|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?