Finally, the asset arket opens. Before households trade in the asset arket, current labor incoe and dividends are paid back in cash to households. Further, governent bonds can be repurchased fro the central bank with cash, i.e. household i can repurchase bonds Bi,t R using oney M i,t R = BR i,t. After repurchase agreeents are settled, oney and bond holdings of household i equal fm i,t = I i,t + M H i,t + P t w t n i,t + P t δ i,t P t c i,t M R i,t, eb i,t = B i,t B c i,t + B R i,t, where w t denotes the real wage rate, n t working tie and P t δ i,t dividends. In the asset arket, households borrow/lend and trade oney and bonds aong each other. They can further buy bonds fro the governent at the price /R t, such that the price of oney in ters of bonds in the asset arket equals R t. Hence, we can suarize the 8

10 asset arket constraint of household i as (B i,t /R t )+E t [q t,t+ D i,t ]+M H i,t eb i,t + D i,t + fm i,t + P t τ t, (5) where P t τ t denotes lup-su governent transfers and q t,t+ is a stochastic discount factor, which will be defined below. Money cannot be issued by the private sector, R Mi,t f di = R Mi,t H di, while the total aount of governent bonds held by the private sector at the end of the period R B i,t di will depend on how any bonds are issued by the fiscal authority or held by the central bank. In what follows we describe the odel in detail. 3.2 Private sector Households Households have identical asset endowents and identical preferences. Household i axiizes the expected su of a discounted strea of instantaneous utilities u : X E β t u (c it,n it ), (6) t= where E is the expectation operator conditional on the tie inforation set, and β (, ) is the subjective discount factor. The instantaneous utility u is assued to satisfy u t =[(c σ i,t ) ( σ) ] γn i,t. Ahouseholdi is initially endowed with oney M i,, governent bonds B i,, and contingent clais D i,. As described above, it faces three constraints in each period. In the oney arket, it can acquire oney I i,t up to the aount of governent bonds carried over fro the previous period B t discounted by Rt. The constraint (3) can be written as I i,t B i,t /R t. (7) The constraint (7) will be called the open (or oney) arket constraint. It should be noted that this odel can also be applied to the case where the central bank withdraws oney fro the private sector I i,t <. For onetary injections to be positive in equilibriu a sufficiently large fraction of oney has to be supplied under repurchase agreeents. Throughout the analysis we will restrict our attention to the case where the central bank supplies oney in a way that ensures I i,t. Households are further assued to rely on cash for transactions in the goods arket. Given that they can first trade with the central bank in open arket operations, the cashin-advance constraint differs fro Svensson s (985) cash-in-advance constraint by I i,t : P t c i,t I i,t + M H i,t. (8) In the asset arket, the governent issues bonds, and households trade oney and bonds 9

11 with each other. They can further borrow and lend using a full set of noinally state contingent clais. Dividing the period t price of one unit of noinal wealth in a particular state of period t + by the period t probability of that state gives the stochastic discount factor q t,t+.theperiodtprice of a payoff D jt in period t +is then given by E t [q t,t+ D jt ]. Substituting out the stock of bonds and oney held before the asset arket opens, B e i,t and fm i,t, in (5), the asset arket constraint of household i reads (B i,t /R t )+E t [q t,t+ D i,t ]+M H i,t +(R t ) I i,t (9) B i,t + D i,t + M H i,t + P t w t n i,t P t c i,t + P t δ i,t + P t τ t, where household i s borrowing is restricted by the following no-ponzi gae condition li E tq t,t+s D i,t+s, () s as well as Mi,t H and B i,t. The ter (Rt ) I i,t easures the costs of oney acquired in open arket operations: The households receive new cash I i,t in exchange for Rt I i,t bonds. Maxiizing the objective (6) subject to the oney arket constraint (7), the goods arket constraint (8), the asset arket constraints (9) and (), for given initial values M i,, B i,,andd i, leads to the following first order conditions for working tie n i,t, consuption c i,t,openarkettradesi i,t, as well as holdings of contingent clais, governent bonds and oney: u i,nt /w t = λ i,t, () u i,ct = λ i,t + ψ i,t, (2) Rt λi,t + η i,t = λi,t + ψ i,t, (3) β λ i,t+ = q t,t+, (4) π t+ λ i,t βe t λi,t+ + η i,t+ π t+ = λi,t /R t, (5) βe t λi,t+ + ψ i,t+ π t+ = λi,t (6) where λ i,t and ψ i,t denote the ultiplier on the asset and goods constraint. The first two conditions () and (2) show that a binding goods arket constraint (ψ i,t > ) distorts the intrateporal consuption-leisure decision in a conventional way, u i,ct + u i,nt /w t = ψ i,t. Cobining () and (2) with (6), discloses the inflation tax on consuption, which is iplied by the cash-in-advance constraint (8): βe t [u i,ct+ /π t+ ]= u i,nt /w t (7) The open arket constraint is associated with the ultiplier η i,t, which easures the liquidity value of bonds. When the goods arket constraint is binding, u i,ct + u i,nt /w t >, therole

12 of oney as a eans of payent is positively valued. Likewise, governent bonds, as a substitute for oney, can also be valued differently fro non-eligible assets; for this, the exchange rate R has to be sufficiently low, as discussed below. Cobining (), (2), and (3), we obtain η i,t = u i,nt w t + u i,ct Rt. (8) The ultiplier on the open arket constraint η i,t, which easures the liquidity value of bonds, tends to decline with the policy rate (see 8), since a higher policy rate reduces the aount of oney for each unit of bonds supplied to the central bank. The bond pricing equation (5) shows that a rise in this ultiplier tends to lower the interest rate on bonds, which can generate a spread between the Euler rate and the bond rate, i.e. a liquidity preiu. The household s investent decisions further links the bond rate to the repo rate. They are willing to hold both assets, oney and bonds, if the rate of return on bonds copensates for the costs of acquiring new oney in the next period. This can be seen by cobining (), (3), (5), and (6) E t /R t+ (ui,ct+ /π t+ ) E t [(u i,ct+ /π t+ )] =/R t, iplying that the interest rate on bonds equals the expected future policy rate up to first order. Throughout, we will repeatedly refer to the rate of return on a noinally risk-free portfolio of clais that deliver one unit of currency in each state. This interest rate Rt d, which corresponds to the Euler rate in section 2, is given by R d t =[E t q t,t+ ]. (9) It should be noted that equation (4), which defines the Euler-rate, can differ fro the standard Euler rate (see ) due to the cash-credit-good friction, λ i,t u i,ct. Finally, the following copleentary slackness conditions are satisfied in the household s optiu i) b i,t π t /R t i i,t, η i,t, η i,t bi,t π t /R t i i,t =, ii) i i,t + H i,t π t c i,t, ψ i,t, ψ i,t ii,t + H i,t π t c i,t =, where H i,t = M i,t H/P t, b i,t = B i,t /P t,andi i,t = I i,t /P t, and (9) and () hold with equality. In equilibriu households are willing to hold both types of oney, i.e. oney held under repurchase agreeents Mi,t R and under outright sales/purchases M i,t H. Changes in the coposition of oney supplied to the private sector ight however affect the distribution of eligible securities between the private sector and the central bank. Production To facilitate a reasonable transission of onetary shocks we will allow for iperfectly flexible prices. We will introduce price stickiness in the standard way following

13 the New Keynesian literature. In particular, we assue that the final consuption good is an aggregate of differentiated goods produced by onopolistically copetitive firs indexed with j [, ]. The CES aggregator of differentiated goods is y t = R y jt dj, with >, where y t is the nuber of units of the final good, y jt the aount produced by fir j, and the constant elasticity of substitution. Let P jt and P t denote the price of good j set by fir j and the price index for the final good. The deand for each differentiated good is y jt =(P jt /P t ) y t,withpt = R P jt dj. A fir j produces good y j eploying the technology: y jt = a t n α jt, where α (, ), a is a stochastic productivity level satisfying a t = a ρ a t exp ε a,t, ρ a, andε a t is i.i.d. norally distributed with E t ε a t =. Hence, labor deand satisfies: w t = c jt αy jt /n jt, (2) where c jt denotes real arginal costs. We consider a noinal rigidity in for of staggered price setting as developed by Calvo (983) and Yun (995). Each period firs ay reset their prices with the probability φ independently of the tie elapsed since the last price