scispace - formally typeset
Search or ask a question
Journal ArticleDOI

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

TL;DR: In this article, the term structure implications of the fiscal theory of price level determination were studied and the intertemporal budget constraint of the government in a general equilibrium model in continuous time was introduced.
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

Summary (3 min read)

1 Introduction

  • This considerations holds both for the finance and macroeconomics literature.
  • The authors also study how the term structure responds to the fiscal parameters.
  • The nominal and real term structure for zero coupon bonds is derived in section 9.

2 The model economy

  • The authors 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.
  • The authors 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], Mt is the nominal money stock providing utility to the representative agent over the interval of length [t−∆, t], and Pt is the price of the consumption good.
  • In equation ( 3), the preference parameter φ must be chosen so that the nominal and real spot rates determined under the assumption of absence of arbitrage opportunities are also equilibrium values .

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.
  • The authors define the money supply aggregate (in nominal terms) as M st = Ht + Ft. (5) In equation (5) they observe that the total money supply is determined by two components.
  • Existing literature consists in the key role for the government budget constraint ∆Dt+∆t + ∆Ft+∆t = ∆it+∆tDt −∆Tt+∆t (11) where Moreover, ∆Tt+∆t is the stochastic process for taxes.
  • Following the fiscal theory of price level, the authors assume that the government sets taxes according to the simple rule rule ∆Tt+∆t = φ1Dt∆t +.

4 The optimal choice problem

  • The investor can choose among one real and one nominal bond (both risk free), and N − 2 equities.
  • The return on bond are it for the nominal bond, and rt for the real bond.
  • Mt, for cash, Ct for consumption and xt for equity holdings.
  • AN,t represent 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 (3) subject to the budget constraint (22).

5 Definition of equilibrium

  • The equality between money demand and supply is stated in equation (29), while equation (30) states that each agent’s demand for equity shares must equal the supply.
  • Equations (32) and (35) state that the representative investor must be indifferent between investing an amount of money equal to PCt in a real risk-free bond and holding the same amount in cash.
  • Monetary policy and the asset market are tied together through equations (35) and (34), which establish the consistency between the money supply and asset markets.
  • The equality (36) rules out bubbles in the price level of any risky asset.

6 The equilibrium in the continuous time limit

  • In this section the authors characterize the equilibrium for the continuous time limit.
  • These results are independent from the assumptions made on the role of fiscal and monetary policies in the determination of the equilibrium.
  • For this reason, the results from this section are similar to Baksi and Chen (1996) Balduzzi (1998).
  • In the continuous time limit equilibrium, the commodity price level is given at time t by 1 PCt =.

7 A specialized economy

  • In what follows the authors lay out a specific model used to derive the stochastic processes for the price level and the other variables.
  • Each type of money supply has two components, a drift term and a stochastic part.
  • The set of assumptions presented earlier allows us to compute the equilibrium price level of the commodity and the inflation process.
  • After solving for the integral, the authors get equation (62).

8 The real spot interest rate

  • In this section the authors derive the dynamics of the real spot interest rate implied by exogenous the dynamics of the technology process.
  • Since this requires solving the differential equation ( 55), xt is Markov and satisfies the necessary technical conditions to apply the Representation Theorem of Feyman-Kac.4.
  • The authors can now follow the partial differential equation — PDE — approach to compute the real spot rate.
  • Given that the Kolmogorov PDE is verified for all t and xt, the authors can divide it into two parts.
  • One part is dependent and the other one is independent from xt.

9 The term structure of real interest rates

  • Here the authors show how to compute the price of zero coupon bonds as a function of time, technology and the real spot rate.
  • The authors follow again a PDE approach because both xt and rt are Markov and satisfy the necessary technical conditions to apply the Representation Theorem of Feyman-Kac.
  • The authors can then find the solution only by using numerical methods.
  • The term structure of real interest rates can be derived from the relation between the price of zero coupon bond and the continuous real interest rate B (t, T ) = e−θR(t,T ). (92) ¥.

