Math based vs rule based

The few books that I've been reading here and here triggered some thoughts about economics and economic models.

1. Economics is as susceptible to fads just as all fields are. When I was in graduate school Sergio Rebelo taught a very popular growth theory class - so popular in fact that as graduate students around 20 of us clustered around the door of the classroom waiting for the class that came before us (an undergraduate class which I don't know what it was) to be over.

Once, the door opened and a young woman gasped at the crowd in front of her and inquired as to what the class was (that was obviously so popular). "Growth theory," someone beside me replied. "What? Gross theory?". We all laughed. "No, no. GROWTH."

In retrospect, maybe gross was a good word to describe the state of growth theory today. Even though it was based on "microfoundations" the implications and its mechanics are too gross to describe the empirics of growth. Back then, countless papers were being generated based on Solow's growth model, extensions of Grossman (yet another gross!) -Helpman and Romer among others. Where is growth theory now? What are the current fads?

2. Back then another fad was DSGE modeling and its offshoot, heterogenous DSGE models (Greenwood, Krusell, Smith, Den Haan, among others). These models were all math based, in particular, they were based on dynamic programming and rested on assumptions of convexity, compactness and continuity. In looking at the dynamics of distributions, ergodicity either had to be assumed or derived.

Despite the limitations of rule-based models as evidenced in the Poundstone book outlined above there is still some hope for these models and its close cousing agent based models. The problem with these models (as was clear in the book) is the somewhat arbitrary starting points that are assumed. (Some may argue that assumptions of convexity and continuity are somewhat arbitrary as well, but mainstream economists probably would not. Or would they?) The same criticism is directed at math based models that have chaotic dynamics.

What reason is there to prefer one starting configuration over another? I would argue that the starting configuration is dictated by the outcome of that configuration. How closely does it match the data? This is not unlike calibrated DSGE models that attempt to replicate the empirical data ("moment matching"). DSGE models would claim to be more "scientific" in that they are "calibrated" independently. Or are they? Many models that I have seen in the past usually had one "free" parameter that the economist could "play" with - one such parameter is the risk aversion parameter. Various values of this parameter would then be used (as "sensitivity analysis") to match the data.

The success of DSGE models has not been surpassed by agent based models and it is perhaps due to the fact that the latter is generally not looked upon as "classical" economics. As such some of the brighter minds that could have made progress in this field have been diverted to more mainstream economics. This is similar to what A. Zee described in the quantum field versus string theory choice that faced graduate students in his day.

One outcome is a hybrid agent math based model where some agents follow rules ("rule-of-thumb" consumers who consume all their income) while others follow some kind of optimization rule. This is the model of Campbell and Mankiw and is already in use. Yet, I think that these models do not go far enough. The advantage of rule-based models is that there is no need to assume convexity or continuity. In fact, it would be more interesting to see what would happen if this were in fact the case. After all, even though time is continuous we do not make decisions in continuous time. We consume discrete amounts and make rule-based decisions - e.g. refinance when interest rate hits x percent (as opposed to a math based rule that would prescribe x.ab percent).

The future of economics and economic models is also dependent on the tools that are available to them. For instance, Einstein was not able to formulate the general theory of relativity without differential geometry or tensor calculus. String theory depended on non-Euclidean geometry. These pure mathematics fields were already on hand to be adopted by physicists for their use. Yet their adoption depended on a few who had the vision to see these tools as being useful.

Does economics suffer from a lack of vision - that vision thing - to use different tools besides dynamic programming and differential equations? Or does it suffer from a lack of tools?