Wednesday, October 21, 2015

Feels Like Science

Figure 1: Evolution of Two State Variables along Two Dynamic Equilibrium Paths

I continue to explore a micro-founded macroeconomic model from Frank Hahn and Robert Solow, generalized to allow a positive rate of growth of households. Hahn and Solow put forth this model as a strawman, to show that even with perfectly flexible prices and wages, markets clearing always, and rational expectations, room for government macroeconomic management can arise. In their book, they then move on to consider imperfectly competitive markets, norms for wages, and so on.

A dynamic equilibrium path, in the model, defines the values of three state variables at the end of each time period in the model. One of these state variables, the real quantity of money in circulation is easily calculated from the other two. The other two, taken here as the real rate of return on corporate bonds and on money, must be found, in general, by solving a recursive system of two equations at each point in time. I found the code I wrote for this post helpful here.

Figure 1 illustrates the evolution of two state variables for two dynamic equilibrium paths. (The model parameters are β = 2/5, ξ = 2.11, and G = 2. The household utility function is of the form specified by Example 1 in Hahn and Solow, with ε = -1/2.) The stationary, dashed-line, path is for a steady state, which is asymptotically approached by the other dynamic equilibrium path. The oscillations seen in this approach are not in the linear approximation about the steady state. One might view these oscillations as a decaying business cycle. One should be clear, however, that even though economic output varies along such a path, neither unemployment nor disappointed plans arise in this model. Households foresee all future variations in prices and quantities along a dynamic equilibrium path.

One could add various complications to make the model more realistic. Households could live for multiple periods more than two, thereby perhaps modifying the time period for the business cycle. One could add leisure into the utility function and model the supply of labor as the result of trading off the earning of wages for consumption and leisure. Employment would then vary along a business cycle; in this theory, recessions are long vacations. One could add noise terms, from known probability distributions, for various terms. So agents would be continually adjusting their plans to accommodate realizations of stochastic processes. One could add imperfect competition, as modeled by Avinash Dixit and Joseph Stiglitz. I suppose one could describe the parameters of utility functions as lying along a continuum, therefore adding a sort of diversity in the model of households. And so on.

I suppose one would find it difficult to add all of these refinements at once. So one could empirically compare a basic model with each refinement. And a model with one refinement might fit better here and with another there. Room for technical innovation for modelling then arises. Can you add two or more refinements, perhaps simplified, where others could could only add one before? Can you take a model that previously was only described for a linear approximation and analyze at least some non-linearities (as I do above)?

I gather I have just briefly outlined the direction of research in mainstream macroeconomics over the last third of a century, albeit the freshwater school did not start, I take it, with overlapping generations models and a Clower constraint.

None of these refinements would even hint at an approach to addressing the question of how economies get into equilibrium. At the end of each year, the economy is automatically in equilibrium in the model, and this instantaneous magic has been foreseen for all time. Head of households and managers of firms have no need to learn a model of the economy. Agents never have disagreements among themselves about what is the true model. And they never change their minds about the structure of the model. J. R. Hicks, the inventor of the model of temporary equilibrium, came to see that it was set in logical time, not historical time. In other words, John Hicks chose to ally himself with Joan Robinson on this theoretical point.

Without an acceptable understanding of disequilibria, mainstream economists should be tolerant of polyvocality in methodology. Why should some economists not be exploring models that are not microfounded? Why not consider the impact and evolution of social norms, without first insisting that they they be justified by methodological individualism? I consider some work in complexity and agent based modeling to be of interest along these lines and not even all that non-mainstream.

Monday, October 05, 2015

A Bifurcation Diagram for Hahn and Solow

Figure 1: Bifurcation Diagram for Hahn and Solow, Example 1, Generalized

I have been writing a draft paper, "A Neoclassical Model of Pension Capitalism in which r > g". In my latest iteration, I have developed the bifurcation diagram shown above. This is a generalization for the overlapping generations model, in which the number of households can grow, but specialized to Hahn and Solow's Example 1. Example 1 specifies the form of the utility function.

One can define dynamic equilibrium paths for the model. And given the values of certain parameters, one can locate a steady state in a certain range of parameters. Always being happy to examine a model, whether it can or cannot ever be instantiated in an actually existing economy, I have identified types of steady states and their stability in certain parameter ranges. I was able to establish analytically the boundary between steady Portfolio Indifferent and Liquidity Constrained States. I located the curved dashed and solid lines towards the south east of the diagram through a mixture of analysis and numeric experimentation. This is also true for my identification of types of stability (saddle-point, locally stable, locally unstable).

I do not fully understand the topological variation in flows for the bifurcations that I have identified. I think I understand the bifurcation, shown by the dashed line, in which a steady Liquidity Constrained State loses stability. This bifurcation most likely results from the steady state ejecting a stable or absorbing an unstable two-period business cycle. The former case is analogous to the logistic equation for a parameter a of 3. I can understand the bifurcation in which the steady state disappears in terms of the diagram in this post. But I find it difficult to understand how dynamic equilibrium paths differ across this bifurcation. And I have not previously gone into the details of the analysis of how two dynamic systems - in this case, for Portfolio Indifferent and Liquidity Constrained States are patched together across a bifurcation. But the linked paper illustrates what I have so far.

More complete details are provided in the linked paper. I provide more details than anybody can want in appendices so as to be able to step through the model myself, if I look at this stuff later.

  • Hahn, Frank and Robert Solow (1995). A Critical Essay on Modern Economic Theory, MIT Press