setting. The fraction φ [, ) of firs is assued to adjust their prices with the steady state inflation rate π, whereπ t = P t /P t, such that P jt = πp H,jt. Ineachperiodaeasure φ of randoly selected firs sets new prices P e jt in order to axiize the expected su of discounted future dividends P P t δ jt =(P jt P t c t ) y jt :axp jt E t s= φs q t,t+s ( P e jt y jt+s P t+s c t+s y jt+s ),s.t. y jt+s = ep jt Pt+sy t+s.forφ>, thefirst order condition is given by ep jt = P E t s= φs q t,t+s y t+s Pt+s + c t+s P E t s= φs q t,t+s y t+s Pt+s. (2) Aggregate output is y t =(Pt /P t ) n α t,where(pt ) = R P jt dj and thus (P t ) = φ Pt + ( φ) P e t. Under flexible prices φ =, real arginal costs are given by c jt = ε 3.3 Public sector The public sector consists of a governent and a central bank. The governent issues debt B T, which is held both by households R B i,t di = B t and by the central bank R Bi,t c di = Bt c : Bt T B t + Bt c. It further receives payents P t τ t fro the central bank and transfers financial wealth P t τ t to the households. Its flow budget constraint thus reads B T t /R t + Pt τ t = B T t + P t τ t. ε. Though taxes are non-distortionary, Ricardian equivalence will not apply when the oney arket constraint is binding. The supply of governent debt will then not be irrelevant for the conduct of onetary policy and for onetary transission. Given that for any central banks only short-ter governent debts are eligible, we focus on the supply of t-bills 2

14 and disregard debt with longer aturity. As we are focusing on onetary policy effects, we assue that the supply of governent bonds is exogenously deterined and that governent debt is issued at a constant growth rate Γ satisfying: Γ >β: B T t = ΓB T t. The central bank supplies oney in exchange for governent bonds in open arket operations in for of outright sales/purchases M H t and repurchase agreeents M R t. Before the oney arket opens, the central bank s stock of governent bonds equals Bt c andthestockof outstanding oney equals Mt H. It then receives an aount of bonds Bc t in exchange for oney I t, and after repurchase agreeents are settled its holdings of bonds reduces by Bt R and the aount of outstanding oney by M R t = B R t. Before the asset arket opens, where the central bank can invest in governent bonds B c t, it holds an aount of bonds equal to eb c t = B c t + B c t BR t. Its budget constraint is given by (B c t /R t )+P t τ t = B c t + B c t B R t + M H t M H t I t M R t. In accordance with the operational practice of central banks we assue that it rolls over its aturing assets (see e.g. Meulendyke, 998, ch.7). Thus, we assue that the central bank also enters the asset arket at the end of each period, and reinvests in bonds to the aount that equals its current stock of aturing debt B c t = e B c t. Further using B R t = M R t and Bt c = Rt I t, the budget constraint can be siplified to (Bt c /R t ) Bt c = M t H Mt H + (Rt ) I t P t τ t. Following coon practice (see Meulendyke, 998), we assue that the central bank transfers interest earnings fro asset holdings to the governent. P t τ t = B c t ( /R t ). Note that these transfers will not be negative in equilibriu, such that the central bank will never rely on funds fro the governent. 7 Accordingly, its bond holdings will evolve according to Bt c Bt c = Rt I t I t Mt H + Mt H. (22) Thus the central bank tends to accuulate ore bonds, i.e. bonds flow fro households to the central bank, when oney supply or the policy rate is high. Regarding the ipleentation of onetary policy, we assue that the central bank conducts onetary policy by using siple instruent rules, which contain a stochastic eleent 7 Note that this is different in odels, where central bank tranfers seigniorage (defined as the change in the onetary base) to the governent in each period. A discussion of governent transfers and central bank independence can be found in Sis (23). 3

