Macroeconomic Stability in a Model with Bond Transaction Services

Abstract

Cochrane (2014) shows that high-powered money balances and short-term government bonds can be considered as perfect substitutes for the U.S economy during the past twenty years. We build on this claim and consider a variant of the standard cashless new-Keynesian model with two types of government bonds, which can be thought of as short- and long-term bonds. The first one has a macroeconomic role in the sense that it provides transaction services in addition to generating a yield. The other type of government bond pays only an interest rate. Consistent with previous findings, the Taylor principle is not a panacea for equilibrium determinacy in a model without money. When the government bond market matters beyond the need for fiscal solvency, monetary policy rules do not need to comply with the Taylor principle for unique equilibria to exist.

1. Introduction

On 10 February 2010, Fed Chairman Ben S. Bernanke testified before the U.S. House of Representatives Committee on Financial Services with a statement titled “Federal Reserve’s exit strategy”. His speech included the following remarks:

“Most importantly, in October 2008 the Congress gave the Federal Reserve statutory authority to pay interest on banks’ holdings of reserve balances. By increasing the interest rate on reserves, the Federal Reserve will be able to put significant upward pressure on all short-term interest rates, as banks will not supply short-term funds to the money markets at rates significantly below what they can earn by holding reserves at the Federal Reserve Banks.”

Interestingly, in Bernanke’s view, the experience from the payment of interest rates on banks’ reserves could provide useful guidance for amending the operational framework for the Fed monetary policy:1

“The authority to pay interest on reserves is likely to be an important component of the future operating framework for monetary policy. For example, one approach is for the Federal Reserve to bracket its target for the federal funds rate with the discount rate above and the interest rate on excess reserves below.”

This type of monetary policy action has gone hand in hand with large-scale asset purchases by the Fed. On 1 December 2008, Bernanke (2008) announced publicly that the Fed “could purchase longer-term Treasury securities (…) in substantial quantities”. By 18 March 2009, the Fed held 300 billion USD of long-term Treasury securities. Hence, a wide array of different policy tools have been put in place since the outbreak of the financial crisis in the U.S. Based on the idea that macroeconomic recovery runs through the ability of the Fed to steer financial markets towards a desired reaction path, many commentators and policy analysts have dwelled on the short-term effects of the Fed decisions (e.g., see Gagnon et al. 2010; Woodford 2012).

One of the consequences of the Fed actions is that high-powered money balances and short-term government bonds have become perfect substitutes for banks. In other words, banks’ reserve balances and short-term U.S. Treasuries pay a similar interest rate. In the experience of ultra-loose liquidity supply by the FED, reserve requirements are far from binding. In terms of macroeconomic outlook, Cochrane (2014) stresses that the U.S. inflation rate has been stable during the past decade, and that there have been small variations in policy and short-term rates. But what are the implications of this macroeconomic landscape for the ability of monetary policy to lead the economy towards a cyclical path of stability? Does that ability that depend only on the decisions by the central bank in a world where the government bond market plays a well-acknowledged role for achieving the objective of macroeconomic stability? These policy-relevant questions are the focus of our paper.

The point of departure of our analysis concerns the interpretation of two facts that have not been fully discussed in the literature yet, and that are centred around the role of government bonds. In a world where short-term bonds can be considered as perfect substitutes for high-powered money, it becomes clear that short-term bonds have an economic property that long-term bonds do not have. This means that the economic differences between short- and long-term bonds are not only about the maturity structure, i.e., about exogenous characteristics.2

The second key to understanding the implications of the perfect substitutability between banks’ reserves and short-term government securities arises from the role of money as a medium of exchange in the provision of ‘transaction services’ that ‘facilitate’ purchases in consumption goods markets.3 In particular, if money and short-term bonds are perfect substitutes, then these bonds can be expected to deliver transaction services in a broad economic sense.

Obviously this discussion is not meant to imply that anyone can go to the grocery and buy, say, apples by posting bonds as a payment instrument or as a guarantee. Inasmuch as economic agents are satiated with cash in modern economies, the macroeconomic role of government bonds is pervasive in enabling the capability of financial markets to finance every day’s economic activities. They are used to perform functions that indirectly facilitate transactions in the real markets, and that contribute to the smooth functioning of final-goods markets. For instance, Aggarwal et al. (2017) suggests that the European government bond lending market allows borrowers to access safe assets and, as such, is an integral part of the short-term funding market.

Our modelling approach for riskless assets as a function of transaction costs has already been considered in the literature. For instance, Vayanos and Vila (1999) study a continuous-time economy with overlapping generations of agents with two assets. One of these carries a transaction cost, while the other is ‘liquid’ in a microeconomic sense. Vayanos and Vila (1999) show that, in equilibrium, the price difference depends on the relative asset supply.

Based on these considerations, we study the ability of a central bank to pin down determine macroeconomic equilibria in a standard New Keynesian model with two types of bonds.4 The first type provides transaction services that are accounted for in the purchase of consumption goods. It pays the policy rate as a yield. The other bond performs the usual function of store of value, and pays an exogenous spread on the policy rate. We can think of the bond-market segment with frictions as representing short-term bonds. The rest of the bond market can be thought of as a longer-term segment. The financial market friction generates a non-trivial form of the aggregate demand curve, which depends on the spread between the rates on the bond with frictions and the frictionless bond. Monetary policy follows a Taylor rule. Government taxes are non-distortionary and evolve according to a simple feedback rule.

