# 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

## 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.

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### Additional excerpts

...We follow Baksi and Chen (1996), Merton (1971) and Grossman and Grossmann and Shiller (1982) by assuming ∆pi,t pi,t = µei,t∆t + σ e i,tΩ e i,t √ ∆t (37) where µei,t and σ e i,t are, respectively, the conditional expected value and the standard deviation of real return on asset i per unit of time....

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### "A continuous-time model of the term..." refers background in this paper

...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....

[...]

...We follow Baksi and Chen (1996), Merton (1971) and Grossman and Grossmann and Shiller (1982) by assuming...

[...]

...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....

[...]

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847 citations

### "A continuous-time model of the term..." refers background in this paper

...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....

[...]

...A bound on φ1 can be established from Sims (1994) by setting φ1 at a value lower than or equal to the discount factor β....

[...]

...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....

[...]

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##### Frequently Asked Questions (2)

###### Q2. What have the authors stated for future works in "A continuous-time model of the term structure of interest rates with fiscal-monetary policy interactions" ?

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.