15 to allow for onetary policy shocks. We consider two alternatives. For the benchark specification of onetary policy, we assue that the central bank sets the repo rate Rt.Itight be set contingent on its own lags and on current inflation to allow for a Taylor-rule-type interest rate setting. To assess the onetary transission echanis, we further consider shocks to the interest rate reaction function R t = R t ρ (R ) ρ (π t /π) ρ π ( ρ) exp ε ρ t. (23) where ρ and ε ρ t is norally i.i.d. with E t ε ρ t =and variance var ερ. The long-run repo rate, R >, and the target inflation rate, π>β, can be chosen by the central bank. Alternatively, we will also assue that the central bank controls the growth rate of oney. In contrast to (standard) odels, where repurchase agreeents are not considered, the central bank has an additional role: It can decide on whether oney is traded in for of outright sales/purchases or in for of repurchase agreeents. For siplicity, we assue that it controls the ratio of oney supply under both types of open arket operations Ω: M R t = Ω M H t, or Mt R Ω = M t +Ω,whereΩ and M t is the total oney supply, M t = Mt H + Mt R. Finally, substituting out central bank transfers in the governent budget constraint shows that the governent transfers revenues fro debt issuance and central bank profits to the households: P t τ t = Bt T /R t B T t + Bt c ( /R t ). 3.4 Rational expectations equilibriu In equilibriu, there will be no arbitrage opportunities and arkets clear, n t = R n jtdj = R n itdi and y t = R y jtdj = R c itdi = c t. Households will not behave differently and aggregate asset holdings satisfy t : R D i,t di =, Z Z Z Z Mi,tdi H = fm i,t di = Mt H, Mi,tdi R = Mt R, B i,t di = B t, Z I i,t di = I t = Mt H Mt H + Mt R, Bt T = B t + Bt c. Since governent bonds are the single eligible security, its distribution between the central bank and the private sector will atter. Given that the governent issues bonds according to a constant growth rate Γ, household bond holdings change according to B t B t = (Γ )Bt T Bc t + Bt c. Further using (22), the evolution of bonds held by households satisfies B t B t =(Γ )Bt T Rt M H t Mt H + Mt R + M R t. (24) Thus, private sector holdings of bonds tend to decrease with a higher price of oney R and to increase with Γ. For a given injection I t households further loose less bonds when the 4

16 fraction of oney held under repurchase agreeents increases. Throughout, we will focus on the case where the central bank sets its instruent such that the goods arket constraint (8) is strictly binding (ψ t > ). 8 A rational expectations equilibriu can then be defined as follows: A rational expectations equilibriu is a set of sequences {c t,n t,y t,w t, t,b t,b T t,rt, Rt d,r t,p t } t= satisfying the firs first order conditions and the production technology, the households first order conditions ()-(6) and the transversality condition, the binding goods arket constraint P t c t = Mt H + Mt R, the open arket constraint b t R t π t R t + H t H t π t, H t H t π t (R t ) R t,forγ = b T t π t /b T t, and b t b t π t =(Γ )b T t π t Rt for a onetary policy satisfying (23) and initial values M, B >, andp >. Note that under a non-binding open arket constraint, b t /π t >Rt R t + H t H t π t the evolution of governent bonds will neither affect the equilibriu allocation nor the associated price syste. If however the open arket constraint is binding, b t / (Rt π t )= R t + H t H t π t, household bond holdings atter and (24) reduces to B t =(Γ ) B T t +M R t. 3.5 Steady state In the following analysis, the two cases of a binding and a non-binding open arket constraint (7) will be treated separately, which facilitates analyzing the echaniss that are responsible for the ain results. 9 Throughout the analysis, we are particularly interested in the case where the oney arket constraint is binding. For this we assue that the central bank conducts onetary policy in a way that ensures the rate of return on governent bonds to be lower on average than the rate of return on private debt in equilibriu. Households then tend to econoize on bond holdings, i.e. they will not hold ore governent bonds than necessary for their oney arket trades. If however both returns are identical, households can borrow and invest in governent bonds without costs such that the oney arket constraint will not be binding. As seen fro Figure, the policy rate has alost always been below the iplied Euler rate, which corresponds to the binding oney arket constraint case. In order to analyze the two regies in a separate way, we first exaine steady states with a binding and a non-binding open arket constraint. We then assue that onetary policy is conducted in a way to ipleent one particular steady state and that aggregate shocks are sufficiently sall, so that we can analyze the properties of the econoy in the neighborhood, 8 In the long-run, this is ensured by the noinal interest rate R being larger than one. 9 The set of equilibriu conditions for both cases can be found in the appendix 8.2. Likewise, if the central bank siply declares both assets as eligible for open arket operations, the private sector can freely create any aount of private debt that can be used in exchange for oney, such that the private sector never runs out of eligible securities. 5

