# The basic stochastic macroeconomic model and the short-term interest rate dynamics

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## Abstract

In this chapter, we will develop a stochastic analogue to the dynamic macroeconomic model already discussed in the last chapter. This is accomplished by modeling the economy’s technology not as a fixed parameter, but as a certain continuous-time stochastic process. As a consequence, holding capital involves the bearing of risk. The intertemporal consumption/asset allocation problem of the private households becomes risky. This risky decision-making reflects the reality of how financial markets form their behavioral rules much better than the deterministic model in the previous chapter. Hence, we can expect our model to ‘produce’ prices for assets traded on the financial market that will generally not be constant but change dynamically as it happens in reality. Specifically, interest rates will no longer be equal to the net return on capital but rather react on changes in the economy, especially on fiscal policy changes. The link between fiscal policy and the term structure of interest rates is a new aspect of this model. In equilibrium, the model in this chapter will produce nonlinear, stochastic dynamics describing the evolution of output, capital, private wealth and public debt. For this reason, this chapter lays the basis for the analysis of all interesting economic aspects to be discussed in subsequent chapters: the relationship between fiscal policy and the term structure of interest rates, the question how economic growth is affected by the financial market, and the dynamics of public debt in the light of stochastically varying interest rates.

## Keywords

Interest Rate Stochastic Differential Equation Fiscal Policy Public Debt Transversality Condition## Preview