Given the macroeconomic role of government bonds, the fiscal theory of the price level determination of Leeper (1991) holds.5 Hence, our discussion touches upon the large field of study on the interaction between fiscal and monetary policy in a world where nominal rigidities bind.6 Consistently with Cochrane (2014) and Woodford (1995) among others, we assume that agents are satiated with liquidity, and abstract from the role of money in business cycles. For this reason, our analysis can be read as an investigation into the results of Canzoneri and Diba (2005), Canzoneri et al. (2008) and Canzoneri et al. (2011). They study equilibrium determinacy in a New Keynesian setting where money balances and government bonds are imperfect substitutes. Both assets contribute to an aggregate measure of transaction services that allows to finance the purchase of consumption goods. This implies that there is a direct channel for government bonds to affect the inflation rate, along with money balances. Hence, the Taylor principle need not hold any longer for determinate equilibria to exist.

We show that a segmented bond market with transaction services delivers non-trivial implications for equilibrium determinacy also in a cashless economy. Simple rules for monetary policy do not necessarily require that the inflation coefficient should be larger than one in order to generate unique macroeconomic equilibria. In other words, the Taylor principle is not sacrosant any longer, even when changes in monetary aggregates do not matter. For instance, when the policy rate responds to inflation in a persistent way, a weak response of the central bank to inflation on one hand, and a weak response of the tax rate to government liabilities on the other are sufficient to pin down the inflation rate. These findings are consistent with the results of Canzoneri and Diba (2005) and their subsequent papers based on monetary economies.

This paper is organized as follows. In the Section 1 we introduce the model forming the basis of the theoretical results. In Section 2 we discuss the equilibrium characteristics of the model with explicit reference to the existence of a deflationary/inflationary equilibrium path. Section 3 reports the main theoretical results of the model together with some simulations. Section 4 presents some concluding remarks. An additional Appendix includes a detailed discussion of the model structure, and a full set of proofs of the analytical results discussed in the main text.

2. A Model with Bond Transaction Costs

We assume that there is a representative consumer that maximizes the following utility function

$$U_t=\sum _{t=0}^{\infty }{\beta }^t\frac{C_t^{\gamma \left(1-\frac{1}{\sigma }\right)}{\left(1-L_t\right)}^{\left(1-\gamma \right)\left(1-\frac{1}{\sigma }\right)}}{\left(1-\frac{1}{\sigma }\right)}$$

(1)

where $C_t$ indicates the amount of consumption, and $L_t$ the amount of labour effort supplied. In (1) $\beta$ indicates the discount factor, while $\sigma$ ($0<\sigma <\infty$) denotes the intertemporal substitution elasticity, the inverse of which is the coefficient of relative risk aversion.

We consider a cashless economy where the consumer can hold a portfolio of two types of government bonds, $B_t^{*}$ and $B_t$. Bonds $B_t^{*}$ perform the standard function of intertemporal store of value. This type of security pays an interest rate $i_{t-1}^{*}$. Bonds $B_t$, instead, provide direct transaction services in place. This is accounted for by the transaction cost function $f\left(F_t\right)$, where $F_t$ plays the role of ‘velocity of circulation of bonds’:

$$F_t=\frac{P_t{C}_t}{B_t}$$

(2)

In addition to the interest rates obtained from investment in bonds ($i_{t-1}{B}_{t-1}$ and $i_{t-1}^{*}{B}_{t-1}^{*}$), the consumer derives funds from supplying labour in quantity $L_t$, paid at the wage rate $W_t$. An additional source of income is given by the participation to the profit ${\mathsf{\Omega }}_t^j$ of a firm producing the final-good variety j. A lump sum tax denoted by $T_t$ is imposed by the government.7 The consumer’s budget constraint takes the form:

$$B_t+B_t^{*}=\left(1+i_{t-1}^{*}\right)B_{t-1}^{*}+\left(1+i_{t-1}\right)B_{t-1}+W_t{L}_t+P_t{\mathsf{\Omega }}_t^j-P_t{C}_t\left(1+\chi g\left(F_t\right)\right)-T_t$$

(3)

We assume that the bond-velocity function $g\left(F_t\right)$ has the following properties:

$$\begin{array}{ccc}\hfill g\left(F_t\right)& =& 0\mathrm{for}{F}_t\le 0\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill g^{\prime }\left(F_t\right)& >& 0\mathrm{and}{g}^{″}\left(F_t\right)\ge 0\hfill \end{array}$$

(5)

Assumption (4) tells us that negative bond holdings do not provide any transaction services. Assumption (5), instead, formalizes the idea that the transaction cost function is increasing and convex in $F_t$. The convexity of $g\left(F_t\right)$ is needed in order to make sure that the utility maximization program delivers a true maximum. If the budget constraint was not convex, we would need to explore if there is - at least locally - the possibility of showing that the first order conditions deliver a true maximum. Overall, the term $\chi g\left(F_t\right)$ identifies the transaction costs in terms of consumption spending, with a constant scale parameter $\chi$.

