نرخ های بهره اسمی و واقعی تعادل عمومی

We derive the general equilibrium short-term real and nominal interest rates in a monetary economy affected by technological and monetary shocks and where the price level dynamics is endogenous. Assuming fairly general processes for technology and money supply, we show that an inherent feature of our equilibrium is that any real variable dynamics, in particular that of the short-term real interest rate, is driven by both monetary and real factors. This money non-neutrality is generic, as it does not stem from any friction such as price stickiness, or from a particular utility function. Non-neutrality obtains because the ex ante cost of real money holdings is random due to inflation uncertainty. We then analyze in depth a specialized version of this economy in which the state variables follow square root processes, and the representative investor has a log separable utility function. The short-term nominal rate dynamics we obtain encompasses most of the dynamics present in the literature, from Vasicek and CIR to recent quadratic and, more generally, non-linear interest rate models. Moreover, our results pave the way to several new nominal term structures.

We propose a general equilibrium of a frictionless monetary economy in which money is an argument of the representative individual's utility function. In a fairly general framework set in continuous time, we first derive and analyze the behavior of macroeconomic aggregates such as consumption, investment and real wealth and devote special attention to the inflation rate and the real and nominal interest rates. In a specialized version of the economy, where the representative agent has a log separable utility function and the state variables follow square root processes, we then provide explicit solutions to our model and derive in particular the implied dynamics for the real and short-term nominal rates. The main characteristic of our economy is that generically money is neither neutral nor superneutral, as monetary policy always affects the level and the dynamics of all real variables.1 The transmission mechanism works as follows. An individual holding real balances faces an opportunity cost that ex ante is the nominal interest rate. However, the effective cost of money holding is not the nominal rate but the sum of the real rate and the inflation rate realized ex post. Under uncertain inflation, the two costs are distinct since, at the beginning of each period, the first one is known while the second is random. Investors' real wealth, and thereby all other endogenous real variables, are affected by this uncertainty. To further investigate the consequences of money non-neutrality, we provide a closed form solution for a specialized economy that can be viewed as the monetary extension of the real economy developed by Cox et al., 1985a and Cox et al., 1985b, hereafter CIR. Our monetary economy turns out to possess original properties as compared to pure real economies or monetary economies in which the real and the nominal sectors are linked artificially or not at all. The abundant and ever growing literature on term structure modeling and interest rate derivatives pricing witnesses the sizeable progress that has been accomplished in recent years both at the theoretical and the empirical levels. The adoption of new parametric and non-parametric techniques to estimate the term structure enhanced our understanding of the behavior of bond market prices and of the shortcomings of standard models. By comparison, relatively little effort has been devoted to providing these new models a sound economic background. The most widely used approach simply consists in assuming on a priori grounds a given dynamics for the short-term nominal rate and then deriving the dynamics of bond prices and/or the price of derivatives. Although the CIR model of the term structure was set in a general equilibrium framework, this was the case for almost none of the extensions proposed thereafter. In addition, the CIR model has been derived for a real, non-monetary, economy and nonetheless was used as if it were applicable to a nominal term structure. Others have followed this tradition. For example, Longstaff and Schwartz (1992) and Ahn et al. (2002) derive their nominal term structure dynamics within a CIR-like purely real economy. More generally, Jin and Glasserman (2001) show that every Heath et al. (1992) arbitrage free model of the term structure can be supported by a real-economy equilibrium a la CIR. This can be true only if interest rates are interpreted as real ones, or, equivalently, if the inflation rate is deterministic. This paper, however, is not the first one to attempt to build a truly monetary economy, and to derive the equilibrium real and nominal term structures. The standard approach, followed by CIR themselves and others, merely consists in adding to CIR's framework an exogenous process for the price level or the inflation rate and then deriving the relevant variables, assuming along the way that money has no real effect. Other authors such as Pennacchi (1991) added artificially some non-neutrality, for instance by assuming on a priori grounds that the drift of the technological process depends on inflation in an otherwise CIR economy, the dynamics of the inflation rate being exogenous. Important progress has however been accomplished by Bakshi and Chen (1996) for a domestic economy and Basak and Gallemeyer (1999) for an international economy. These authors built monetary economies in which the price level is found endogenously within a money-in-utility framework with a representative agent. However, both papers proposed a partial equilibrium framework in which output and consumption are in fact exogenous. Not surprisingly then, these models exhibit money superneutrality in equilibrium. Therefore, the present model of a truly monetary economy that leads to money non-neutrality without “ad hoc” assumptions fills an obvious gap. In particular, our dynamics for the real and nominal interest rates are dramatically different from each other. 2 The first main contribution of this paper thus is to introduce a consistent framework of a monetary economy in which money is held because it provides utility and cannot in general be neutral in equilibrium, regardless of the shape of the utility function. Moreover, both expected and non-anticipated changes in the money supply rate of growth affect the level and the growth rate of all relevant variables. Such non-neutrality is achieved without introducing the imperfections (such as price stickiness and/or wage rigidities) characteristic of today standard models. The key is the correct modeling of the representative investor's wealth dynamics. This has important bearings on monetary theory and policy, term structure modeling and asset pricing. The second main contribution concerns the behavior of the real interest rate in equilibrium. The short-term (in fact, continuous) real rate is equal to the expected return on real investment adjusted by a risk premium. The latter has two components, one related to consumption risk and the other to real balances risk. Since consumption and real money holdings are affected by monetary factors, so is the risk premium. This is the first transmission mechanism of monetary impulses to the real rate. Consumption risk has three components, technology risk, monetary risk, and the risk associated with changes in the proportion of total wealth devoted to real investment. Since the latter risk is itself related to monetary risk, this compounds the impact of monetary policy on the real interest rate and makes its relationship to monetary factors highly non-linear. In the particular case of log separable utility functions, we solve completely for both the real rate level and its dynamics. They are shown to be very different from both what CIR obtained in their purely real economy and what is obtained in frictionless monetary economies in which money is in fact neutral (for a representative example, see Bakshi and Chen, 1996). To the best of our knowledge, such results for the short-term real rate are novel. Another contribution concerns the behavior of the equilibrium nominal rate of interest. We recover the well-known result that this rate is equal to the marginal rate of substitution between real money balances and consumption. Although it is in general affected by both technological and monetary parameters, it is solely influenced by monetary factors in the log utility case. Its dynamics does nevertheless encompass most interest rate models offered in the literature, which therefore obtain as special cases of ours. First and foremost, we can recover CIR's square root interest rate model with the crucial provision that the latter was derived for the real rate, not the nominal rate as here. This provides a sound theoretical background to the numerous papers that used CIR's model as if it was obtained in a monetary economy and vindicates its adoption as the nominal interest rate model of a truly monetary economy in general equilibrium. Second, not only well known affine models of the term structure but also more complicated ones such as the non-linear models of Ahn and Gao (1999) or the log normal model of Miltersen et al. (1997) can be derived as particular cases. Third, our model also embeds the quadratic term structure model recently developed by Leippold and Wu (2002) and Ahn et al. (2002). Finally, our model has obvious implications as to the potential factors explaining time series and cross-sectional features of nominal bond prices. In general, the factors found in the literature are related to the properties of the term structure itself such as the general level, steepness and convexity of the curve, or the volatility of the interest rates. This is similar to explaining the cross-section of asset returns by the return on the market portfolio, its volatility, skewness and kurtosis, and/or by the returns of particular “ad hoc” portfolios deemed to reflect common exposures to (generally) non-specified risks. Thus, bond returns are not related to fundamental economic risks. Our paper identifies exactly the factors affecting the short-term real and nominal rates. While three factors are needed to explain the level and dynamics of the real rate, namely the technology, the money supply and the investment/wealth ratio, one factor only, the money stock, plays a role for the determination of the nominal rate. Our identification thus is parsimonious and provides theoretical support to the many papers that showed that one factor in general explains about 90% of nominal bond price fluctuations (see for instance Chapman and Pearson, 2001). Moreover, estimating the level and dynamics of the short-term nominal rate does not require the use of consumption data. The remainder of the article is organized as follows. Section 2 presents the monetary economy under investigation. Section 3 derives the equilibrium in the general economy and characterizes all the relevant variables, either real or nominal. Section 4 derives explicit solutions for all endogenous processes and variables in a specialized version of our economy in which the state variables follow square root processes, and the representative investor has a log separable utility function. Section 5 concludes. A mathematical appendix, which is not provided to save space but is available upon request to the authors, gathers all proofs and technical derivations.