17 of this steady state. A steady state value of an endogenous variable x t will not carry a tie index, x. To exaine the two cases, we cobine (4), (5), and (9), to give the following steady state condition η/λ = ³ R d R /R. (25) The spread between the debt rate R d and the bond rate R thus deterines if the ultiplier on the open arket constraint is positive η >, which indicates a binding open arket constraint. Before exaining the differences between both steady states, we look at coon properties. Throughout the paper, we assue that the central bank successfully ipleents its inflation target π in the long-run. Hence, the steady state Euler rate is as usual deterined by (4) and (9), R d = π/β. In the steady state, consuption is then given by c σ+/α = ε α R d ε γ where we used (7), (2), and the c = n a. Real balances are then deterined by: = c, = H + R, and R = Ω H.Thus,forafixed inflation target π, the steady state values R d, c,, h and R are independent of η, i.e. do not depend on the tightness of the oney arket constraint. The latter only atters for the steady state values of the bond rate and debt. i.) If the central bank sets the average repo-rate R equal to the debt rate R d in a steady state, R = π/β, the interest rate on governent bonds R also equals R d,ascanbe seen fro (6). The ultiplier on the open arket constraint will then be equal to zero η =(see 25) and the steady state is characterized by R = R d = R, while bonds are neutral. ii.) If however the central bank chooses an average repo-rate R that is strictly saller than R d,whichrequiresr <π/β, there exists a steady state with a binding open arket constraint, η> (see 25) satisfying R = R, and b Γπ = R Γπ. b R π = H π + R, (26) In the case ii.), the condition b Γπ = R Γπ together with (26) would only be consistent with Γ 6= π and Mt H for deflationary equilibria, thus we restrict our attention to the case where the growth rate of bonds equals the steady state inflation rate Γ = π. For this, we assue that the central bank chooses its inflation target and eventually 6

18 adjusts the set of eligible assets if the growth rate of bonds differs fro the inflation target, which is not considered in this paper. If, for exaple, Γ <π, the central bank ight accept also a fraction of private debt in open arket operations. If Γ >π, it ight accept only a fraction of governent bonds in open arket operations. Thus, by deciding on the set of eligible securities, the central bank actually decides on the axiu aount of oney that can be traded in open arket operations. 4 Interest rates and spreads In this section, we exaine the relation between the three interest rates, i.e., the repo or policy rate R, the bond rate R, andthedebtrater d. The bond rate R t and the repo rate R t are closely related to each other as can be seen fro (6). The spread between these two rates, i.e. a risk preiu, will be exained below. Before, we will take a look at the spread between the debt rate Rt d and the bond rate R t, i.e. a liquidity preiu. For the analysis of both spreads we will use siple versions of the odel, to facilitate the derivation of analytical results. Throughout this section, we assue that production is linear α =, the production sector is perfectly copetitive, and that prices are perfectly flexible φ =. We further siplify public policy by assuing that oney is only supplied under repurchase agreeents Ω and that the supply of governent bonds is constant Γ =. 