Modeling Monetary Policy

Transcription

1 Modeling Monetary Policy Sauel Reynard Swiss National Bank Andreas Schabert University of Dortund Deceber 3, 28 Abstract Models currently used for onetary policy analysis equate the onetary policy interest rate instruent to the consuption Euler rate which is related to expected consuption growth and inflation, i.e. the two variables onetary policy is designed to control. This specification however fails badly on data: both rates are negatively correlated and the policy rate co-oves negatively with the spread between these two rates. We propose a ore realistic odel of onetary policy, consistent with these epirical co-oveents, where the central bank affects noinal spending by influencing the value of assets which the private sector directly uses to obtain eans of payent for consuption via open arketoperations. Theliquiditypreiu of these assets, i.e. the spread between a standard Euler rate and their yield, varies according to how uch the private sector values the transaction service they provide. In addition, our odel iplies a new onetary transission echanis and can be used to analyze the effects of changes in aggregate risk and liquidity shocks on oney arket interest rates and policy. JEL classification: E52; E58; E43. Keywords: Monetary Policy; Open arket operations; Liquidity preiu; Money arket rate; Consuption Euler rate; Monetary 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, Jordi Gali, Dale Henderson, Stephanie Schitt-Grohe, Pedro Teles, as well as Buba/CFS/ECB, Gerzensee, SNB, and SSES 28 Lausanne seinar/conference 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 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 the two variables onetary policy is designed to control, i.e. consuption 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 reconciling the epirical relationship between the policy rate and variables deterining the Euler rate, i.e. 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 rate), an interest rate on governent bonds (the bond rate), and the Euler rate. The odel can explain systeatic oveents of the spreads with the onetary policy stance and with aggregate uncertainty, and it can generate an unabiguous liquidity effect. The liquidity preiu on bonds, i.e. the spread between Euler and bond rates, varies endogenously according to how uch the private sector values the transaction service of these assets. Consistent with Atkeson and Kehoe s (28) evidence, our analysis shows that changes in the policy (repo) rate affect 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 with sticky prices, where oney deand is introduced by a cash-in-advance constraint. It 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 an interest rate set by the central bank, i.e., the repo rate. Bonds 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 ean spread between these rates increases with aggregate uncertainty and investors relative risk aversion. 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 assuption 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 ore liquid by investors, 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 latter rate, which actually corresponds to the above entioned consuption Euler rate, thus differs fro the other rates, the bond rate and the repo rate, while the spreads depend 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 received oney per unit of bonds supplied to the central bank, which 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 can exert an additional effect of onetary policy on the private sector behavior. In this paper, we further exaine the transission of onetary policy shocks, either odelled as shocks to a siple interest rate rule or as oney growth shocks. When the constraint in open arket operations ( discounted value of bonds held by the private sector 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 new oney ) is binding, the odel s predictions leads to substantial deviations 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. Taking into account that prices are adjusted in an iperfectly flexible way, 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. Due to the third assuption (s.a.), the rise in the repo rate further affects consuption through its ipact on the distribution of eligible securities. If, for exaple, onetary policy is tightened by an increased repo rate, agents have to supply a relatively larger aount of 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, a higher repo rate leads to a decline in 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). Now suppose that the central bank controls the growth rate of oney. When the open (oroney)arketconstraintisbinding,itiplies a siple 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. At the sae tie, the debt 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. Section 6 concludes. 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 a policy rate R, an interest rate R on an asset that the central bank accepts (at a discount) in exchange for oney in its open arket operations, which easures the relative price of oney outside open arket 4 This is of course due to the firstaboveentionedassuptiononthetiingoffinancial arkets. 3

5 operations, and the Euler rate R d. Spread between Euler and policy rates. This sub-section copares the epirical behavior of two interest rates that standard odels equate, i.e. the Euler and policy rates. 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 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 that consuption, 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 t, in percent. The Euler rate averages at.4 percent, whereas the federal funds rate averages at 6.5 percent; thus the 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 replicating the Sets and Wouter s (27) iplied Euler rate. Thus, the unexplained wedge between the federal funds rate and the 4