We have derived the general equilibrium dynamics of the main real and nominal aggregate variables in a monetary economy affected by technological and monetary shocks. The level and dynamics of any real variable, in particular the short-term real rate of interest, is inherently driven by both monetary and real factors. Money non-neutrality thus is generic, as it does not stem from any friction such as price or wage stickiness, asymmetric information and restricted participation or from a particular utility function. Non-neutrality obtains because the ex ante cost of real money holdings is random due to inflation uncertainty. In a specialized version of this economy in which the state variables follow particular processes, and the representative investor has a log separable utility function, we have explicitly derived the level and dynamics of the short-term real and nominal interest rates. These two kinds of rates in fact behave in very different manners, as the inflation risk premium is diversely affected by real and nominal shocks. The processes obtained for the nominal interest rate encompass most of the dynamics offered in the literature, from the standard affine models to the recent quadratic and non-linear models, and lead to new, more general, nominal term structures. The proposed setting is sufficiently flexible so that many interesting issues of interest for both academics and practitioners can be addressed. At the microeconomic level, these range from fixed income instrument pricing, option pricing and hedging, asset–liability management, value-at-risk assessment and other interest risk measurements to the valuation of floaters, interest rate and currency swaps, forwards and futures, and swaptions, to name a few. At the macroeconomic level, they range from the conduct of monetary policy and its impact on real income, investment, consumption and wealth and on the inflation, interest and exchange rates to the relationship between asset returns and inflation, the equity premium puzzle, the current hotly debated stock return predictability issue, and, last but foremost, the term structure estimating issue. This work could be extended in a number of ways. First, instead of adopting the money-in-the-utility approach, one could consider a cash-in-advance economy with a credit good in addition to the cash good. Second, an explicit reaction function on the part of the monetary authorities could be modeled rather than assuming an exogenous process for the money supply. This could allow for the comparative study of the influence of various policy rules on the aggregate variables deemed relevant. Third, a government sector with autonomous expenditures and nominal taxes could be introduced, for instance to assess the impact of fiscal policies on the real and nominal yield curves.