4. The liquidity preiu Households are willing to hold governent bonds even if the bond rate is lower than the debt rate, since bonds exhibit an additional liquidity value. Due to lower interest earnings, households will econoize on bond holdings such that the oney arket constraint is binding. This property has already been used for the steady state analysis (see 25). The central bank can ipleent a long-run equilibriu with a binding oney arket constraint if the repo rate R issetatavaluelowerthanr d = π/β. Outside the steady state, the debt-bond rate spread will not be constant over tie and will in particular depend on the onetary policy stance, since the value of liquidity will depend on the oney arket conditions. To facilitate the analysis of the liquidity preiu, we focus on the case of an exogenous interest rate policy ρ π =. Since the current bond rate is affected by toorrow s repo rate rather than today s repo rate, we further assue that the repo rate sequence exhibits inertia ρ>. A rise in the repo rate Rt, i.e. in the relative price of oney, has two iediate effects. It reduces noinal consuption for a given stock of household bond holdings B t Under perfectly flexible prices both rates, R d and R, will be constant if ρ =. This will not be the case under sticky prices (see section 5). 7

19 (see 7 and 8). It further leads to lower end-of-period noinal bond holdings B t (see 24, which in the siplified version reads B t = B t /Rt ). Thus, both effects tend to reduce inflation. Since the repo rate is raised in an inertial way, inflation is also expected to be lower in the following period, thus households deand a lower debt rate Rt d (see 4). The effects can easily be shown for the case where utility is logarithic in consuption σ =. The results are suarized in the following proposition. Proposition Consider the case, where σ = and the repo rate satisfies ρ π = and R < /β, such that the open arket constraint is binding. The debt rate Rt d and the ratio Rt d /R t decrease with i.) the current level of the repo rate if ρ> and ii.) with the variance of repo rate innovations ε ρ t. Proof. See appendix 8.3. The spread, i.e., the liquidity preiu, depends on the ability of bonds to be convertible into eans of payents in open arket operations before the goods arket opens. If these costs of exchanging bonds against oney Rt are high or ore uncertain, the liquidity value of bonds and thus the liquidity preiu declines (see 8). According to the standard Fischer effect, the debt rate falls in response to an increase in the repo rate, when inflation is expected be to lower than average inflation rate in the subsequent period, which requires ρ>. It should be noted this inflation response is also responsible for an increase in consuption, since the inflation tax on cash goods is then lowered (see 7). This counterfactual consuption response will disappear when prices are assued to be iperfectly flexible (see section 5). 4.2 The risk preiu As discussed in the previous section, the interest rates on bonds and debt only differ when the open arket constraint is binding. In contrast, there can be a spread between the repo rate and the bond rate, regardless whether the open arket constraint is binding or not. This can be seen fro the household optiality condition (6), which can by using () and (2) be rewritten as /R t = E t /R cov t /R t+ + t+, (uct+ /π t+ ) (27) E t [u ct+ /π t+ ] Households are willing to hold both, oney and bonds, if the rate of return on bonds copensates for the costs of converting bonds against oney in next period s open arket operations. Up to first order, the current bond price /R t in the asset arket equals the expected future oney-price of bonds in open arket operations E t /R t+. However, the price of a governent bond /R t will be saller than E t /R t+, if the covariance on the RHS of (27) is negative, i.e., if the real repo rate Rt+ is positively related to the arginal utility of consuption divided by the inflation rate, u ct+ /π t+. 8

20 The spread between the bond rate and the repo rate then tends to be positive and increases with the easure of relative risk aversion σ. It can therefore be interpreted as a risk preiu on the noinal rate of return on bonds copared to the expected repo rate. A risk-averse agent who considers investing in bonds in the asset arket will ask for a price /R t that is lower than the oney-price of bonds in next period s open arket arket, if a lower real repo rate (and thus a higher real pay-off fro bonds) is associated with higher consuption. To establish this result, we again apply a siplified version (α = Γ =, φ =,, and Ω ). We now allow for varying degrees of relative risk aversion, σ>, andwefocus on i.i.d. technology shocks, ρ a =, as the only source