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Enjoying your posts on the topic as a macro guy trying to work on similar things. Thinking about the trade-offs between preferences and agglomeration effects, and trying to solve spatial dsge model so finding that this type of multiple equilibria and instability a little annoying (but interesting)!

Thanks Jon. It's that parallel between Urban and Macro, with the first being cross-section and the second being time-series. Who was the macroeconomist who told the story about looking out a train window: lots of economic activity....little economic activity...lots....little? Booms and busts; cities and countryside.

People are conserved and your model presupposes they live somewhere on a finite X axis. Ground state, where no person has a viable move, is uniform distribution across the finite X segment because each person would have to see the same trade left right trade. Except for the end point, that person must have a non zero energy, it sees no move at one side so a move to the center is always biased. Ground state is not zero, the distribution would be uniform random.

Just gabbing, no real thought here. Make the problem a disc, like real cities.

After a bit of thought. You created an inherently quantum mechanical model because the utility function is derived from local congestion. Multi-equilibria with super position.

This result because the utility function is uncertain, error bound by travel time up and down the axis. Superposition means congestion patterns can change, suddenly, like going from the morning commute to the lunch commute, superposition. Any given congestion state is a bit out of balance, but as they superimpose, the imbalances redistributed. So restaurant deliveries come earlier than morn commute; and tranacion space is conserved. But we carry a surplus of food until lunch.

There's some good New Economic Geography thinking that formalises this, which starts with some good work by Fujita and Krugman, if you want to dig in: https://mitpress.mit.edu/books/spatial-economy

Benjamin: that is something I ought to dig into, instead of doing my dilettante thing with Urban economics. But is anything there as simple?

I don't think land rents would change the conclusion (unless they get determined in some weird way) qualitatively. Assume that the land rent is proportional to the population at a given location. At 50/50, the land rent will be the same at both so your two curves will still cross at that point. That means that at that point total utility will still be higher than at the stable points (which would indeed move closer to the center). Tricky part is including those rents in overall surplus, since somebody somewhere does get them.

Or to put it in a fancy way, your non-pecuniary externality (congestion and agglomeration) is non-monotonic, it flips sign, but the pecuniary externality (land rents) is monotonic, so unless you really rig it somehow, one is not going to offset the other.

It might change your conclusion however if there is some intrinsic difference between the two locations (like quality of the weather, California vs...... Canada) In that case the two utilities can still have the same general shape but not be symmetric. And one of them, say U(E), could cross three times on the downward sloping portion of the other. In that case you still got multiple equilibria, but one of them will be a good one and one will be a "poverty trap".

notsneaky: I think you might be right. But my head is not totally clear on it yet.

I'm glad to see you've gone full Communist and called for a "more even distribution of the population". ;P

Looking into the historical examples of population distribution probably would help guide you though. There are the infamous instances, such as the Red Khmers, but as a Texan and a Western fan my first thought would be the (largely unsuccessful) Homestead Acts.

https://en.wikipedia.org/wiki/Homestead_Acts

This makes me ask the big question: does this shift your allegiance in Shane towards or away from the Staretts?

This reminds me of the T. Schelling's residential segregation problem taught in Phd Micro (though not in the econ dept but in the public policy school). https://lectures.quantecon.org/jl/schelling.html . A good introduction to models with multiple equilibria.

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