The first order conditions with respect to $C_t$, $L_t$, $B_t$ and $B_t^{*}$ are, respectively,

$$\begin{array}{ccc}\hfill \gamma C_t^{\gamma \left(1-\frac{1}{\sigma }\right)-1}{\left(1-L_t\right)}^{\left(1-\gamma \right)\left(1-\frac{1}{\sigma }\right)}& =& P_t{\lambda }_t\left[1+\chi g\left(F_t\right)+\chi g^{\prime }\left(F_t\right)F_t\right]\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill \left(1-\gamma \right)C_t^{\gamma \left(1-\frac{1}{\sigma }\right)}{\left(1-L_t\right)}^{\left(1-\gamma \right)\left(1-\frac{1}{\sigma }\right)-1}& =& {\lambda }_t{W}_t\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill \beta E_t\frac{{\lambda }_{t+1}}{P_{t+1}}\left(1+i_t^{*}\right)& =& \frac{{\lambda }_t}{P_t}\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill \beta E_t\frac{{\lambda }_{t+1}}{P_{t+1}}\left(1+i_t\right)& =& \frac{{\lambda }_t}{P_t}\left[1-\chi g^{\prime }\left(F_t\right)F_t^2\right]\hfill \end{array}$$

(9)

In (6)–(9) ${\lambda }_t$ indicates the Lagrange multiplier associated to constraint (3). Equation (6) defines the intertemporal choice of consumption, where the effects of first order due to the transaction technology appear a critical element in the definition of intertemporal trade-offs. Equation (7) defines the optimal labour supply choice and equates the disutility from work effort to the real wage weighted by the marginal utility of consumption. Equation (8) indicates the optimal allocation of bonds $B_{t-1}^{*}$, while Equation (9) represents the optimal allocation of bonds $B_t$. In particular, from Equation (9) we observe that the presence of transaction costs creates a wedge in the Euler equation between the left and the right-hand side.

We make the assumption that the transaction cost function $g\left(F_t\right)$ is given by

$$g\left(F_t\right)=\frac{F_t^2}{2}$$

(10)

Function (10) fulfils all the requirements stated in (4) and (5) and allows a tractable derivation of the equilibrium conditions. Therefore, using (10) into (6) and (9) and rearranging terms, we can derive the demand for short term bonds

$$b_t={\left[\chi \frac{\left(1+i_t\right)}{i_t^{*}-i_t}\right]}^{1/3}{C}_t$$

(11)

where $b_t=B_t/P_t.$ We can see that the short-term bond demand (11) is increasing with respect to $i_t$ and $C_t$, and (ii) decreasing with respect to $i_t^{*}$. In fact, if the return on un-starred bonds increases, they become more attractive with respect to starred bonds. If consumption increases, then the demand for short term bonds increases, because of the need to finance more transactions. Finally, if the return $i_t^{*}$ on competing assets increases, the demand for $B_t$ falls.8 The relation of imperfect substitutability between the two types of bonds is already reflected by the transaction role attached to short-term bonds.

For simplicity, we assume that the relationship between the interest rates on short- and long-term bonds takes the form:

$$i_t^{*}={\zeta }_t{i}_t$$

(12)

where ${\zeta }_t$ is a term that determines the comovements between $i_t^{*}$ and $i_t$. This follows a first-order autoregressive process with an exogenous shock ${\epsilon }_{\zeta t}$:

$$log{\zeta }_t=\left(1-{\rho }_{\zeta }\right)\zeta +{\rho }_{\zeta }log{\zeta }_{t-1}+{\epsilon }_{\zeta t}$$

(13)

The rest of the model follows the standard new-Keynesian tradition. To complete the demand side of the economy, we assume that there is a large number of differentiated goods indexed over the real line between 0 and 1. This allows each firm to have a control of the price of her final good to be sold, since output becomes demand determined. Following the approach by Dixit and Stiglitz (1977), we assume that the consumption bundle $C_t$ demanded by the consumer is a CES type aggregate of all the $j\in \left[0,1\right]$ varieties of final goods produced in this economy

$$C_t={\left[{\int }_0^1{c}_t{\left(j\right)}^{\frac{\theta -1}{\theta }}dj\right]}^{\frac{\theta }{\theta -1}}$$

(14)

where $\theta$ is the elasticity of substitution between different varieties of goods produced by each firm j.

On the supply side, the source of nominal rigidities consist in sticky prices modelled via the menu costs approach of Rotemberg (1982). A firm pays a cost in terms of output $Y_t$ each time the price level of her final goods differs from steady state inflation rate $\pi$. We assume that price adjustments take place subject to the function according to

$$A{C}_t^p=\frac{{\varphi }_p}{2}{\left[\frac{P_t\left(j\right)}{P_{t-1}\left(j\right)}-\pi \right]}^2{Y}_t$$

(15)

where $\pi$ is the steady-state inflation rate. The production function of the firm for each variety j is given by

$$Y_t\left(j\right)=A_t{L}_t^{\alpha }\left(j\right)$$

(16)

All firms producing j varieties are subject to an homogenous technological shock $A_t$, which follows the standard autoregressive process of order one. This problem structure implies that each firm faces a downward demand curve, and chooses the optimal quantity of labour input $L_t\left(j\right)$ by maximizing the expected stream of real profits subject to the demand function, the production function (16) and the price adjustment cost function (15).

The government budget constraint is

$$B_t+B_t^{*}=\left(1+i_{t-1}\right)B_{t-1}+\left(1+i_{t-1}^{*}\right)B_{t-1}^{*}+G_t-T_t$$

(17)

In (17) the primary deficit (surplus) $G_t-T_t$ plus interest rate proceedings paid by the government to the owner of government debt (both short and long debt) $i_{t-1}{B}_{t-1}$, $i_{t-1}^{*}{B}_{t-1}^{*}$ are financed by issuing new debt: $B_t-B_{t-1}$, $B_t^{*}-B_{t-1}^{*}$.

