Saturday, December 31, 2016
The emphasis on this blog, however, is mainly critical of neoclassical and mainstream economics. I have been alternating numerical counter-examples with less mathematical posts. In any case, I have been documenting demonstrations of errors in mainstream economics. My chief inspiration here is the Cambridge-Italian economist Piero Sraffa.
In general, this blog is abstract, and I think I steer clear of commenting on practical politics of the day.
I've also started posting recipes for my own purposes. When I just follow a recipe in a cookbook, I'll only post a reminder that I like the recipe.
Comments Policy: I'm quite lax on enforcing any comments policy. I prefer those who post as anonymous (that is, without logging in) to sign their posts at least with a pseudonym. This will make conversations easier to conduct.
Tuesday, November 24, 2015
Herbert Scarf died this 15th of November. I think of Scarf as the economist who first demonstrated that general equilibria need not be stable. Something more, some special case assumption or another approach entirely, is needed.
From his Wikipedia page, I learned that have been exposed to more of Scarf's work than I knew. Long ago I took a course in Operations Research, in which we were taught queuing theory and how to find policies for optimal inventory management. Apparently, that approach to the study of inventory policies comes from Scarf.
I did not find the New York Times obituary enlightening. I wish they had mentioned that his algorithm was for finding so-called Computable General Equilibrium (CGE). I have never quite got CGE models. The ones I have seen do not have the dated commodities of the Arrow-Debreu model of intertemporal equilibrium. I have never been sure that they really belong with that tradition, or, like Leontief's model, really fit with a revival of classical economics. Perhaps they are an example of temporary equilibria, as put forth by J. R. Hicks in Value and Capital.
Quite some time ago, Rajiv Sethi discussed Duncan Foley's appreciation of Scarf as a teacher.
(Unrelated to the above, Cameron Murray recently comments on economists confusion about what is meant by "capital".)
Wednesday, November 18, 2015
"...I spent the evening walking round the streets, especially in the neighbourhood of Trafalgar Square, noticing cheering crowds, and making myself sensitive to the emotions of passers-by. During this and the following days I discovered to my amazement that average men and women were delighted at the prospect of war. I had fondly imagined what most pacifists contended, that wars were forced upon a reluctant population by despotic and Machiavellian governments. I had noticed during previous years how carefully Sir Edward Grey lied in order to prevent the public from knowing the methods by which he was committing us to support France in the event of war. I naïvely imagined that when the public discovered how he had lied to them, they would be annoyed; instead of which, they were grateful to him for having spared them the moral responsibility..."
Meanwhile, I was living at the highest emotional tension. Although I did not foresee anything like the full disaster of the war, I foresaw a great deal more than most people did. The prospect filled me with horror, but what filled me with even more horror was the fact that the anticipation of carnage was delightful to something like ninety percent of the population. I had to review my views on human nature. At that time I was wholly ignorant of psychoanalysis, but I arrived for myself at a view of human passions not unlike that of the psychoanalysts. I arrived at this view in an endeavour to understand popular feeling about the War. I had supposed until that time that it was quite common for parents to love their children, but the War persuaded me that it is a rare exception. I had supposed that most people liked money better than almost anything else, but I discovered that they liked destruction even better. I had supposed that intellectuals loved truth, but I found here again that not ten per cent of them prefer truth to popularity. Gilbert Murray, who had been a close friend of mine since 1902, was a pro-Boer when I was not. I therefore naturally expected that he would again be on the side of peace; yet he went out of his way to write about the wickedness of the Germans, and the superhuman virtue of Sir Edward Grey. I became filled with despairing tenderness towards the young men who were to be slaughtered, and with rage against all the statesmen of Europe. For several weeks I felt that if I happen to meet Asquith or Grey I should be unable to refrain from murder. Gradually, however, these personal feelings disappeared. They were swallowed up by the magnitude of the tragedy, and by the realization of the popular forces which the statesmen merely let loose.
-- Bertrand Russell (1951). The Autobiography of Bertrand Russell: The Middle Years: 1914-1944
Tuesday, November 03, 2015
I have updated my paper, "A Neoclassical Model of Pension Capitalism in Which r > g". Changes include:
- Deletion of the claim that, in general, inequality increases in a steady state when the real rate of return on finance exceeds the rate of growth.
- Deletion of states of portfolio indifference, in which the real rates of return on money and on bonds are equal, from the model.
- Addition of illustrations of the solution to the (nonlinear) model with some graphs of some state variables along dynamic equilibrium paths.
- Inclusion of a description of one method for finding such solutions numerically.
- Many minor corrections and rewording.
In general, I try to write papers so anybody, including me several months hence, can follow all the details all they want. I realize in submissions to publication, my appendices would have to be drastically shortened or deleted altogether. My typesetting of the mathematics in this paper needs modification, but it is kind to those with old eyes.
