A Continuous-Time Model of the Term Structure of Interest Rates with Fiscal-Monetary Policy Interactions

Abstract

We study the term structure implications of the fiscal theory of price level determination. We introduce the intertemporal budget constraint of the government in a general equilibrium model in continuous time. Fiscal policy is set according to a simple rule whereby taxes react proportionally to real debt. We show how to solve for the prices of real and nominal zero coupon bonds.

Full text

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Massimiliano Marzo – Silvia Romagnoli – Paolo Zagaglia
A continuous-time model of the
term structure of interest rates with
fiscal-monetary policy interactions
Bank of Finland Research
Discussion Papers
25 • 2008

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Suomen Pankki
Bank of Finland
PO Box 160
FI-00101 HELSINKI
Finland
?? +358 10 8311

http://www.bof.fi

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Bank of Finland Research
Discussion Papers
25 • 2008

Massimiliano Marzo* – Silvia Romagnoli** –
Paolo Zagaglia***

A continuous-time model of the
term structure of interest rates
with fiscal-monetary policy
interactions

The views expressed in this paper are those of the authors and
do not necessarily reflect the views of the Bank of Finland.

* Università di Bologna. E-mail:
massimiliano.marzo@unibo.it.
** Università di Bologna. E-mail:
silvia.romagnoli@unibo.it.
*** Bank of Finland and Stockholm University. E-mail:
paolo.zagaglia@bof.fi. Corresponding author.

We are grateful to Juha Kilponen and Jouko Vilmunen for
many suggestions.

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http://www.bof.fi

ISBN 978-952-462-466-4
ISSN 0785-3572
(print)

ISBN 978-952-462-467-1
ISSN 1456-6184
(online)

Helsinki 2008

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A continuous-time model of the term structure of
interest rates with fiscal-monetary policy interactions
Bank of Finland Research
Discussion Papers 25/2008
Massimiliano Marzo – Silvia Romagnoli – Paolo Zagaglia
Monetary Policy and Research Department


Abstract
We study the term structure implications of the fiscal theory of price level
determination. We introduce the intertemporal budget constraint of the
government in a general equilibrium model in continuous time. Fiscal policy is set
according to a simple rule whereby taxes react proportionally to real debt. We
show how to solve for the prices of real and nominal zero coupon bonds.

Keywords: bond pricing, fiscal policy, mathematical methods

JEL classification numbers: D9, G12

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Hintatason fiskaalinen teoria ja korkojen
aikarakenteen määräytyminen jatkuva-aikaisessa
mallissa
Suomen Pankin keskustelualoitteita 25/2008
Massimiliano Marzo – Silvia Romagnoli – Paolo Zagaglia
Rahapolitiikka- ja tutkimusosasto


Tiivistelmä
Tässä työssä tarkastellaan tuottokäyrän eli korkojen aikaranteen dynamiikkaa, kun
talouden hintataso määräytyy osana julkisen sektorin velan rahoitusratkaisua eli
olettaen, että hintatason fiskaalinen teoria on voimassa. Työssä käytettyä jatkuva-
aikaista yleisen tasapainon mallia täydennetään julkisen sektorin intertemporaali-
sella budjettirajoitteella, joka kuvaa julkisen sektorin kunkin ajankohdan velkaan-
tumispotentiaalia annetulla talouden hintatasolla. Finanssipolitiikka puolestaan
määräytyy yksinkertaisen verosäännön mukaan säätelemällä verotuksen kireyttä
suorassa suhteessa julkisen velan reaaliarvon muutoksiin. Työn tutkimuksellinen
anti on pääasiallisesti metodinen, koska työssä osoitetaan, miten rahassa ja kulu-
tuksessa mitattujen eli nimellisten ja reaalisten nollakuponkilainojen hinnat voi-
daan tässä tapauksessa ratkaista.