Following the prescription of the fiscal theory of price level determination, the comparative evaluation of alternative monetary policy rules has to be coupled with a fiscal policy rule setting the level of taxes in reaction to the outstanding path of real debt. Therefore, a ‘tax backing’ of public debt is needed in order to prevent the increase of the prices. We assume that this is implemented according to the simple fiscal policy rule

$$T_t={\psi }_0+{\psi }_1\frac{B_{t-1}^{*}}{P_t}+{\psi }_2\frac{B_{t-1}}{P_t}$$

(18)

According to (18), the level of fiscal revenues is set to react to the outstanding level of real debt. Our final assumption concerns the evolution of public expenditure $G_t$, for which we assume the following AR(1) process

$$log{G}_t=\left(1-{\rho }_G\right)G+{\rho }_{G}log{G}_{t-1}+{\epsilon }_{Gt}$$

(19)

where ${\epsilon }_{Gt}$ is an i.i.d. random variable distributed normally, with zero mean and variance given by ${\sigma }_g^2$.

To close the model, we assume that the central bank sets the policy rate according to variants of the well-celebrated rule proposed by Taylor (1993) and largely used in the literature (e.g., see Clarida et al. 2000).:

$$i_t=i{\left(\frac{{\pi }_t}{\pi }\right)}^{{\varphi }_{\pi }}{\left(\frac{y_t}{y}\right)}^{{\varphi }_y}{\left(\frac{i_{t-1}}{i}\right)}^{{\varphi }_i}$$

(20)

with ${\varphi }_{\pi },{\varphi }_y,$ and ${\varphi }_i$ as positive reaction coefficients.

Qualitative Model Properties

In order to provide intuition on our variant of the New Keynesian model, we discuss the impulse responses of the model to the term spread shocks ${\epsilon }_{\zeta t}$.9 For that purpose, we calibrate the model on quarterly U.S. data over the sample 1959:1-2007:4. The values of the non-policy parameters are reported in

Table 1. Calbration of non-policy parameters.

$\mathit{\beta }$	$\mathit{\chi }$	$\mathit{\gamma }$	$\mathit{\theta }$	$\mathit{\sigma }$	${\mathit{\psi }}_0$	${\mathit{\psi }}_1$	$\mathit{\alpha }$
0.997	0.22	0.76	10	0.1	8.31	0.05	0.77

. The nominal interest rate has been set equal to 1.6 per cent per quarter, as recovered from sample observations. In order to simplify algebra, the long run inflation rate (in gross terms) has been set equal to 1. The share of consumption in GDP at the steady state is assumed to be 0.57. The value of $\chi$ has been set to match a level of transaction costs equal to 2 per cent per year. The parameter ${\varphi }_0$ in the fiscal policy reaction function has been set to match the steady state ratio of debt to GDP equal to 0.44. For the analysis of this section, we fix the parameters of the Taylor rule to ${\varphi }_{\pi }=1.5$, ${\varphi }_y=0.5$ and ${\varphi }_i=0.7$.10 We also assume that the fiscal policy rule depends on ${\psi }_0=0.1$ and ${\psi }_1={\psi }_2=0.05$. Finally, we set the autocorrelation ${\rho }_{\zeta }$ of the term spread shock to 0.98.11

Figure 1 reports the impulse responses to a one standard deviation of the term spread shock ${\epsilon }_{\zeta t}$ in percentage points. This implies that the long-term interest rate increases by more than the policy rate. Since the supply of nominal bonds is fixed, the inflation rate must increase for the real bond holdings to fall, and for markets to clear. The positive response of the two interest rates on impact generates proceeds on the pre-existing stock of debt. This allows households to increase consumption while working less. The negative response of output depends on the interaction between two forces of opposite sign. The rise in consumption and the drop of the holdings of short-term real debt produces an increase in transaction costs, which drives output above steady state. However, the impact of the drop in labour supply prevails.

3. Determinacy of Rational Expectations Equilibria

In this section we study the stability of a system with three equations for ${\tilde{\pi }}_t,$ ${\tilde{y}}_t$ and ${\tilde{b}}_t$. One of these equations is related to the policy rate, which is set according to alternative formulations of the general reaction function (20).12

3.1. Taylor Rule

Let us assume that monetary policy follows the standard rule

$${\tilde{i}}_t={\varphi }_{\pi }{\tilde{\pi }}_t+{\varphi }_y{\tilde{y}}_t$$

(21)

This formulation implies that the nominal interest rate reacts contemporaneously with respect to current inflation and output gap. Using Equation (21), we can obtain a closed-form three-equation system in ${\tilde{\pi }}_t,$ ${\tilde{y}}_t$, ${\tilde{b}}_t$ represented in matrix form as13

$$A{Z}_{t+1}=B{Z}_t$$

(22)

or, after multiplying matrix B by $A^{-1}$,

$$Z_{t+1}=\mathsf{\Gamma }{Z}_t$$

(23)

With the monetary policy rule (21), the matrix $\mathsf{\Gamma }$ is defined according to

$$\begin{array}{ccc}\hfill \mathsf{\Gamma }& =& \left[\begin{array}{cc}{\mathsf{\Gamma }}_{11}^{2\times 2}& {\mathsf{\Gamma }}_{12}^{2\times 1}\\ {\mathsf{\Gamma }}_{21}^{1\times 2}& {\mathsf{\Gamma }}_{22}^{1\times 1}\end{array}\right]\hfill \end{array}$$

(24)

$$\begin{array}{cc}=& \left[\begin{array}{ccc}{\beta }^{-1}\left(1+{\mu }_i{\varphi }_{\pi }\right)& \beta \left({\mu }_i{\varphi }_y-{\mu }_y\right)& 0\\ {\alpha }_{21}& {\alpha }_{22}& 0\\ {\alpha }_{31}& {\alpha }_{32}& {\beta }^{-1}-{\psi }_1\end{array}\right]\hfill \end{array}$$

