Bifurcation analysis of Zellner's Marshallian Macroeconomic Model
Introduction
With the formulation of the Marshallian Macroeconomic Model (MMM) by Veloce and Zellner (1985) for a single sector, and later extension to a multi-sector model by Zellner and Israilevich (2005), Zellner and his co-authors attempted to incorporate sectoral dynamics and its effects on aggregates and vice versa. One of the novel features of this model was explicitly to formulate the dynamics of firm entry and exit within industries. With this in mind the basic MMM is described by sectoral demand, supply, and entry/exit equations. In the later version of the model in Zellner and Israilevich (2005), factor markets, the government, and a monetary sector were added to complete the model.
The entry/exit behavior modeled in the MMM can be described by the equation , i.e. the growth rate of firms in the industry is proportional to the difference in current industry profitability, , and the long-run future profitability in the industry, Fe. The speed of adjustment is determined by the parameter, . Zellner and Israilevich (2005) describe the emergence of rich dynamics in key variables, such as price and output at the sectoral, as well as at the aggregate, level once an entry/exit equation for each industry is introduced into the model. In the simulation exercises conducted by Zellner and Israilevich (2005), and Fe were fixed parameters. Varying these parameters would change the equilibria and could possibly cause changes in the nature of the equilibria, such as the number of solutions and the stability properties of the equilibria. In this paper, we undertake this task of examining the model's characteristics with respect to the entry/exit parameter Fe by searching for a bifurcation within the theoretically feasible parameter space.
Examining the existence of bifurcations in dynamic economic models has important consequences from a theoretical as well as an empirical perspective. Grandmont (1985) showed that it was possible for even the most classical dynamic, general equilibrium, macroeconomic models to demonstrate stable solutions or more complex solutions in the form of cycles or chaos. The reason behind such disparate behavior was not a difference in the structure of the model, but the fact that the parameter space of such models was stratified into subsets or bifurcation regions, each of which supported a very different kind of dynamics. As Barnett (2000) pointed out, it is possible for economists having different policy views to agree on structurally similar or identical models, but with the parameters being in different bifurcation subsets of the parameter space. This conclusion is in contrast with the earlier view that different policy views must imply different structural models.
Bifurcation analysis of parameter space stratification is a fundamental and frequently overlooked approach to exploring model dynamic properties. Basic properties of any dynamic system are stability and the nature of its disequilibrium dynamics. Just as it is important to know for what parameter values a system is stable or unstable, it is equally important to know the nature of stability (e.g. monotonic convergence, damped single periodic convergence, or damped multi-periodic convergence) or instability (periodic, multi-periodic, or chaotic). Informally we say that a system has undergone a bifurcation if a small, smooth change in a parameter value(s) produces a sudden topological change in the nature of singular points and trajectories of the system.
When the values of system's parameters are not known with certainty, bifurcation analysis can provide meaningful insights into solution dynamics of the system. If a confidence region around a parameter estimate includes a bifurcation point, then various kinds of dynamics can be consistent with the parameter being within the confidence interval. In such cases the robustness of inferences about dynamics is damaged.
Bifurcations can be local or global. Local bifurcations are examined through linearization of a non-linear system around its equilibrium, since in general non-linear systems tend to behave in the same manner as the linear system in a close neighborhood of the equilibrium. At a bifurcation point, the number of equilibrium may change. There may also be changes in the stability properties of equilibrium points and/or changes in the nature of orbits near the equilibrium. Examples of local bifurcations include saddle-node bifurcation, transcritical bifurcation, pitch-fork bifurcation, period-doubling(flip) bifurcation, and Hopf bifurcation.
Examining the existence of bifurcation has important consequences for theoretical and empirical model building in economics. Boldrin and Woodford (1990) have given an extensive survey of developments in dynamic, general equilibrium theory and conditions under which endogenous fluctuations are possible. Benhabib and Nishimura (1979) show that the optimal growth path becomes a closed orbit in a multi-sector model for some discount rate values within the theoretically feasible region. Benhabib and Day (1982) and Grandmont (1985) have also shown the possibility of chaotic behavior in general equilibrium models.
More recent work on detecting bifurcations in macroeconomic models have been undertaken by Barnett and his co-authors. Barnett and He (2002) show the existence of a transcritical and Hopf bifurcation for different policy parameters in the dynamic, continuous time macroeconometric model of Bergstrom et al. (1992). Furthermore, Barnett and He, 2004, Barnett and He, 2006, Barnett and He, 2010 find the existence of a singularity induced bifurcation within the empirical parameter space of the Leeper and Sims (1994) Euler equations model for the US economy. Barnett and Duzhak, 2008, Barnett and Duzhak, 2010 recently found the presence of period-doubling and Hopf bifurcation in New Keynesian models.
Section 2, describes the MMM and the derivations of the dynamic equations governing the path of output in each sector. In Section 3 we discuss the possibility of cyclical behavior in MMM and present our result of a Hopf bifurcation. Finally, Section 4 concludes the paper and indicates some future extensions.
Section snippets
The model
We consider a two sector, continuous time version of the Marshallian Macroeconomic Model (MMM), as outlined in Zellner and Israilevich (2005). Each sector is characterized by an aggregate demand function for its output, an aggregate supply function, and aggregate input demand functions for labor and capital. We also include the government that collects taxes on output, purchases output from the two sectors, and inputs from the factor markets. Although Zellner and Israilevich (2005) include
Stability and bifurcation analysis of equilibrium
The dynamics in this two sector MMM, in terms of convergence to the equilibrium given (2.14), can be described by generalizing the analysis of the one sector MMM by Veloce and Zellner (1985). Using the entry/exit equation assumed in Zellner and Israilevich (2005) and solving the one sector model as in Veloce and Zellner (1985) would yield the following differential equation, which is equivalent to (2.12) for the one sector model: where a depends on the structural parameters. Here the
Conclusions and extensions
In this paper we study a special, nested case of the two sector MMM and investigated the possibility of cyclical behavior, which could arise due to certain combinations of own price, cross price, and income elasticities. We also showed that a Hopf bifurcation exists within the theoretically feasible parameter space, giving rise to stable cycles. Our choice of F1 as the candidate for bifurcation parameter re-emphasizes the importance of a dynamic entry/exit equation in models of this class.
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