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## Notes

- 1.Turnovsky (1995, Chapter 14) points out that this specification of a production function is the stochastic analog to the deterministic linear A-K production technology as was already used in Eaton (1981).Google Scholar
- 2.They can also be interpreted as the instantaneously expected marginal product of capital and instantaneous variance of the marginal product of capital.Google Scholar
- 3.Note that we have omitted time subscripts for the Brownian motion B, output Y and capital stock K and that we will continue to do so for all the other variables in the further course of the model, mainly for notational convenience. However, it should be clear from the formulation that these variables are time-dependent.Google Scholar
- 4.For a more detailed discussion of Brownian motion, see General Appendix 1.Google Scholar
- 5.A diffusion process is a Markov process with almost surely (i.e. probability one) continuous sample paths. For a brief discussion of diffusion processes, see General Appendix 2.Google Scholar
- 6.For a brief discussion of the notions ‘drift term’ and ‘diffusion term’, see General Appendix 2.Google Scholar
- 7.If we had not assumed that the price process (and thus the rate of return process, too) of government bonds has a vanishing diffusion term (i.e. that government bonds are not locally riskless’) then we would have ended up with infinitely many equilibrium price processes. The reason for this outcome is the fact that then the financial market would have been incomplete which usually prevents equilibrium price processes to be unique.Google Scholar
- 8.Note that the symbol „<>“ denotes the so-called ‘quadratic variation’ of a diffusion process. For a detailed explanation of it as well as an introduction to stochastic integration and Ito’s Lemma, see Durrett (1996). For a short discussion, see General Appendix 2.Google Scholar
- 9.For a lucid exposition of these considerations, see Merton (1971).Google Scholar
- 10.Detailed descriptions of this method as well as economic applications can be found, among others, in Malliaris/Brock (1982), Dixit/Pindyck (1994) and Turnovsky (1995). More rigorous mathematical treatments on dynamic control theory for Markovian processes can be found in Reming/Rishel (1975) or Fleming/Soner (1993). A brief introduction of the idea can also be found in General Appendix 3.Google Scholar
- 11.For a detailed derivation see Appendix 3.1.Google Scholar
- 12.As we will see later on in Chapter 4, this condition finally yields, together with equation (3.2c) and the diffusion process describing the stochastic evolution of the equilibrium short-term real interest rate, a valuation equation for the pricing of private bonds in the form of a parabolic partial differential equation of second order. The solution of this equation will provide the base in Chapter 4 for the examination how fiscal policy influences the term structure of interest rates.Google Scholar
- 13.The risk premium term would become positive only if γ < 0 held, i.e. when the representative household is risk loving. The household would then require higher returns on safe bonds than on risky capital. Besides, the risk premium can get zero when γ equals zero. In this case, the household is said to be risk neutral and appreciates capital as much as safe bonds. This case leads to the same rates of return as in Chapter 2. Both cases are excluded for we assumed γ>1.Google Scholar
- 14.Ingersoll (1987, p. 257-258) calls this consumption behavior ‘myopic’.Google Scholar
- 15.The question whether the sign of private wealth is always positive has to be postponed to Proposition 3.4 in Section 6. The reason is that in order to answer this question we first need to determine the short-term interest rate dynamics. We will thus assume here that private wealth remains always positive.Google Scholar
- 16.Since (3.18) shows that r
_{D}is a smooth, twice differentiable function of the two diffusion processes for K and W, calculating the SDE for r_{D}using Ito’s Lemma is the proper procedure to derive the dynamics.Google Scholar - 17.The sign of γ
_{0}can be derived using the tax policy constraint (3.23), other signs are obvious. We have omitted the sign when it was ambiguous.Google Scholar - 18.This feature could probably help to better understand observed interest rate behavior which remains unexplained when using the usual linear ad-hoc specifications of the short-term interest rate. These shortcomings recently prompted some researchers (for example Ait-Sahalia (1995)) to call for nonlinear specifications of stochastic interest rate dynamics. Remaining in the tradition of mathematical finance, such nonlinear extensions had to be introduced arbitrarily without getting conscious of their economic meaning. Our model, however, makes clear where such nonlinearities may come from and is thus an important alternative to the usual ad-hoc specification.Google Scholar
- 19.He proposes perturbation type methods to obtain approximate formulas. The crucial thing with this approach is that one cannot simply assess whether a special approximation, that still has to be simple enough for comparative statics or dynamics purposes, performs well in terms of being a good approximation to the original problem.Google Scholar
- 20.In so doing, one has to discretize the SDE according to some suitable discretization scheme so that a stochastic difference equation results. Unfortunately, this is a very special method since it requires the numerical specification of all parameter values and hence looses generality.Google Scholar
- 21.Such a case is the stochastic analog to the notion of a fixed point in deterministic dynamics.Google Scholar
- 22.Note that this could also be shown more rigorously according to the lines of Proposition 3.2 and 3.3.Google Scholar
- 23.See the General Appendix 2 for a general definition and discussion of the meaning of the scale function. An additional remark may be due here: if the interest rate process is in Ii, then the first two terms inside the integral of the scale function become negative for any x ∈ I
_{1}. Since this could yield complex-valued outcomes instead of real-valued ones, we will, for the use in the later Propositions 3.3 and 3.4, normalize (3.26) appropriately so that the scale function remains real-valued. Such a normalization, which is equivalent to a multiplication with a scalar, does not falsify the conclusions drawn later on since the only thing that really matters is whether the scale function is unbounded (∞) or bounded (< ∞) for specific boundaries.Google Scholar - 24.Note that, if (3.27) holds as an equality, then the short-term interest rate starts at the right boundary. Owing to (3.28), however, it leaves it immediately and never reaches it any more.Google Scholar
- 25.We have chosen these parameter values in order to guarantee that the state space of the short-term interest rate becomes reasonable given empirical observations. Moreover, these parameter values do not violate the fiscal policy restrictions (3.23) and (3.28).Google Scholar
- 26.For both trajectories an identical series of stochastic shocks is used. The necessary calculation is done by firstly discretizing the original SDE (3.24a) according to the so-called ‘Euler-scheme’ (see Kloeden/Platen (1992), Kloeden/Platen/Schurz (1994)) which yields a stochastic difference equation. We then simulate this equation week by week using the same parameter values as above together with a sample of standard normally distributed numbers
^{26}(one for each week with 254 weeks at total) and an initial value for r_{D}= 4 %. This delivers the ‘benchmark’ short-term interest rate trajectory. Then, we calculate a second trajectory with a temporary tax rate decrease which means that in the first time period the tax rate is reduced to 6 % and from then on again set to 7 %. Finally, we calculate a third trajectory with a permanent tax rate decrease happening in the first time period where the tax rate is permanently reduced to 6 %.Google Scholar - 27.In order to simplify notation, we omit the arguments of the indirect utility function V and denote its partial derivatives with subscripts. Likewise, we write in the subscripts t for t
_{0}and W for W_{0}.Google Scholar - 28.Note that we are omitting the argument of Z and denoting derivatives of Z with respect to W using primes.Google Scholar
- 29.The fraction itself would tend to minus infinity but since it has to be multiplied with-2, it finally tends to plus infinity.Google Scholar
- 30.‘Y’ here denotes an arbitrary diffusion process and has nothing to do with output Y! Additionally, we distinguish the process Y from values y out of the state space of Y in order not to confuse the process with its values.Google Scholar
- 31.p(y, tlyo, 0) should be read and understood as follows: the probability that the process Y attains a value of y at time t when it started with a value of y
_{0}at initial time 0.Google Scholar - 32.Note that the signs of Ψ and ξ are determined by the definitions (3.24b) and the government expenditure ratio constraint (3.28).Google Scholar