(25)

where ${\alpha }_{21}=\frac{\left({\lambda }_{\pi }-{\lambda }_i{\varphi }_{\pi }\right)\left(1+{\mu }_i{\varphi }_{\pi }\right)}{\beta \left(1+{\lambda }_i{\varphi }_y\right)}+\frac{{\gamma }_i{\varphi }_{\pi }}{1+{\lambda }_i{\varphi }_y}$, ${\alpha }_{22}=\frac{\left({\lambda }_{\pi }-{\lambda }_i{\varphi }_{\pi }\right)\left({\mu }_i{\varphi }_y-{\mu }_y\right)+\beta \left(1+{\gamma }_i{\varphi }_{\pi }\right)}{\beta \left(1+{\lambda }_i{\varphi }_y\right)}$, $b_{31}=\frac{\left({\gamma }_y+{\gamma }_i{\varphi }_y\right)\left({\lambda }_i{\varphi }_{\pi }-{\lambda }_{\pi }\right)}{\beta \left(1+{\lambda }_i{\varphi }_y\right)}-\frac{\left({\gamma }_{\pi }+{\gamma }_i{\varphi }_{\pi }\right)}{\beta }$, $b_{32}=\frac{{\gamma }_y+{\gamma }_i{\varphi }_y}{1+{\lambda }_i{\varphi }_y}$, ${\alpha }_{31}=b_{31}\left(1+{\mu }_i{\varphi }_{\pi }\right)+b_{32}{\gamma }_i{\varphi }_{\pi }+{\theta }_i{\varphi }_{\pi }$, ${\alpha }_{32}=b_{31}\left({\mu }_i{\varphi }_y-{\mu }_y\right)+b_{32}\left(1+{\gamma }_i{\varphi }_{\pi }\right)+{\theta }_y+{\theta }_i{\varphi }_y$.

Proposition 1.

Given ${\varphi }_{\pi }\ge 0$, ${\varphi }_y\ge 0$. For a model with contemporaneous inflation and output targeting interest rules, necessary and sufficient conditions for a unique rational expectations equilibrium are: (i) either

$${\varphi }_{\pi }<{\underline{\varphi }}_{\pi },{\varphi }_y>{\overline{\varphi }}_y\mathit{and}{\beta }^{-1}-1<{\psi }_1<{\beta }^{-1}+1$$

(26)

(ii) or

$${\varphi }_{\pi }>{\underline{\varphi }}_{\pi }\mathit{or}{\varphi }_y<{\overline{\varphi }}_y\mathit{and}{\psi }_1<{\beta }^{-1}-1;{\psi }_1>{\beta }^{-1}+1$$

(27)

where

$$\begin{array}{ccc}\hfill {\underline{\varphi }}_{\pi }& =& \frac{{\mu }_y\left(1+{\lambda }_{\pi }\right)-2\left(1+\beta \right)-{\varphi }_y\left({\lambda }_i+{\lambda }_{\pi }{\mu }_i+{\gamma }_i\left(1+\beta \right)\right)}{2{\mu }_i+{\lambda }_i{\mu }_y}\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill {\overline{\varphi }}_y& =& \frac{{\varphi }_{\pi }{\lambda }_i{\mu }_i+\beta -1+{\mu }_y\left(1-{\lambda }_{\pi }\right)}{{\gamma }_i-{\lambda }_i-{\lambda }_{\pi }{\mu }_i-\beta {\gamma }_i}\hfill \end{array}$$

(29)

Proof. 

See Appendix E. □

Even in this case we obtain the results presented for pure inflation targeting. From condition (26) we see that a combination of active monetary policy with passive fiscal policy delivers a determinate equilibrium. Since matrix $\mathsf{\Gamma }$ is once again lower triangular to guarantee a determinate equilibrium we need to have two roots outside the unit circle. Because of the format of matrix $\mathsf{\Gamma }$, the requirement of active monetary policy can be violated, but the equilibrium is still determinate if fiscal policy is properly set to keep the price level determinate.

To investigate the nature of the bounds described in the proposition above, we compute numerical model solutions for intervals of ${\varphi }_{\pi }$ and ${\varphi }_y$ described in (28)–(29). We calibrate the model parameters to the values outlined earlier. Figure 2 reports the numerical bounds for the region of determinate equilibria. The continuous line refers to the bound (29) for ${\varphi }_y$ as function of ${\varphi }_{\pi }$, and the dotted line represents the evolution of ${\varphi }_{\pi }$ (28) as function of ${\varphi }_y$. Each parameter varies over a range of values largely accepted by current empirical evidence lying between 0 and 10. The direction of various arrows follows the inequalities (28)–(29). The graph in Figure 2 indicates that determinacy can be obtained in a wide range of parameter, but there is an upper bound for inflation targeting coefficient ${\varphi }_{\pi }$, and a lower bound for ${\varphi }_y$. In the model considered here there is an upper bound for the inflation targeting coefficient. It is not difficult to realize that this is almost entirely due to the presence of liquidity services modelled in a non-trivial way. Fiscal policy parameters have the role to pin down determinacy conditions even when bounds (28)–(29) are not respected. When, instead the conditions are given by (27), the regions with determinacy/indeterminacy is reversed, with respect to what has been represented in Figure 2. In this case, monetary policy need not be active, provided that fiscal policy evolves according to (27).