10 The term structure of nominal interest rates

  • The analytical expression for the nominal zero coupon bond, and for the nominal term structure of interest rates.the authors.
  • From the expressions for (93), (94) and (95), the authors observe that the values of the rates for the nominal term structure are higher than those for the real variables if (61) is higher than one.

11 Simulation results

  • The authors now calibrate the model and run numerical simulations to get a better understanding of the relations involved.
  • The intertemporal substitution coefficient β has been set equal to 0.998.
  • Figure 1 shows that the higher the reaction of the tax rate to real debt, the lower µ∗M , and the lower the position of the nominal curve in the plan.
  • Are aware of the limitation of this choice.
  • This is consistent with the results form the fiscal theory of the price level.

12 Concluding remarks

  • In this paper the authors study a simple intertemporal model for the determination of the nominal and real term structure where the interaction between fiscal and monetary plays a key role.
  • In particular, the authors investigate the relation between the term structure of interest rate and the fiscal theory of price level determination.
  • In so doing, the authors move beyond the standard finance models where monetary and technological factors are the sole determinants of the term structure of interest rates.
  • This is likely to shadow more light on the role of monetary policy expectations when also fiscal policy matters.

Did you find this useful? Give us your feedback

Content maybe subject to copyright    Report

A continuous-time model of the term structure of
interest rates with fiscal-monetary policy interactions
Massimiliano Marzo, Silvia Romagnoli and Paolo Zagaglia
July 2008
Abstract
We study the term structure implications of the fiscal theory of price level
determination. We introduce the intertemporal budget constraint of the gov-
ernment 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: D9, G12.
Marzo: Department of Economics, Universit`a di Bologna; massimiliano.marzo@unibo.it. Ro-
magnoli: Department of Mathematics, Universit`a di Bologna; silvia.romagnoli@unibo.it. Za-
gaglia: Research Unit, Bank of Finland, and Department of Economics, Stockholm University;
Paolo.Zagaglia@bof.fi. We are grateful to S. Alliney for many suggestions received on the first draft
of this paper. The views expressed here are those of the authors and should not be attributed to
the Bank of Finland.
1

1 Introduction
The theory of price level determination advocated by Leeper (1991), Sims (1994),
Woodford (1996) and Cochrane (1998) has brought to the attention of macroe-
conomists 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 govern-
ment. If the expected primary surplus is not sufficient to comply with the intertem-
poral 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 substan-
tial debate on the capability of fiscal and monetary policy to affect the price level,
study has considered its potential implications for asset prices. This considera-
tions 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 U.S. data. They find significant responses of the term
structure of interest rates to the deficit-GDP ratio. The macroeconomic restric-
tions 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 for the
nominal and real interest rates. We also study how the term structure responds to
the fiscal parameters.
2

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 discus-
sion 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 char-
acterization of the equilibrium. Section 6 outlines the continuous time limit of the
equilibrium relationships in discrete time presented in the previous sections. In sec-
tion 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.
2 The 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 consump-
tion 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
X
t=0
e
βt
E
0
½
u
µ
C
t
,
M
t
P
t
¶¾
t (1)
where β is the discount factor. In equation (1), C
t
indicates the level of consumption
over the interval [t, t + t], M
t
is the nominal money stock providing utility to the
representative agent over the interval of length [t , t], and P
t
is the price of the
consumption good. Real money balances M
t
/P
t
enter the utility function of the
household. The utility function is twice continuously differentiable and concave in
3

both consumption and real balances
u
c
> 0, u
m
> 0, u
cc
< 0, u
mm
< 0, u
cm
< 0, u
cc
u
mm
(u
cm
)
2
> 0 (2)
where the subscript to u indicates the partial derivative. In what follows, we make
the following functional assumption on the utility function
u
µ
C
t
,
M
t
P
t
= φ log C
t
+ (1 φ) log
µ
M
t
P
t
(3)
This type of utility function is used in Stulz (1986). In equation ( 3), the preference
parameter φ must be 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
Y
t
Y
t
=
Y
t+∆t
Y
t
Y
t
= µ
Y,t
t + σ
Y,t
Y,t
t. (4)
The terms µ
Y,t
and σ
Y,t
are, respectively, the conditional expected value and the
standard deviation of output per unit of time and {
Y,t
t = 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). This approach states that a tight fiscal policy is a strictly necessary
complement to ensure price stability.
1
A more realistic law of motion for output is introduced in section 7.
4

