# How can I model my problems with math?

I want to ask a question about mathematical economics. When I read an article about economics, I see lots of mathematical equations. I can solve them without any help. But I can't create my own mathematical equations. For example, this guy created his own equation. How can I learn this skill? I know how to solve some advanced equations like differential equations. But I want to create my own problems' equations. Please give me some advice.

The best way how to learn how to create new models is to see how other people create new models and learn from that. If you are already good at math then you could learn this by observing how models are created from some graduate textbook focused on theory such as MWG Microeconomic Theory.

1. It will help you to see what is the thinking process of an author that motivates model development.

2. Most new models are created just by tweaking old ones. In order to do that you already need to have an excellent knowledge of the existing models.

Next, the point of models is to create a simplification of a reality which will allow you to think about problem analytically. So when you try to create new model you first need to find some new problem, that has not been modeled yet, and then try to capture it in mathematical form.

For example, let us take equation from that linked Math.SE question as an example and let us think about how it was derived. First we can observe that people in cities seem to be more productive, they usually earn more money even when controlled for difference in price levels, also commercial and industrial activity seems to be located at or around various cities. Cities are by definition places where many people live (each country might have different definition of what city is but usually it will be defined in terms of demography and mainly population), so how could we model such situation?

Well a natural and logical way is to assume that peoples productivity $$P$$ depends on both some innate productivity $$P_0$$ and then has also some component which depends to proximity of other people so we can add term that tells us how productivity varies with proximity to other people $$\int_APf(r)dA$$ which will then gives us the expression derived by user Daniel:

$$P=P_0+\int_APf(r)dA$$

Moreover, this can also serve to illustrate my previous point. Now that we learned about this model we can also try to tweak it and in process come up with a new model. For example, what else does productivity depends on? I would guess that pollution will also be one thing that affects productivity. And in turn what does level of pollution depend on? Well historically again proximity to other people, cities are more polluted than villages. So based on this observation we can come up with 'new' equation:

$$P=P_0+\int_A Pf(r) - Pg(r)dA$$

where $$-Pg(\cdot)$$ will be term that captures how productivity is negatively affected by pollution which will again depend on proximity to other people. So we created new model from the old one (although I am not sure if this model already does exist or not in the literature, perhaps we just rediscovered some already existing model but for didactic purposes this serves as an example).

So in conclusion in order to learn how to make new models you should observe how other people create new models. Then you should find problem you are interested in and try to translate the model into math like in example above. Also this can usually be done just by tweaking the old models, most theoretical research does not start from scratch but builds on some preexisting work.

You should read Hal Varian's "How to Build an Economic Model in Your Spare Time".