6 Standard Euler interest rate Federal funds rate Spread Figure : Euler and federal funds rates (%) Euler rate cooves with the federal funds rate in a substantial way. 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 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. Figure 2 displays the sae variables as in Figure but HP filtered. The opposite of the federal funds rate has been plotted in order to draw attention to the fact that there is a very close atch between fluctuations of the spread and of the opposite of the policy rate. Also, the Euler and policy rates are negatively correlated at business cycle frequency. Table Epirical correlations Standard Euler equation Our odel s Euler equation corr(s,r).98.9 corr r d,r Table presents the (unconditional) correlations between the federal funds rate r, the Euler 5

7 Standard Euler interest rate (HP filtered) Federal funds rate (opposite, HP filtered) Spread (HP filtered) Figure 2: HP-filtered Euler and federal funds rates rate r d, and the spread s, using standard Euler equations as well as our own odel s Euler equation. 5 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 in Canzoneri et al. (27) in the case of real rates. 6 The correlations presented in Table are relatively siilar for both Euler rates. Spread between policy and oney arket rates. 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 lie slightly above the policy rate (see appendix 9). The average spread between the federal funds rate and the overnight (3-onth) LIBOR has been 7 (25) basis points, when the spreads fro the recent financial crisis period (back to the st August 27) are oitted. 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 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 gains of financial interediation and assue that households directly trade with the central bank in open arket operations. Households deand for oney is induced by assuing that goods arket transactions cannot be conducted by using credit. This is odelled, for siplicity, by a cash-in-advance constraint, i.e. by assuing that households have to hold oney for goods arket purchases. Asset arkets are separated. Households can get oney fro the central bank only in exchange for securities in open arket operations. They further 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. To give a preview, financial arkets separation will lead to a spread between the bond and policy (repo) rates, whereas the spread between the Euler and bond rates will be due to the special role of bonds in open arket operation. Throughout the paper, upper case letters denote noinal variables, lower case letter real variables, and variables without an index (i or j) aggregate (or econoy-wide) 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 : M H i,t + B i,t + D i,t, where M H denotes holdings of oney, B governent bonds, and D private debt.. Aggregate shocks aterialize, labor is supplied by households, and goods are produced by firs. 2. Households enter the oney arket, where oney can be traded only in exchange for eligible securities. We assue that the central bank supplies oney via outright sales/purchases and via repurchase agreeents. The relative price of oney R t (for both types of trades) is controlled by the central bank and will be called repo rate: B c i,t/r t = I i,t, 7

9 where I i,t is the aount of oney delivered to the 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 open arket its bonds holdings equal B i,t B c i,t. 3. Households enters 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. Thus, the price of oney in ters of bonds in the asset arket equals R t. Hence, we can suarize the asset arket constraint of household i as (B i,t /R t )+E t [q t,t+ D i,t ]+M H i,t e B i,t + D i,t + f M 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 and held by the central bank. In what follows we describe the odel in detail. 8

10 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 only 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 + Mi,t. H (8) In the asset arket, the governent issues bonds, and households trade oney and bonds 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 canbewrittenas (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, 9

11 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. 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,consuptionc i,t, open arket trades, as well as holdings of contingent clais, bonds and oney: u i,nt /w t = λ i,t, () u i,ct + u i,nt = ψ w i,t, t (2) u i,ct Rt + u i,nt w t = η i,t, (3) β π t+ λ i,t+ = q t,t+, (4) λ i,t λi,t+ + η i,t+ βe t π t+ =/R t, (5) λ i,t h R E i t t+ λi,t+ + ψ i,t+ π t+ E t λi,t+ + ψ i,t+ π =/R t, (6) t+ where λ i,t, ψ i,t,andη i,t denote the ultiplier on the asset, goods, and open arket constraint. It should be noted that the ultiplier on the open arket constraint, which easures the liquidity value of bonds, tends to decline with the policy rate (see 3), since a higher policy rate reduces the aount of oney for each unit of bonds supplied to the central bank. As can be seen fro the bond pricing equation (5), a rise in this ultiplier tends to lower the interest rate on bonds. This can generate a spread between the Euler and bond rates. Equation (4) defines the Euler-rate (see below) that differs slightly fro the standard Euler rate (see ) due to the cash-credit-good friction, which is easured by ψ i,t. Equation (6), which is derived fro the first order condition on oney holdings, iplies that households are indifferent between oney and bonds, i.e., between both assets that facilitates goods arket transactions. These effect will be analyzed below in detail. 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 H i,t /P t, b i,t = B i,t /P t,andi i,t = I i,t /P t, and (9) and () hold with equality.