3.2. Pure Current-Inflation Targeting

In this section, we assume that monetary policy follows the simple rule

$${\tilde{i}}_t={\varphi }_{\pi }{\tilde{\pi }}_t$$

(30)

The first-difference system is defined by the matrices

$$Z_t=\left[\begin{array}{c}{\tilde{\pi }}_t\\ {\tilde{y}}_t\\ {\tilde{b}}_t^{*}\end{array}\right];A=\left[\begin{array}{ccc}\beta & 0& 0\\ {\lambda }_i{\varphi }_{\pi }-{\lambda }_{\pi }& 1& 0\\ a_1& {\gamma }_y& 1\end{array}\right];B=\left[\begin{array}{ccc}1+{\mu }_i{\varphi }_{\pi }& -{\mu }_y& 0\\ {\gamma }_i{\varphi }_{\pi }& 1& 0\\ b_3& b_2& {\beta }^{-1}-{\psi }_1\end{array}\right]$$

with $a_1\equiv {\gamma }_i{\varphi }_{\pi }+{\gamma }_{\pi };$ $b_2\equiv {\gamma }_i{\varphi }_y+{\gamma }_y;$ $b_3\equiv {\theta }_i{\varphi }_y+{\theta }_y$, and

$$\mathsf{\Gamma }=\left[\begin{array}{ccc}{\beta }^{-1}\left(1+{\mu }_i{\varphi }_{\pi }\right)& -\beta {\mu }_y& 0\\ {\gamma }_{21}& {\gamma }_{22}& 0\\ {\gamma }_{31}& {\gamma }_{32}& {\beta }^{-1}-{\psi }_1\end{array}\right]$$

(31)

where the terms ${\gamma }_{21}$, ${\gamma }_{22},{\gamma }_{31}$, ${\gamma }_{32}$ in (31) are defined as follows

$$\begin{array}{ccc}\hfill {\gamma }_{21}& =& {\beta }^{-1}\left(1+{\mu }_i{\varphi }_{\pi }\right)\left({\lambda }_{\pi }-{\lambda }_i{\varphi }_{\pi }\right)+{\gamma }_i{\varphi }_{\pi }\hfill \\ \hfill {\gamma }_{22}& =& 1-{\beta }^{-1}\left({\lambda }_{\pi }-{\lambda }_i{\varphi }_{\pi }\right)\hfill \\ \hfill {\gamma }_{31}& =& \left[{\gamma }_y\left({\lambda }_i{\varphi }_{\pi }-{\lambda }_{\pi }\right)-{\gamma }_i\right]\left(1+{\mu }_i{\varphi }_{\pi }\right)+b_3-a_1{\gamma }_i{\varphi }_{\pi }\hfill \end{array}$$

From (31), we observe that the structure of the system is block-triangular. Therefore, to study determinacy we can restrict our attention to the $2\times 2$ submatrix $\mathrm{\Delta }$ given by

$$\mathrm{\Delta }=\left[\begin{array}{cc}{\beta }^{-1}\left(1+{\mu }_i{\varphi }_{\pi }\right)& -\beta {\mu }_y\\ {\gamma }_{21}& {\gamma }_{22}\end{array}\right]$$

(32)

To achieve a fully-determinate system, two eigenvalues should be outside the unit circle and one inside, since public debt is a predetermined variable. This allows a richer configuration of determinacy conditions rather than in cases without an explicit role for the government budget constraint.

Proposition 2.

Let ${\varphi }_{\pi }\ge 0$. Under contemporaneous pure inflation targeting rule, necessary and sufficient conditions for a rational expectations equilibrium to be determinate are:

(a) either

$${\varphi }_{\pi }>{\varphi }_{\pi 1}\mathit{and}{\beta }^{-1}-1<{\psi }_1<{\beta }^{-1}+1$$

(33)

(b) or

$${\varphi }_{\pi }<{\varphi }_{\pi 1}\mathit{and}{\psi }_1<{\beta }^{-1}-1;{\psi }_1>{\beta }^{-1}+1$$

(34)

where ${\varphi }_{\pi 1}=\frac{{\mu }_y{\lambda }_{\pi }-2\left(1+\beta \right)}{\left[2{\mu }_i+{\mu }_y\left({\gamma }_i+{\lambda }_i\right)\right]}.$

Proof. 

See Appendix F. □

According to the terminology of Leeper (1991), condition (33) identifies a combination of active monetary and passive fiscal policy, while condition (34) identifies a combination of passive monetary and active fiscal. In the sense of Leeper (1991), an active monetary policy is defined when monetary authority sets nominal interest rate (or money supply) in order to keep under control the inflation rate. A passive monetary policy is when, instead, the interest rate (or money supply) are left free to adjust. In the recent monetary policy literature, condition (34) is identified with a combination of active fiscal policy and passive monetary policy. In particular, if ${\varphi }_{\pi }>{\varphi }_{\pi 1}$ and ${\psi }_1<{\beta }^{-1}-1$, ${\psi }_1>{\beta }^{-1}+1$, we have three roots inside the unit circle. In this case, we have three converging roots and a continuum of solution. In this case the price level is converging, but indeterminate. A combination of passive fiscal policy with passive monetary policy determines an indeterminacy of order 1. As condition (34) shows, determinacy can be also reached if the burden of price stability is entirely on behalf of fiscal policy authority. Hence, monetary policy can be also set independently from the inflation rate and determinacy is ensured by fiscal policy.