Avainsanat: joukkovelkakirjojen hinnoittelu, finanssipolitiikka, matemaattiset
menetelmät

JEL-luokittelu: D9, G12

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Contents
Abstract .................................................................................................................... 3
Tiivistelmä (abstract in Finnish) .............................................................................. 4

1 Introduction ...................................................................................................... 7

2 The model economy .......................................................................................... 8

3 Fiscal and monetary policy .............................................................................. 9

4 The optimal choice problem .......................................................................... 13

5 Definition of equilibrium ............................................................................... 14

6 The equilibrium in the continuous time limit .............................................. 16

7 A specialized economy ................................................................................... 19

8 The real spot interest rate .............................................................................. 22

9 The term structure of real interest rates ...................................................... 24

10 The term structure of nominal interest rates .............................................. 26

11 Simulation results ........................................................................................... 27

12 Concluding remarks ....................................................................................... 28

References .............................................................................................................. 29

Appendix A ............................................................................................................ 30

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1Introduction
The theory of price level determination advocated by Leeper (1991), Sims
(1994), Woodford (1996) and Cochrane (1998) has brought to the attention of
macroeconomists the role of interactions between fiscal and monetary policy. In a
nutshell, the idea is that the price level is determined by the degree of solvency of
the government. If the expected primary surplus is not sufficient to comply with
the intertemporal budget constraint of the government, then part of the public
debt should be inflated away if it is default-free.
Although the fiscal theory of price level determination has generated a
substantial debate on the capability of fiscal and monetary policy to affect the
price level, study has considered its potential implications for asset prices. This
considerations holds both for the finance and macroeconomics literature. For
instance, the continuous-time model of the term structure of interest proposed
by Buraschi (2005) includes lump-sum taxes, but disregards the implications
of the government’s budget constraint. Dai and Philippon (2005) estimates a
no-arbitrage ane term structure model with fiscal variables on US data. They
find significant responses of the term structure of interest rates to the deficit-GDP
ratio. The macroeconomic restrictions they impose to identify the structural
responses are fairly different from those implied by the fiscal theory of the price
level (see Sala, 2004).
The available finance models the term structure of interest rates consider
an explicit role for only two crucial factors, output growth and monetary policy,
which is typically expressed as a diffusion process for the growth of money supply.
In this paper, we consider a general-equilibrium model with money where the ow
budget constraint of the government plays an active role. This provides a link
between monetary and fiscal policy because lump-sum taxes are adjusted as a
function of real debt. We solve the structural model, and derive the law of motion
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for the nominal and real interest rates. We also study how the term structure
responds to the fiscal parameters.
This paper is organized as follows.The first two sections introduce the
reader to the framework employed to develop the analysis, together with a brief