12 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 is given by R d t =[E t q t,t+ ]. (7) 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 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 between these differentiated goods. 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.afir j produces good y j eploying a technology which is linear in labor: y jt = a t n α jt,wherea is a stochastic productivity level satisfying a t = a ρ a t exp ε a,t, ρ a and ε 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, 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 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. (8) 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 ε.

13 3.3 Public sector The public sector consists of a governent and a central bank. The governent issues bonds B T, which are either held by households R B i,t di = B t or by the central bank R Bi,t c di = Bt c : Bt T R B i,t di + R Bi,t c di. It further receives payents P tτ 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. The supply of governent bonds will not be irrelevant for the conduct of onetary policy and for onetary transission. In order to specify bond supply in a neutral way, we assue, for siplicity, that governent bonds are 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 R t + B c t + M H t M H t I t M R t. Following the operational practice of central banks we assue that it rolls over their 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 Bt c = B e t c. Further using Bt R = Mt R and Bt c = Rt I t,the budget constraint can be siplified to (Bt c /R t ) Bt c = M t H Mt H +(R t ) 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 bot be negative in equilibriu, such that the central bank will never deand funds fro the governent. 7 Accordingly, its bond holdings will evolve 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). 2

14 according to B c t B c t = R t I t I t M H t + M H t. (9) 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 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. The repo rate ight be set contingent on its own lags and on current inflation to allow for inertia and a Taylor-rule-type interest rate setting. It ight further change in an unpredictable way R t = R t ρ (R ) ρ (π t /π) ρ π ( ρ) exp ε ρ t. (2) where ρ and ε ρ t is norally i.i.d. with E t ε ρ t =and variance var ερ. The longrun 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, where the growth rate is allowed to be serially correlated and ight change in an unpredictable way. 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 exogenously controls the ratio of oney holdings under both types of open arket operations Ω: Mt R = Ω Mt H, 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 behave in an identical way 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 bonds holdings change according to B t B t = 3

15 (Γ )Bt T Bc t + Bt c. Further using (9), 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. (2) Thus, private sector holdings of bonds tend to decrease with a higher repo-rate and to increase for a given injection I t with a larger fraction of oney held under repurchase agreeents. Throughout, we will however focus on the case where the central bank sets its instruent such that the goods arket constraint (8) is strictly binding (ψ t > ). 8 Arational 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 a given onetary policy 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 (2) 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 especially 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 induces 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 bonds than necessary for their oney arket trades. If however both returns are identical, households can borrow and invest in bonds without costs such that the oney arket constraint will not be binding. In order to analyze the two regies in a separate way, we first briefly exaine steady states with a binding and a non-binding open arket constraint. We then assue that, 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. 4

16 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 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 (7), to give the following steady state condition ³ η/λ = R d R /R. (22) 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. Suppose that the central bank conducts oney policy in a way that the average repo-rate R approaches the debt rate R d (π/β) in a steady state. As can be seen fro (6), which leads to the condition R = R in a steady state, the interest rate on governent bonds R will then be identical to the interest rate on private debt R d in the steady state and the ultiplier on the open arket constraint will then be equal to zero η =. The steady state is then characterized by R = R d = R, π/β = R d, c σ/α = R d (γ/α) ε/ (ε ), = c, = H + R, R = Ω H. (23) If however the central bank chooses an average repo-rate R that is strictly saller than R d = π/β, there exists a steady state with a binding open arket constraint satisfying b R π = H π + R, (24) (23), R = R,andb Γπ = R Γπ. Since the latter condition together with (24) would only be consistent with Γ 6= π and Mt H for deflationary equilibria, 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 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. 5