of aggregate uncertainty, such that var ε ρ =, while the repo rate will endogenously be adjusted according to ρ π > and ρ =. The following proposition suarizes the ain results. Proposition 2 Considerthecasewhereσ > and ρ a =, while the repo rate satisfies ρ π >, ρ =and R < /β, such that the open arket constraint is binding. The current price of governent bonds is saller than the expected future oney-price of bonds /R t < E t /R t+. The average bond rate Rt further increases with the households relative risk aversion and with the variance of productivity shocks. Proof. See appendix 8.4. The covariance ter in (27) is strictly negative under a binding open arket constraint, where a higher repo rate tends to reduce current consuption ties inflation b t /Rt = c t π t. Hence, the bond rate tends to exceed the repo rate and further increases for a given repo rate, if aggregate uncertainty, var(ε a ) or the relative risk aversion σ increases. In both cases households only want to invest in bonds at a higher rate of return. 5 Nuerical analysis In this section we apply a nuerical analysis of a less siplistic odel, using a second order approxiation at the deterinistic steady state (see Schitt-Grohé and Uribe, 24) and standard paraeter values as far as possible (see table A in appendix 8.5). 2 In the first part, we re-exaine the behavior of the interest rates. In the second part we look at the transission of onetary policy shocks. To allow for ore realistic dynaics we now use a sticky price version of the odel. For ost of the odel s paraeter we apply standard values for quarterly data, naely, σ =2, α =.66, φ =.8, ρ (a) =.9, and =6. To atch the average interest rate values found in the data, we apply an inflation rate of π =.8 (for an annual rate of 4.4%, see section 2), a low discount factor β =.984, and a target repo rate equal to R =.5, leading to a steady state spread R d R equal to 2 basis points per quarter and a spread of 53 basis 2 For the coputation we used dynare. 9

21 points per year. We further set the inflation feedback ρ π either equal to zero or equal to.5. 3 Finally, we set the value for the ratio between repo-oney and oney supplied outright Ω equal to Interest rate behavior In this section we again look at the relations between the interest rates, which have already been analysed qualitatively in section 4. To facilitate coparisons with the latter results, we present nuerical results for only one type of shock. Liquidity preiu Table 2 presents values for the average spread between the debt rate and the bonds rate, E s,t = E R d t R t. Starting with a steady state value of 2 basis points, it decreases with larger variances of repo rate innovations ε ρ t,whichaccordstoclai i.) in proposition. When the repo rate is endogenously adjusted (ρ π =.5), such that its variance is not only directly affected by the innovations ε ρ t,thiseffect is less pronounced. Table 2 Average spread Es,t under interest rate shocks var (ε ρ )=. var (ε ρ )=.5 var (ε ρ )=. ρ π = 477 b.p. 342 b.p. 72 b.p. ρ π = b.p. 4 b.p. 288 b.p. Table 3 further presents the correlation between the debt rate and the repo rate as well as the correlations between the spread s,t and the repo rate. The coluns refer to only one type of shock. Both, the debt rate and the spread are found to be highly negatively correlated with the repo rate, while the correlations are slightly saller under technology shocks. Overall, these finding supports clai ii.) ade in proposition. The correlations of the spread further accord to the epirical results presented in section 2 and in other studies (see Atkeson and Kehoe, 28, and Canzoneri et al., 27). Table 3 Unconditional correlations Interest rate shocks Technology shocks ρ π = ρ π =.5 ρ π =.5 corr(s,t,rt ) corr Rt d,rt The odel overstates the negative correlation between the debt rate and the repo rate copared to the nubers presented in the epirical analysis (see section 2). Nevertheless, we can conclude that the debt rate hardly iics the policy rate in all cases. 3 In contrast to standard sticky price odels a passive interest rate policy does not give rise to local equilibriu indeterinacy when the oney arket constraint is binding. The reason is that noinal debt serves a noinal anchor like a constant oney supply. A local deterinacy analysis for a siplified odel version can be found in Schabert (24). 2