I really like this post. I have been playing with (or -- threatening to play with) a nearly identical model myself. There are a number of things to prove about this Markov chain, including what the stationary distribution is.
One additional effect to consider would be something like a "blur kernel," which could describe the flow of wealth within the economy to "nearby" individuals (e.g., the coffee shops in the finance district might be able to charge more, and so wealth will tend to diffuse to them). I think that effects like this are, at least in theory, what are supposed to prevent the excessive concentration of capital. Modeled on a graph, one could ask questions like, "How does income inequality (e.g., Gini coefficient) change as a function of graph connectivity (e.g., Fiedler number)?" The obvious story then would be that, as the economy becomes increasingly centralized in a topological sense (with some appropriate graph-theoretic measure of centralization), or as people become less dependent on commerce with their peers, distributions sharpen and inequality grows.
Finally, I'll add that there are other models of social phenomena, like armed conflict, with similar "inequality exacerbating" properties; see e.g. Lanchester's square law of combat. In fact, I think RTS games like Starcraft are an interesting model for this kind of thing -- they're really-abstracted models of violent societies -- and, between Lanchester's square law, and the exponential growth of the economy (the rate at which you can build SCVs is the rate at which you can mine minerals is the rate at which you can build SCVs...), you see the same kind of positive feedback there. Of course, I don't think you need a full-on video game to demonstrate this; the old board game Monopoly was invented by a Quaker woman to exhibit the same property and so demonstrate the evils of capitalism (...and everyone missed the point). In short, I think it's easy, and interesting, to construct mathematical systems that broadly have these properties.
Of course, to turn this kind of mathematical play into something with actual predictive power -- to do science -- would be a whole 'nother undertaking.