3.3. Expected Inflation Targeting

Now we consider a generalization of expected inflation targeting

$${\tilde{i}}_t={\varphi }_{\pi }{E}_t{\tilde{\pi }}_{t+1}$$

(35)

The resulting system of first-order difference equations with the form (22) depends on the matrices14

$$A=\left[\begin{array}{ccc}1& 0& 0\\ a_{21}& a_{22}& 0\\ a_{31}& a_{32}& 1\end{array}\right];B=\left[\begin{array}{ccc}{b}_{11}& -{\mu }_y{b}_{11}& 0\\ 0& 1& 0\\ 0& {\theta }_y& \left({\beta }^{-1}-{\psi }_1\right)\end{array}\right]$$

where $a_{21}=\frac{{\lambda }_i{\varphi }_{\pi }-{\lambda }_{\pi }\left(\beta -{\mu }_i{\varphi }_{\pi }\right)-{\gamma }_i{\varphi }_{\pi }\left(\beta -{\mu }_i{\varphi }_{\pi }\right)}{\left(\beta -{\mu }_i{\varphi }_{\pi }\right)}$; $a_{22}=\frac{\beta -{\varphi }_{\pi }\left({\mu }_i+{\lambda }_i{\mu }_y\right)}{\left(\beta -{\mu }_i{\varphi }_{\pi }\right)}$; $a_{31}={\gamma }_{\pi }+\frac{{\gamma }_i{\varphi }_{\pi }}{\beta -{\mu }_i{\varphi }_{\pi }}-{\theta }_i{\varphi }_{\pi }$; $a_{32}={\gamma }_y-\frac{{\gamma }_i{\varphi }_{\pi }{\mu }_y}{\beta -{\mu }_i{\varphi }_{\pi }}$; $b_{11}={\left(\beta -{\mu }_i{\varphi }_{\pi }\right)}^{-1}$. The structure of matrix $\mathsf{\Gamma }$ is given by

$$\begin{array}{ccc}\hfill \mathsf{\Gamma }& =\hfill & \left[\begin{array}{cc}{\mathsf{\Gamma }}_{11}^{2\times 2}& {\mathsf{\Gamma }}_{12}^{2\times 1}\\ {\mathsf{\Gamma }}_{21}^{1\times 2}& {\mathsf{\Gamma }}_{22}^{1\times 1}\end{array}\right]=\hfill \\ & =\hfill & \left[\begin{array}{ccc}{b}_{11}& -{\mu }_y{b}_{11}& 0\\ -\frac{a_{21}{b}_{11}}{a_{22}}& \frac{{\mu }_y{b}_{11}{a}_{21}+1}{a_{22}}& 0\\ c_{31}{b}_{11}& c_{32}-c_{31}{\mu }_y{b}_{11}+{\theta }_y& {\beta }^{-1}-{\psi }_1\end{array}\right]\hfill \end{array}$$

(36)

where $c_{31}=\left(a_{21}{a}_{32}-a_{22}{a}_{31}\right)/\beta$; $c_{32}=-{\beta }^{-1}{a}_{32}$. From (36) we observe that the structure of the system is block-triangular. The conditions for determinacy are collected in the following proposition.

Proposition 3.

Given ${\varphi }_{\pi }\ge 0$. For a model with expected inflation targeting interest rules, necessary and sufficient conditions for an equilibrium to be determinate are:

(i) either

$${\overline{\varphi }}_{\pi 1}<{\varphi }_{\pi }<{\overline{\varphi }}_{\pi 2}and{\beta }^{-1}-1<{\psi }_1<{\beta }^{-1}+1$$

(37)

(ii) or

$${\overline{\varphi }}_{\pi 1}<{\varphi }_{\pi }<{\overline{\varphi }}_{\pi 2}and{\psi }_1<{\beta }^{-1}-1;{\psi }_1>{\beta }^{-1}+1$$

(38)

Proof. 

See Appendix G. □

Proposition 3 suggests that it is possible to reach determinacy conditions without satisfying the Taylor principle. This is possible because of a non-trivial specification of the AS and IS curve, due to the different way by which liquidity services are modelled in this context.

3.4. Backward Inflation Targeting

Another variant of the Taylor rule is based on the idea of backward-looking inflation targeting

$${\tilde{i}}_t={\varphi }_{\pi }{\tilde{\pi }}_{t-1}$$

(39)

Our results are collected in the following proposition

Proposition 4.

With a backward-looking rule, necessary and sufficient conditions for a unique equilibrium are such that

$$\begin{array}{ccc}\hfill {\varphi }_{\pi }& >& {\overline{\varphi }}_{\pi 1}\hfill \end{array}$$

(40)

$$\begin{array}{ccc}\hfill {\varphi }_{\pi }& >& {\overline{\varphi }}_{\pi 2}\hfill \end{array}$$

(41)

and

$${\beta }^{-1}-1<{\psi }_1<{\beta }^{-1}+1$$

(42)

where

$$\begin{array}{ccc}\hfill {\overline{\varphi }}_{\pi 1}& =& \frac{{\lambda }_{\pi }}{{\lambda }_i-{\gamma }_i}\hfill \end{array}$$

(43)

$$\begin{array}{ccc}\hfill {\overline{\varphi }}_{\pi 2}& =& \frac{1+2\left(1+\beta \right)+{\mu }_y{\lambda }_{\pi }}{2{\mu }_i+{\mu }_y\left({\lambda }_i+{\gamma }_i\right)}\hfill \end{array}$$

(44)

Proof. 

See Appendix H. □

With our baseline model calibration, the reader can easily check that bounds (43)–(44) are not binding, if ${\varphi }_{\pi }>0$. As a result, the following holds:

Corollary 5.

With our model calibration, necessary and sufficient conditions for a unique equilibrium are ${\varphi }_{\pi }>0$ and (42).

This result emerges because the bounds established in Proposition 4 are negative. Obviously the determinacy regions depend on the parameter values of the Taylor rule. It should be stressed, though, that the Taylor principle does not need to hold strictly to generate determinate equilibria.

3.5. Expected Inflation and Current Output

In this section, we focus on the monetary policy rule

$${\tilde{i}}_t={\varphi }_{\pi }{E}_t{\tilde{\pi }}_{t+1}+{\varphi }_y{\tilde{y}}_t$$

(45)

Rule (45) is a representation for the monetary policy strategy of expected inflation targeting outlined by Svensson (2003). It is based on a pre-emptive response of the monetary authority with respect to the inflation rate.

