A very simple model of too much city

100 identical individuals choose to live in one of two identical locations. The only thing they care about is how many people live in the same location. Let W individuals choose to live in the West, so 100-W choose to live in the East. The Utility of living in the West is U(W), and the Utility of living in the East is U(100-W), but it's the same U(.) function for both locations. Individuals have zero moving costs, and choose the location with the highest Utility.

Draw a curve for U(W). Then take the mirror-image of that curve (so East and West are transposed) and superimpose it on the first curve. Any point at which the two curves cross is an equilibrium, with the same Utility of living in each of the 2 locations, so no individual has an incentive to move. There is always an equilibrium at W=50, but it may be a stable or unstable equilibrium, and there may be other equilibria.

If the curve always slopes down, U'(W) < 0, the W=50 equilibrium is unique and stable. People dislike crowding, so move away from the more crowded location, until both locations are equally crowded. But that equilibrium does not fit the fact that cities exist.

If the curve always slopes up, U'(W) > 0, the W=50 equilibrium is unstable. People like crowding (there's some sort of Increasing Returns for jobs or amenities), so move towards the more crowded location, until everyone lives in the same location. There are two "corner solution" equilibria: one where everyone lives in the West; a second where everyone lives in the East. But those equilibria do not fit the facts either, because not everyone lives in the city.

So assume the curve initially slopes down, then slopes up, and is upward-sloping when W=50, and then eventually slopes down again, as W gets bigger. We get 3 equilibria: one unstable equilibrium where W=50; and two stable equilibria -- one in which most people live in the West, and one in which most people live in the East. Those two stable equilibria fit the facts; a city exists, but not everyone lives there. So that is the version of the model I will stick to.

The two stable equilibria are identical, except that East and West are transposed. They have the same level of Utility.

Here is the important question: Does the stable equilibrium (where the city exists) have a higher or lower level of Utility than the unstable equilibrium (where half the individuals live in each location)? It could go either way. It all depends on how you draw the curve.

Here is an example where cities are Bad Things, even though most people choose to live in the city:

Here is the intuition: Start in the unstable equilibrium where W=50. Now suppose that one individual randomly moves from East to West. That creates a positive externality for those already living in the West, and a negative externality for those remaining in the East. So that encourages a second individual to move West.

It is the (positive) sign of the difference between those two externalities that determines whether or not there is strategic complementarity (so the second individual wants to follow the first). Which depends on the sign of U'(W). But it is the (positive or negative) sign of the sum of those two externalities (weighted by populations in West and East) that determines whether it is a Good Thing or a Bad Thing for individuals to move West. Which depends on the sign of U''(W).

"Network externality" is a term that fudges that important distinction: the kid who gets a cellphone may create a positive externality for other kids who have cellphones, but a negative externality for other kids who don't have cellphones. Following the herd might be individually rational but collectively irrational. And following the herd might even create a negative externality for those in the herd, but an even bigger negative externality for those not in the herd, which induces others to join the herd but makes everyone worse off. It depends.

The biggest flaw in my very simple model is not that everyone is identical and the locations are identical; it's that I have ignored land rents. Land rents are a way to transfer utility from those who live in the city to those who live outside the city (because you don't need to live in the city to own land in the city).

Thanks to Robert Waldmann on Twitter for fixing my mathos. Robert has a related but more complicated post, but Photobucket has lost his pictures. Alex Tabarrok on Twitter notes that my model is related to one used to allocate cars between two roads. The difference is that drivers always prefer the road to be less crowded (U' < 0), and so the two roads problem is only interesting if the roads are not identical (one road is longer than the other). I don't know how original my model is; because as I keep telling myself, I don't do Urban economics.

Comments

[Jonswarbrick]

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)!

[Nick Rowe]

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.

[Matthew Young]

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.

[Matthew Young]

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.

[Benjamin Dachis]

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

[Nick Rowe]

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

[notsneaky]

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.

[notsneaky]

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.

[notsneaky]

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".

[Nick Rowe]

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

[C Trombley]

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?

[David Pringle]

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.