17 4 Interest rate spreads In this section, we exaine the relation between the three interest rates, i.e., the repo rate R, the bond rate R, andthedebtrater d. The bond rate R t and the repo rate Rt are closely related to each other as can be seen fro (6). The spread between these two rates, which depends on second order effects and is relatively sall, will be exained below. Before, we will take a look at the relation between the debt rate Rt d and the bond rate R t,whichwill differ whenever the oney arket constraint is binding. Otherwise, the spread equals zero. For the analysis of both spreads we will use siple versions of the odel, to facilitate the derivation of analytical results, as well as nuerical exaples, which are based on paraeter values (given in table A, see appendix 8.3) and coputed by using a second order approxiation at the deterinistic steady state (see Schitt-Grohé and Uribe, 24). 4. Bond rate vs. debt rate 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 22), where the central bank can ipleent a long-run equilibriu with a binding oney arket constraint if the reporateissetatavaluelowerthanπ/β. Outside the steady state, the debt-bond spread will not be constant over tie and will in particular depend on the onetary policy stance, since the valuation of liquidity will depend on the oney arket conditions. This property can be shown by applying a siplified version (A) of the odel without technology shocks and with flexible prices, constant supply of governent bonds, a linear production technology, perfect copetition, preferences satisfying σ =, oney being supplied under repurchase agreeents only and an exogenous repo rate. This version A is thus characterized by φ = ρ π = var(ε a )=, Γ = α = σ =, ρ>, andω =. Proposition Consider version A of the odel where the open arket constraint is binding, R <π/β. The spread between the debt rate Rt d and the bond rate R t decreases with i.) the variance of repo rate innovations ε ρ t and ii.) the current level of the repo rate. (Details can be found in appendix 8.4.) The debt-bond spread s,t = Rt d R t is a easure for the liquidity value of bonds and can also be interpreted as a liquidity preiu. It particularly depends on the ability of bonds to be converted into eans of payents, i.e. oney, in open arket operations. 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 3). To get soe nubers for the spread s,t, we use a sticky price version of the odel with soe standard paraeter values for quarterly data (σ =2, α =.66, φ =.8, ρ (a) =.9, and For the coputation we used dynare. 6

18 =6). To roughly 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, a target repo rate equal to R =.5, leading to a steady state spread R d R equal to 2 basis points per quarter iplying a spread of 53 basispointsperyear. We further set the inflation feedback ρ π either equal to zero or equal to.5. 2 Finally, the ratio between repo-oney and oney supplied outright Ω, which we found to vary substantially between different saple periods, is set equal to.5. 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 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,whilethiseffect is less pronounced when the repo rate is endogenously adjusted (ρ π =.5). The nuerical results thus support clai i.) in proposition. 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 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 the clai ii.) ade in the proposition. The correlations of the spread further correspond to the epirical results presented in section 2 and in other studies (see Atkeson and Kehoe 28, and Canzoneri et al., 27). Though the odel overstates the negative correlation between the debt rate and the repo rate copared to the nubers in section 2, we can conclude that the debt rate hardly iics the policy rate in all cases. 2 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 of a siplified odel version can be found in Schabert (24). 7

19 4.2 Repo rate vs. bond rate 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 will in general 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+ ) (25) E t [u ct+ /π t+ ] In order to hold both, oney and bonds, households deand the rate of return on bonds to copensate for the costs of converting bonds against oney in next period s open arket operations. Up to first order, the current bond price equals the expected price of oney. However, the price of a governent bond /R t will be saller than the expected future price of oney E t /R t+, if the covariance on the RHS of (25) 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+. This covariance can be shown to be strictly negative under a binding open arket constraint, where a higher repo rate tends to reduce current consuption and inflation. To show this, we apply a siplified version of the odel. Here, we again assue that prices are flexible, the supply of governent bonds is constant, production is linear, firs are perfectly copetitive, and that oney is supplied under repurchase agreeents only. In contrast to version A, we now allow for higher degrees of relative risk aversion, σ>, and assue that the central bank endogenously adjusts the repo rate in a non-inertial way, ρ π >, ρ =,andvar ε ρ =and with i.i.d. technology shocks. This version B is thus characterized by φ = ρ (a) =, Γ = α =, σ>, ρ π >, var ε ρ =,andω =. Proposition 2 Consider version B of the odel where the open arket constraint is binding, R <π/β. The price of governent bonds is saller than the expected future price of oney (/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. (Details can be found in appendix 8.5.) While the covariance is strictly negative, the bond rate further increases for a given repo rate, if aggregate uncertainty or the relative risk aversion increases. In both cases investors want to be copensated by a higher bond rate. Table 4 Spread E s 2,t for technology shocks σ =2 σ =5 var (ε a )=..34 b.p..75 b.p. var (ε a )=.2.68 b.p..5 b.p. 8