discussion on the fiscal and monetary policy rules adopted. Section 4 and 5,
respectively, discuss the optimization process form the representative investor’s
side and the characterization of the equilibrium. Section 6 outlines the continuous
time limit of the equilibrium relationships in discrete time presented in the
previous sections. In section 7, we consider a specialized economy with a more
realistic set of assumptions for the model. In section 8 we present the solution
for the real spot rate. This is extended in section 9 for the pricing of the entire
real term structure. The nominal and real term structure for zero coupon bonds
is derived in section 9. Since the solution does not admit a closed form, we use
numerical simulations in section 10 to generate some qualitative results on the
shape of the term structure. Section 11 reports some concluding remarks.
2The model economy
We study an economy populated by a representative agent that maximizes over
the composition of her portfolio along the lines of the traditional literature
on consumption and asset pricing.We model the economy at discrete time
intervals of length ∆t. The representative agent chooses its portfolio holdings
by maximizing the following utility function
∞
?
t=0
e−βtE0
?
u
?
Ct,Mt
Pt
??
∆t
(1)
where β is the discount factor. In equation (1), Ct indicates the level of
consumption over the interval [t,t+ ∆t], Mtis the nominal money stock providing
utility to the representative agent over the interval of length [t − ∆,t], and Ptis
the price of the consumption good. Real money balances Mt/Ptenter the utility
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function of the household. The utility function is twice continuously differentiable
and concave in both consumption and real balances: uc> 0 and um> 0, ucc< 0,
umm< 0, ucm< 0 and uccumm− (ucm)2> 0, where the subscript to u indicates
the partial derivative. We make the following functional assumption on the utility
function
u
?
Ct,Mt
Pt
?
= φlogCt+ (1 − φ)log
?Mt
Pt
?
(2)
This type of utility function is used in Stulz (1986). In equation (2), the preference
parameter φ is chosen so that the nominal and real spot rates determined under
the assumption of absence of arbitrage opportunities are also equilibrium values
(see Corollary 1 in the Appendix).
As a working hypothesis to derive the first order conditions, we consider a
model of pure endowment economy where output growth evolves as
∆Yt
Yt
=Yt+∆t− Yt
Yt
= µY,t∆t + σY,tΩY,t
√∆t.
(3)
The terms µY,t and σY,t are, respectively, the conditional expected value and
the standard deviation of output per unit of time and {ΩY,tt = 0,∆t,...} is a
standard normal process.1
3 Fiscal and monetary policy
The main point of this paper is to examine the impact of the interaction between
monetary and fiscal policy on the the term structure of interest rates. We think
of ‘interactions’ in the sense captured by the “fiscal theory of the price level” of
Leeper (1991), Sims (1994), Woodford (1996), and recently extended by Cochrane
(1998, 1999), which suggests that a tight fiscal policy is a necessary complement
to ensure price stability.
1A more realistic law of motion for output is introduced in section 7.
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We define the money supply aggregate (in nominal terms) as
Ms
t= Ht+ Ft.
(4)
In equation (4) we observe that the total money supply is determined by two
components. Htis the so called ‘high powered money’ (or monetary base). Ft
represents the amount of money needed by the government to budget its balance.
Basically, Ftis an additional financing source for the government apart from taxes
and debt2.
We assume that Htand Ftfollow the processes described by
∆Ht
Ht
=Ht+∆t− Ht
Ht
= µH,t∆t
(5)
∆Ft
Ft
=Ft+∆t− Ft
Ft
= µF,t∆t + σF,tΩF,t
√∆t
(6)
where µH,tand µF,tare, respectively, the mean of the stochastic process of the
monetary base and of the financing to public debt. In (5), the stochastic process
for Ht does not have a standard error term, implying that the monetary base
possesses only a deterministic component. The process leading Ft, instead, has
a standard deviation term σF,t, where {ΩF,tt = 0,∆t,...} are standard normal
random variables.
From (4), (5) and (6), we can write the stochastic process for the total money
supply Ms
∆Ms
Ms
t
t
=Ms
t+∆t− Ms
Ms
t
t
= µM,t∆t + σM,tΩM,t
√∆t
(7)
µM,t= µH,t+ µF,t
(8)
σM,tΩM,t= σF,tΩF,t.