Wednesday, October 21, 2015
|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
|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.Reference
- Hahn, Frank and Robert Solow (1995). A Critical Essay on Modern Economic Theory, MIT Press
Wednesday, September 23, 2015
|Figure 1: Steady States As Function Of Effective Return On Savings|
I have previously said I am not thrilled about arguments about whether or not assumptions are realistic. In this post, I describe some analysis I have done with a model of a world that does not exist and analysis I may do in the future with some variation on such a world. The title of this post refers to this quote from Bob Solow, talking about how to respond to Robert Lucas and the new "classical" school:
"Suppose someone sits down where you are sitting right now and announces to me that he is Napoleon Bonaparte. The last thing I want to do with him is to get involved in a technical discussion of cavalry tactics at the battle of Austerlitz." -- Robert Solow2.0 Generalization of Hahn and Solow's Model of Overlapping Generations
I have previously outlined a micro-founded macroeconomic model of overlapping generations, presented in Hahn and Solow (1995). They use this model to show that claims, from new classical economists and their followers, of the desirability of perfectly flexible prices and wages are unjustified, even on their own theory. They do not think of this model as a good empirical description of any actually existing economy. Hahn and Solow present another model as a prototype of the direction in which they thought macroeconomics should have developed.
Hahn and Solow consider case where one household is born at the start of each year. Under their assumptions, a stationary state is characterized by an equality between a certain function of the effective rate of return on savings and certain model parameters:
g(Q) = [ξ/(ξ - 1)] [β/(1 - β)]
The parameter ξ relates to the Clower cash-in-advance contraint. The parameter β is for the aggregate Cobb-Douglas production function. Parameters and the form of the utility function are embodied in the function g.
I consider a slight modification to this model. Suppose the number of households born each year is no longer constant. Specifically, let the number of households born at the start of year t, ht, grow at the rate G:
ht = Gt,
G ≥ 1.
I have worked through this model somewhat. A steady state exists if only if the following equality holds for the effective rate of return on savings:
g(Q) = G [ξ/(ξ - 1)] [β/(1 - β)]
Along a steady state growth path, the nominal price of corn declines so as to maintain a constant real money supply. Hahn and Solow also have that the supply of money is a fixed quantity. They need this assumption, I guess, for their abstract discussion of policy responses to a shock to make sense.3.0 Other Generalizations
Here are some other possible generalizations and explorations one might make to the model:
- Household lives more than two years.
- Endogenous supply of labor, with leisure entering the utility function.
- Introduction of a bequest motive.
- Heterogeneous households.
- Non-homothetic preferences.
- Various specific forms of utility functions.
- Multiple sectors in production, instead of the production of a single good.
- Introduction of fixed capital (with radioactive depreciation), instead of only circulating capital.
- Various specific forms of production functions.
- Introduction of stochastic noise.
- Analysis of reactions to different kind of shocks.
- Introduction of government, foreign trade.
- More detailed analysis of money, finance, and banks.
The above outlines a research program, not necessarily original. Econometricians can go through models in this family in the literature, trying to find the best fit for some time period and country. From what little I know, one can find models with one generalization and not another, or vice versa. A theoretician might want to try to develop a model that combines some generalizations, thereby advancing the field.4.0 Empirical Applicability of Generalized Model?
This program entails lots of work, some of it empirical. How could an outsider have standing to criticize this approach?
Truthfully, the mathematics is mostly tedious algebra, only not at a high school level because of the length of the derivations. I suppose the concepts I am applying here are deeper than that. Sometimes one gets to the level of high school calculus, what with LaGrangians and all. (If I can develop a fairly comprehensive and interesting bifurcation diagram for some models, I will consider myself to be approaching advanced mathematics.) Some conventional concepts from economics (marginal conditions, excess demand functions, Walras' law, steady states) help organize the approach.
One who has learned the details of such a program might react negatively to criticism. The supposedly unrealistic assumptions you object to are maintained for analytical tractability. Past developments have supposedly shown us how to relax assumptions. One can be confident that future developments will continue to show us how to generalize the models and how to remove more scaffolding, leaving the building untouched. And, if analytical developments, such as tractable models of imperfect competition, lead to widescale changes, we will adopt them if empirical data shows such changes to be warranted.
But are there some assumptions that are untouched by such a program, that are always maintained, and that render all models (admittedly, internally consistent) developed along these lines forever empirically inapplicable?4.1 How Are Dynamic Equilibrium Paths Found?
Under the assumption of perfect competition, prices and wages are assumed to be flexible. This is assumed to imply that markets in each period instantaneously clear. I do not understand why anybody up-to-date on economic theory should believe this?4.2 No Keynesian Uncertainty
Households and firms are assumed to know what the usual range of interest rates, for example, will be in 60 years, in only probabilistically. This does not seem to be plausible to me.5.0 Conclusions
I intend to pursue some generalizations suggested above. (I could be distracted by trying to develop a bifurcation diagram by a Hahn and Solow model in a later chapter.) The point of the mathematics is to tell a story of some fantasy or science fiction world. This sort of project, to me, does not to make empirical claims. Rather I am interested in whether qualitatively similar stories can be told with some complications. Which, if any, generalizations undermine such stories?