We define the money supply aggregate (in nominal terms) as
M
s
t
= H
t
+ F
t
. (5)
In equation (5) we observe that the total money supply is determined by two com-
ponents. H
t
is the so called ‘high p owered money’ (or monetary base). F
t
represents
the amount of money needed by the government to budget its balance. Basically,
F
t
is an additional financing source for the government apart from taxes and debt
2
.
We assume that H
t
and F
t
follow the processes described by
H
t
H
t
=
H
t+∆t
H
t
H
t
= µ
H,t
t (6)
F
t
F
t
=
F
t+∆t
F
t
F
t
= µ
F,t
t + σ
F,t
F,t
t (7)
where µ
H,t
and µ
F,t
are, respectively, the mean of the stochastic process of the mon-
etary base and of the financing to public debt. In (6), the stochastic process for
H
t
does not have a standard error term, implying that the monetary base possesses
only a deterministic component. The process leading F
t
, instead, has a standard de-
viation term σ
F,t
, where {
F,t
t = 0, t, . . .} are standard normal random variables.
From (5), (6) and (7), we can write the stochastic process for the total money
supply M
s
M
s
t
M
s
t
=
M
s
t+∆t
M
s
t
M
s
t
= µ
M,t
t + σ
M,t
M,t
t (8)
µ
M,t
= µ
H,t
+ µ
F,t
(9)
σ
M,t
M,t
= σ
F,t
F,t
. (10)
At a first glance, these expressions stress that the central bank is assumed to target
money growth.
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
2
F
t
can be thought of as the demand for money balances expressed by the government.
5

Citations
More filters
Journal ArticleDOI
TL;DR: In this article, the authors investigate the impact of bank competition on the use of collateral in loan contracts and show that the presence of collateral is more likely when bank competition is low.
Abstract: We investigate the impact of bank competition on the use of collateral in loan contracts. We analyze asymmetric information about the borrowers’ type in a Salop model in which banks choose between screening the borrower and asking for collateral. We show that the presence of collateral is more likely when bank competition is low. We then test this prediction empirically on a sample of bank loans from 70 countries. We perform logit regressions of the presence of collateral on bank competition, measured by the Lerner index. Our empirical tests corroborate the theoretical predictions that bank competition reduces the presence of collateral. These findings survive several robustness checks.

80 citations

Journal ArticleDOI
TL;DR: In this article, the authors estimate a new-Keynesian DSGE model with the cost channel to assess its ability to replicate the price puzzle, i.e., the inflationary impact of a monetary policy shock typically arising in VAR analysis.
Abstract: We estimate a new-Keynesian DSGE model with the cost channel to assess its ability to replicate the price puzzle ie the inflationary impact of a monetary policy shock typically arising in VAR analysis. In order to correctly identify the monetary policy shock, we distinguish between a standard policy rate shifter and a shock to trend inflation ie the time-varying inflation target set by the Fed. While offering some statistical support to the cost channel, our estimated model clearly implies a negative inflation reaction to a tightening of monetary policy. We offer a discussion of the possible sources of mismatch between the VAR evidence and our own.

