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Love this step-by-step reasoning.

The big picture concept you're grasping is "spatial equilibrium", the idea that migration equalises quality of life. Quality of life has pull factors like "friends, jobs, museums, and nightlife" (agglomeration) and push factors like rents, travel times, and crowding disamenity. Basically, quality of life equals wages plus fun things minus costs and not-so-fun things, and this residual equalises across space. Imagine water levels equalising when the tide changes.

BTW there's two parts to "equalising". One is the direction: people migrate from worse to better. That's your "perpetual motion machine". Another is the impact: in-migration bids up rents, bids down wages, increases congestion, and worsens crowding. That's what stops the machine.

Example: Oakland gets cheaper and induces in-migration. More people = more bars = more fun, yay! In-migration continues. Bartender wages fall. Rents rise. Traffic gets slower. Bar fights increase. People stop arriving. Oakland is bigger. Maybe it's more expensive, or maybe it's cheaper if the growth was managed badly and had high amenity costs. Falling rents can represent falling demand just as much as they can represent rising supply.

This is spatial equilibrium, and it eats up housing price gains faster than we can induce housing suppliers to deliver them.

We totally need better write-ups of this concept. It's critical for zoning debates, and very underappreciated. Many people assume upzoning will change supply without changing demand. But that's wrong. (Some economists even embed this in models of housing density by assuming a "closed city")

Cameron Murray's stuff covers so many parts of this.

Example: stable rates of rent to disposable income suggest you can't sustain lower rents for long (unless you've ruined amenity at the same time). https://www.fresheconomicthinking.com/p/why-is-the-rent-to-income-ratio-flat

And the parallels between spatial equilibrium in housing and the Downs paradox:

https://www.fresheconomicthinking.com/p/downs-thomson-housing-paradox

And a cracker on AMM models which assume a "closed city":

https://www.fresheconomicthinking.com/p/its-time-to-throw-out-the-standard

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