(9)
At a first glance, these expressions stress that the central bank is assumed to
target money growth.
2Ft can be thought of as the demand for money balances expressed by the government.
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The subsequent building block of the model assigns a proper macroeconomic
role to the government. The innovation introduced in this paper with respect
to the existing literature consists in the key role for the government budget
constraint
∆Dt+∆t+ ∆Ft+∆t= ∆it+∆tDt− ∆Tt+∆t
(10)
where Dtindicates the stock of public debt, and ∆it+∆tis the stochastic process of
the nominal spot interest rate, whose endogenous law of motion will be computed
later. Moreover, ∆Tt+∆tis the stochastic process for taxes. We assume that the
government does not face any form of public spending. Recall that ∆Dt+∆t=
Dt+∆t− Dt, ∆Ft+∆t = Ft+∆t− Ft. Basically, the government can use taxes,
money and debt to finance its budget.
Following the fiscal theory of price level, we assume that the government sets
taxes according to the simple rule rule
∆Tt+∆t= φ1Dt∆t + φ1DtσT,tΩT,t
√∆t
(11)
According to (11), the government sets as a function of the outstanding amount
of public debt. This means that if the stock of debt issued rises, taxes must
change accordingly with a marginal elasticity equal to φ1. A bound on φ1can be
established from Sims (1994) by setting φ1at a value lower than or equal to the
discount factor β.
To close the model, we assume that the process for the nominal spot interest
rate is
∆it+∆t= µi∆t + σi,tΩi,t
√∆t,
(12)
for values of the mean and the standard deviations to be determined later. By
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plugging (12) and (11) into (10), we can recover the flow budget constraint of the
public sector
∆Dt+∆t+ ∆Ft+∆t= (µi− φ1)Dt∆t + Dtσi,tΩi,t
√∆t − φ1DtσT,tΩT,t
√∆t. (13)
In order to obtain a semi-closed form solution, we assume that the quantity
of newly-issued public debt follows a deterministic process with mean µD
∆Dt
Dt
=Dt+∆t− Dt
Dt
= µD,t∆t.
(14)
Thus, the flow budget constraint becomes
µD,t∆t + µF,t∆t + σF,tΩF,t
√∆t = (µi− φ1)Dt∆t
+ Dtσi,tΩi,t
√∆t + −φ1DtσT,tΩT,t
√∆t.
(15)
To get intuition on these relations, we focus on their deterministic part. Assume
that the government aims to maintain a constant ratio of nominal bond to
governmental money, ie, ψ = D/F.Therefore, by applying Ito’s Lemma to
the definition of ψ, we can write the relationship between the mean of the public
debt and money
µD= µF− σ2
F.
(16)
From the equality between the deterministic and the stochastic terms of Ft,
ψµD+ µF
=(µi− φ1)ψ
ψ (σi− φ1σT).
(17)
σF
=(18)
Finally, using the definition of µDinto (17), we get the semi-closed solution for
the mean of the stochastic process of the governmental money
µF=
?µi− φ1+ σ2
F
?ψ
1 + ψ
.
(19)
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Therefore, (18) and (19) represent the full equilibrium relationship in the
economy.By using (19), it is clear that the mean of the stochastic process
leading money is
µM= µH+
?µi− φ1+ σ2
F
?ψ
1 + ψ
.
(20)
4 The optimal choice problem
The representative agent’s budget constraint is
Mt+?Pz,t+ PC
tyt∆t?zt+ PC
+ Pz,tzt+∆t+ PC
ta1,t+ a2,t+
N
?
i=3
Pi,tai,t= PC
tCt∆t + Mt+∆t
t
a1,t+∆t
1 + rt∆t+
a2,t+∆t
1 + it∆t+
N
?
i=3
Pi,tai,t+∆t
(21)
The investor can choose among one real and one nominal bond (both risk free),
and N − 2 equities. Each bond is issued at time t and has maturity at time
t+∆t. The return on bond are itfor the nominal bond, and rtfor the real bond.
Pi,tis the price (inclusive of dividends) of asset i at time t. The representative
agent demands Mt, for cash, Ct for consumption and xt for equity holdings.
a1,t,a2,t,...aN,trepresent the unit of financial asset held from (t − ∆t) to t.3
The choice problem of the representative investor consists in the maximization
of the utility function (2) subject to the budget constraint (21). The first order
conditions for Ct, a1,t, a2,t, Mtand ai,tare, respectively,
uc(Ct,mt) = λtPC
t
(22)
Et
?
e−β∆tλt+∆tPC
t+∆t(1 + rt∆t)
?
= λt
= λtPC
t
(23)
Et
?
e−β∆tλt+∆t(1 + it∆t)
?
(24)
Et
?
e−β∆tλt+∆t+ um(Ct+∆t,mt+∆t)
1
PC
t+∆t
?
= λt
(25)
3This setup above described is similar to that of Baksi and Chen (1996), who use this model
to study the impact of monetary policy and inflation on financial asset.
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