Proposition 6.

Under rule (45), we have the following necessary and sufficient conditions for a unique equilibrium:

$$\begin{array}{ccc}\hfill {\varphi }_y& <& \frac{{\lambda }_i{\mu }_y}{{\gamma }_i{\mu }_i}\hfill \end{array}$$

(46)

$$\begin{array}{ccc}\hfill {\varphi }_{\pi }& >& \frac{\beta }{{\mu }_i}\hfill \end{array}$$

(47)

$$\begin{array}{ccc}\hfill {\varphi }_{\pi }& >& \frac{\beta \left({\gamma }_i-{\lambda }_i\right)-{\mu }_i{\lambda }_{\pi }}{{\gamma }_i{\mu }_i}\hfill \end{array}$$

(48)

$$\begin{array}{ccc}\hfill {\varphi }_y& <& \frac{{\varphi }_{\pi }\left[{\mu }_i+{\mu }_y\left({\lambda }_i-{\gamma }_i\right)\right]-\left(1-\beta \right)+{\mu }_y{\lambda }_{\pi }}{\left(1-\beta \right){\lambda }_i+{\mu }_i{\lambda }_{\pi }-{\varphi }_{\pi }{\gamma }_i{\mu }_i}\hfill \end{array}$$

(49)

and

$${\beta }^{-1}-1<{\psi }_1<{\beta }^{-1}+1$$

(50)

Proof. 

See Appendix I. □

Figure 3 plots the numerical regions consistent with the bounds (48)–(49) for determinate equilibria. These are based on the model calibration discussed earlier. The continuous line refers to bound (48) for ${\varphi }_{\pi },$ while the dashed line refers to (49) for ${\varphi }_y$. We observe that the region for determinacy is located within an area for high values for ${\varphi }_{\pi }$ and low values for ${\varphi }_y$. In this case, the conditions outlined above comply fully with the Taylor principle.

3.6. Interest-Rate Smoothing

Here we consider a Taylor rule with interest rate smoothing:

$${\tilde{i}}_t={\varphi }_{\pi }{\tilde{\pi }}_t+{\varphi }_i{\tilde{i}}_{t-1}$$

(51)

The results are collected in the following proposition

Proposition 7.

Under Rule (51), the following necessary and sufficient conditions hold:

$${\varphi }_i<\frac{1}{\beta }$$

(52)

$${\varphi }_{\pi }>\frac{1+\beta {\varphi }_i{\mu }_y{\lambda }_{\pi }}{\beta \left({\mu }_i+{\mu }_y{\gamma }_i\right)}$$

(53)

$${\varphi }_{\pi }>\frac{{\mu }_y{\lambda }_{\pi }\left(1+\beta {\varphi }_i\right)-2\left(1+\beta \right)-2\beta {\varphi }_i\left(1+\beta \right)}{{\mu }_i\left(1+\beta \right)+{\mu }_y\left({\lambda }_i+\beta {\gamma }_i\right)}$$

(54)

$${\varphi }_i<\frac{{\varphi }_{\pi }\beta \left({\mu }_i+{\mu }_y{\gamma }_i\right)-1}{\beta {\mu }_y{\lambda }_{\pi }}$$

(55)

and

$${\beta }^{-1}-1<{\psi }_1<{\beta }^{-1}+1$$

(56)

Proof. 

See Appendix J. □

Figure 4 plots the bounds described in Proposition 7 for the baseline calibration of the model. The horizontal line consists of the bound (52) for ${\varphi }_i$. Proposition 7 establishes also two bounds for ${\varphi }_{\pi }$. These follow from Equations (53) and (54) and are represented, respectively, by the= dash-dotted the dashed lines. The continuous line corresponds to the bound (55). From Figure 4 we observe that there are determinate equilibria for values of ${\varphi }_{\pi }$ generally bigger than one (as stated in the current empirical literature), and for values for ${\varphi }_i$ compatible with (52), when fiscal policy lies within the range defined by (56).

4. Conclusions

In this paper, we discuss the determinacy of equilibria in a cashless environment where government debt plays an important macroeconomic role. We consider a variant of the standard New Keynesian framework with two types of government bonds. We introduce a short-term bond that provides transaction services the purchase of consumption goods, and a long-term bond that acts as a store of value. We show that, when fiscal policy is designed according to the prescriptions of fiscal theory of the price level, determinacy is achieved by following the Taylor principle (see Leeper and Leith 2016). In order to pin down the price level, fiscal policy need not be passive in the sense of Leeper (1991). These findings confirm with those of Canzoneri and Diba (2005) and Canzoneri et al. (2008), who use a model with liquidity services from money and bonds to suggest that the Taylor principle is not a panacea for determinacy.

Our analysis can be extended along a number of relevant directions. First of all, we can drop the simplifying assumption about exogenous spread between the interest rates of the two types of bonds. An endogenous spread would introduce an additional element that might strengthen the macroeconomic role of government bonds and, possibly, alter the conditions for equilibrium determinacy. The introduction of different forms of distortionary taxation would add new transmission channel that can change the results greatly.

Vayanos and Vila (2009) has proposed a preferred-habitat theory of term structure of interest rates. Their work suggests that there is a non-negligible class of investors that have preference motives for holding government bonds with a desired maturity. But this opens up a specific scope for bonds in general equilibrium, which is independent from the role of transaction or liquidity services. In future work, we plan to consider an extended framework of substitutability or complementarity between consumption and government bonds.