47 citations

Journal ArticleDOI
TL;DR: In 2007, the UK experienced its first bank run of any significance since the reign of Queen Victoria as discussed by the authors, and unlike most runs in banking history, it was a run only on a bank called Northern Rock.
Abstract: In autumn of 2007 Britain experienced its first bank run of any significance since the reign of Queen Victoria. The run was on a bank called Northern Rock. This was extraordinary, for Britain had been free of such episodes because by early in the third quarter of the 19th century the Bank of England had developed techniques to prevent them. A second extraordinary aspect of the affair was that it was the decision to provide support for the troubled institution that triggered the run. And thirdly, unlike most runs in banking history, it was a run only on that one institution. This paper considers why the traditional techniques for the maintenance of banking stability failed - if they did fail - and then considers how these techniques may need to be changed or supplemented to prevent such problems in the future. The paper starts with a narrative of the events, then turns to banking policy before the event and to the policy responses after it. We suggest both why the decision to provide support triggered the run and why the run was confined to a single institution. That prepares the way for our consideration of what should be done to help prevent the recurrence of such episodes in the future.

32 citations

Journal ArticleDOI
TL;DR: In this article, the authors investigate the hypothesis that institutional form is related to performance of central banks and conclude that institutional forms play an important role in central bank performance, but little empirical work has been done to investigate this hypothesis.
Abstract: Over the last decade, the legal and institutional frameworks governing central banks and financial market regulatory authorities throughout the world have undergone significant changes. This has created a new interest in better understanding the roles played by organisational structures, accountability, and transparency in increasing the efficiency and effectiveness of central banks in achieving their objectives and ultimately yielding better economic outcomes. Although much has been written pointing out the potential role that an institutional form can play in central bank performance, little empirical work has been done to investigate the hypothesis that institutional form is related to performance. This paper attempts to help fill this void.

27 citations

Journal ArticleDOI
TL;DR: In this paper, the authors provide an answer to the basic question of why in the United Kingdom the traditional techniques for the maintenance of banking stability appeared to fail in the Northern Rock episode, and also consider how the techniques may need to be changed or supplemented to prevent such problems in the future.
Abstract: This paper provides an answer to the basic question of why in the United Kingdom the traditional techniques for the maintenance of banking stability appeared to fail in the Northern Rock episode. It also considers how the techniques may need to be changed or supplemented to prevent such problems in the future. We propose the following actions to make the banking system robust. First, there should be arrangements for prompt and orderly closure of a bank as it approaches problems, before it would otherwise be forced to close by either insolvency or illiquidity. Second, there should be reform of deposit insurance, such that whatever sum is guaranteed is completely guaranteed, and can be accessed without any significant delay. Third, arrangements need to be made such that customers retain access to all core banking services either through speedy transfer of all accounts or the continued operation in some guise of the troubled bank.

24 citations

References
More filters
Posted Content
Marko Melolinna1
TL;DR: In this article, a methodology for identifying oil supply shocks in a restricted VAR system for a small open economy is introduced, which is applied to Finland and Sweden in illustrative examples in a simple 5-variable model.
Abstract: This paper introduces a methodology for identifying oil supply shocks in a restricted VAR system for a small open economy. Financial market information is used to construct an identification scheme that forces the response of the restricted VAR model to an oil shock to be the same as that implied by futures markets. Impulse responses are then calculated by using a bootstrapping procedure for partial identification. The methodology is applied to Finland and Sweden in illustrative examples in a simple 5-variable model. While oil supply shocks have an inflationary effect on domestic inflation in these countries during the past decade or so, the effect on domestic GDP is more ambiguous. Keywords: oil futures, partial identification, macroeconomic shocks JEL classification numbers: C01, E32, E44

31 citations

Journal ArticleDOI
TL;DR: The authors reviewed the first thirty years of finance research and education in Finland, starting with publication of the first dissertation in finance in 1977 and showed how Finnish financial education and research developed from a humble beginning to a level that brought international acclaim, largely attributed to a number of talented and hard-working individuals but also to the decision for collaboration among the Finnish universities, as a means to overcome some of the problems of a small country.
Abstract: This paper reviews the first thirty years of finance research and education in Finland, starting with publication of the first dissertation in finance in 1977 That was also the year when the first department of finance was established in Finland - among the first in the Nordic countries This review shows how Finnish financial education and research developed from a humble beginning to a level that brought international acclaim This can be largely attributed to a number of talented and hard-working individuals but also to the decision for collaboration among the Finnish universities, as a means to overcome some of the problems of a small country