20 Applying the paraeter values fro above (with ρ π =.5, see table A), we find sall positive nubers for the spread s 2,t = R t Rt. As shown in table 4, they lie in between.3 basis-points and.5 basis-points, where the latter is obtained for a high variance of the technology shock var (ε a ). 3 The results for different values for σ and for var (ε a ) support the clais ade in the second part of proposition 2. Overall, the odel is able to explain the positive spread between the policy rate and the bond rate, thought the average spread between the federal funds rate and the 3onth-libor presented above (25 b.p.) is uch larger than the odel s predictions. 5 Monetary transission In this section we exaine responses to repo rate innovations and oney supply shocks to disclose the onetary transission echanis in our odel. Throughout the analysis we report results for the case where the open arket constraint is binding, unless the opposite is explicitly entioned. 5. Responses to interest rate shocks Consider a positive innovation to the repo rate satisfying (2) with ρ =.9. Figure presents the ipulse responses of interest rates and acroeconoic aggregates for the case where the repo rate is exogenously set (blue solid line: ρ π =) and for the case where it follows a Taylor type feedback rule (ρ π =.5, green arked line). An increase of the repo rate by % fro its steady state value leads to a rise in the bonds rate by less than one percent, which accords to (25). The debt rate decreases on ipact and is closely followed by the rate R-Euler, which is the rate iplied by a standard Euler equation, βe t [u c,t+ /(u c,t π t+ )] = /Rt Euler ; the latter has no eaningful role in our odel and is only coputed to facilitate coparisons (see section 2). The spread between the debt rate and the bond rate decreases, as predicted in proposition. The ipact response of the spread alost equals the size of its steady state value. 4 Regarding the responses of acroeconoic aggregates, figure 3 further shows that inflation, real balances, and output decline in a hup-shaped way, which is qualitatively consistent with standard VAR evidence. It should be noted that hup-shape ipulse responses are usually not generated by siple sticky price odels (like the version of our odel without the oney arket constraint). Hup-shaped ipulse responses, which are also found in the data, can also be generated by considering additional frictions or rigidities (see Christiano et al., 25). Here, it is ainly driven by the dynaics of households real bond holdings b H t, which falls in response to the onetary contraction. 3 The variances are sall enough so that the ultiplier on the open arket constraint reains positive after a productivity shock hits the econoy. 4 The ultiplier on the open arket constraint is thus positive after the interest rate shock. 9

21 exogenous Taylor-rule.5 R R. R d R-Euler 5 R d -R. Pi r y.2 b Figure 3: Responses (in % dev. fro st.st.) to an interest rate shock On the one hand, the real value of governent bonds should increase, since inflation fall. Yet, the aount of bonds held by the central bank tends to rise by higher interest rates and by less repo oney (see 9). Thus, a onetary tightening does not only lead to contractionary effects on ipact, but subsequently shifts the distribution of bond holdings towards the central bank. With depleting eligible securities, households can acquire less oney in the subsequent periods, such that the initial contraction in consuption will even be enhanced. Thus, the dynaics of bond holdings affects the transission of onetary policy shocks, which relies on the assuption that the central banks does not transfer its wealth to the household at the end of each period. For a saller fraction of repo oney, Ω = Mt R /Mt H, the ipact of an interest rate shock, in particular, the responses of the acroeconoic aggregates, are less pronounced. (The ipulse responses to interest rate shocks for Ω =. are given in the appendix.) Thus, our odel predicts that the size of interest rate shock effects depends on the way the central bank conducts open arket operations. Figure 4 shows ipulse responses to a one percent repo rate innovation for a version of the 2