## Friday, March 5, 2010

### Simple versus Complicated

When you're modeling a phenomenon, you face a tradeoff between complexity and ease of use. If you build a model with extremely accurate predictions, it's probably very complex and difficult to use. Conversely, if you build a simple model that's easy to use, you won't be able to capture as much of the behavior of the phenomenon you're modeling.

Here are two examples: long-run growth models and orbital dynamics.

Long-run growth models. Your typical long-run growth model deals in three variables: A, Y, K, and N - technology, GDP, capital, and employment, respectively. You can crudely model long-term growth trends by knocking these two variables together. However, it's very difficult to capture any details; for example, not all forms of capital are the same - some is used for long-term projects, some for short-term projects, a highway here might be more efficient than a highway there, etc. To make the predictions correct, you have to generally use A as a "fudge factor"; the model still predicts a particular pattern in the data, but it's not good enough to get the pattern from more basic principles.

Similarly, a more complicated growth model would require knowledge of the time-distribution and geographic distribution of capital, population and inclination distributions of workers, various forms of technology, and so forth. It would be significantly more difficult to use.

Orbital dynamics. Let's think about the Sun and the Earth. Newton's laws can explicitly be solved for a two-body problem, giving the motion of each body about the center of mass of the system. However, the solution is rather complicated. Since the Sun is much, much, much bigger than the Earth, we may as well assume the Sun is stationary. The problem, instead of dealing with the motion of two bodies, only has to describe the motion of the Earth in a central force field -- much easier.