Practical Tips for Mathematical Modeling: Look first for Beauty

Here are the notes for tomorrow's PhD student talk on beauty in mathematical modeling. I'll bring a handout to class of other modeling tips also.

Practical Tips for Mathematical Modeling: Look first for Beauty

Stephen Kinsella, UL, 26.1.2010.

stephen.kinsella@ul.ie
Building models is part of any PhD, regardless of discipline. You mightn't come up with something brand new, you might apply some one else's work to new data, you might even just read other people's work and interpret it, but when you do, you'll be using a conceptual framework to do so, and that can be called a model for the purposes of this talk.
Secondary to the need for a theoretical lens through which to view the piece of the world you've decided to stick under a microscope for 3-5 years, you'll need, most likely, to get data to get this thesis thing finished. I'm not going to go into data collection today, but trust me, it is much easier looking for data after you've read enough into an area, or thought enough about the questions you're going to ask. So looking hard at your model before coming up with the questions that will generate your search for, or production of, your data, will save you oodles of time, stress, and maybe hair.
How do you choose which existing model to use, and where do you get ideas for new models? That's the subject of this talk. I'll argue it makes sense to go for beauty as a disciplining criterion first. What 'beauty' is for you-and your discipline-we'll have to discuss.
1. Ideas

Begin from the basics. You are not a genius, and you don't need to be. You've decided to find something out about the world. Grand. Your area of study will, normally, determine the types of things you look at for ideas. Personally I look at my own life, read the paper, and talk to people, long before going anywhere near journals. It makes sense to get a strong idea together before going to the journals, as Hal Varian has noted several times, because even the act of putting models together in rough form gets you better at making models later--practice makes, if not perfect, then better for having practiced. I'm a novice at this myself, and expect to remain so for the rest of my career as an economist. There are loads of tips and tools for coming up with new ideas, but maybe the best place to start looking is here.
2. Constraints

You only have so much time, and only so many permutations a model can take before it starts becoming overly complicated or just plain silly. Here's a great example of modeling gone mad. Place constraints on your work by using GICOD: good idea, cut off date. Begin by writing a story, or producing a numerical example, or drawing a simple picture of what's going on, before going looking for data or anything else. Many cups of coffee should be consumed in this part of the process.
3. Derivation/Explanation

Now comes the science bit. You need to specify the model in a manner acceptable to your discipline. In economics that means you need agents, a macroeconomy maybe, a bit of asymmetric information, an odd market structure, and prices and quantities of things. It depends on the sub-field one works in. This bit takes quite sometime, but it is widget-grinding, in the sense that if you are working on a variant of, say, the Melitz model of international trade, or expanding something Wynne Godley worked on, then the steps are pretty straightforward. If you are working on something completely new, it must have a relationship to the real world, and most likely it does. Use the story you've written or you numerical examples or simulations to get a good handle on how to model the elements you'd like to talk about.
4. Exposure and Re-exposure

OK, you've derived an equilibrium, or have a set of precepts to work from, or whatever. Now this model should start throwing up all sorts of interesting questions: what if wages go up? how will trade be effected by autarky? how does value get added down a disaggregated supply chain? and so forth. Together with your supervisor, answer these questions as best you can using the data you've had in the back of your mind for the past while. The data that allowed you to write the story or draw the diagram at the start of this process suggest the data that, fundamentally, you need to explain in your model later on. Gather, and expose your model to that data, improve the model, and re-expose your model. This might mean getting out the econometrics packages you've been dreading and working with panels or time series, it might mean going and doing surveys or interviews or experiments--whatever it is, you'll need to prove your model is justified by, and justifies, the data you are trying to explain.
5. Eh, where's the beauty?

You may have noticed none of this has anything to do with 'beauty'. That's because beauty is something personal, and something professional. I love reading models that I'd class as beautiful. Here's one, and here's another. They are both complicated-looking models that contain enough generality to be really useful when applied to several contexts, as well as providing a framework for thinking about related problems by extending or warping the models themselves. Also, they are cool to try and simulate on a program like Mathematica. When thinking about which direction to go when building a model, I tend to go for the direction I find most beautiful. That said, one of my next papers has 270 equations. But there's a lot in them, and beauty isn't always synonymous with simplicity.
Now, let's take a look at some of your favourite models, and ask what makes them beautiful (or not).