30 citations

Journal ArticleDOI
Kari Kemppainen1
TL;DR: In this paper, the authors discuss the economic consequences associated with the formation of the Single Euro Payments Area (SEPA) and reveal that the introduction of SEPA may not lead to a fully competitive and integrated retail payment markets.
Abstract: Using a spatial competition model of retail payment networks, this paper discusses the likely economic consequences associated with the formation of the Single Euro Payments Area (SEPA). The model considers an expansion of positive network externalities on the demand side and adjustment cost on the supply side and reveals that the introduction of SEPA may not lead to a fully competitive and integrated retail payment markets. This is especially the case when the markets are segments before the introduction of SEPA. In such a scenario, the post-integrated markets are likely to remain segmented or will be characterised by a kinked equilibrium where no significant price competition takes place. In both outcomes, SEPA leads to increased prices, larger network sizes (ie increased number of customers) and a higher consumer surplus. Additionally, if the SEPA-induced adjustment costs for payment networks are not prohibitively high, SEPA may also lead to an increase in both profits and social welfare.

29 citations

Posted Content
TL;DR: In this paper, the authors deal with optimal payment systems and present a short review of calculation methods and empirical results for a sample of countries, including Finland, which is to an extent one of the front-runners in payment technology and institutional design in payment systems.
Abstract: This paper deals with optimal payment systems. The issue boils down to how large are the costs of different payment media, which can be interpreted as a question of the efficiency of the means of payment. However, there are other qualifications related to the choice of payment media. Here, at least three issues can be distinguished. First is the question of optimal payment medium for each individual payment (size, location, EFTPOS etc.). This choice is not independent of the individual characteristics of the payer and payee. Secondly, there is the question of cost effectiveness of payments for different institutions and sectors. The final issue concerns the social optimum for each payment medium. These issues have been particularly controversial in the case of cash, which is still the dominant payment medium in most euro countries. Part of the controversy arises from the fact that the costs and benefits of different payment media affect different market participants in quite different ways, so that a possible social optimum might not correspond eg to the optima for different firms. The paper contains a short review of calculation methods and empirical results for a sample of countries. It also provides new evidence from Finland, which is to an extent one of the front-runners in payment technology and institutional design in payment systems. This shows up in relatively low overall costs of payments. Our estimate of total costs of payment media is 0.3 per cent of GDP, which is very low by international standards.

28 citations

Posted Content
TL;DR: In this article, the use of multiple payment media is directly related to consumer awareness and not controlling for the endogeneity of awareness can bias its effect downwards, which may explain the rise of debit card use around the world.
Abstract: In the market for payment media, some consumers use only one medium when paying for their point-of-sale transactions, while others use many. This pattern reflects the diffusion of new payment media, because a payment method innovation is typically first used simultaneously with the established methods. We study the use of multiple payment media by employing data on young Finnish consumers. We find that the use of multiple payment media is directly related to consumer awareness and that not controlling for the endogeneity of awareness can bias its effect downwards. These results suggest that increasing consumer awareness may have been underlying the rise of debit card use around the world. It could also speed up the adoption of new means of payment, such electronic money and mobile payments. To the extent that antitrust concerns in the market for payment media stem from the lack of information, improving consumer awareness could be a remedy. Keywords: payment media, consumer awareness, adoption of financial technology JEL classification numbers: G200, E590

28 citations

Frequently Asked Questions (2)
Q1. What are the contributions mentioned in the paper "A continuous-time model of the term structure of interest rates with fiscal-monetary policy interactions" ?

The authors study the term structure implications of the fiscal theory of price level determination. The authors introduce the intertemporal budget constraint of the government in a general equilibrium model in continuous time. The authors show how to solve for the prices of real and nominal zero coupon bonds. 

A number of interesting avenues of future work can be considered. The model presented in this paper should be taken to the data to study how inflation risk premia are affected by fiscal determinants.