Question: Why do cities exist? Why isn't population density the same everywhere inside any country?

Why might the demand curve for X slope up? One reason might be strong strategic complementarities, like network effects. The more people who own a phone, the bigger will be the benefit to me of having a phone, and the greater the price I would be just willing to pay to buy a phone.

If the demand curve for X slopes up, a rightward shift in the supply curve of X (caused by relaxing legal restrictions on producing X) will cause a rise in the price of X.

Take this post with a truckload of salt. This is a second in my series in which " Lost Macro Farmboy tries to get his head around Urban Economics ". Think of it as sceptical pushback. I might easily be wrong, but those who know a lot more Urban Economics than I do should be able to explain why I'm wrong.

For simplicity, consider a simple game with identical agents. My utility U is a function of my action X and your actions Y, U=U(X,Y). A symmetric Nash equilibrium is a pair of actions {X*,Y*} where Ux(X*,Y*)=0 (I am choosing X to maximise my utility taking your action Y as given), and X*=Y* (so it's symmetric), and Uxx<0 (for the second order condition).

[Since "owning a phone" and "living in the city" are sort of binary yes/no choices, and I'm ignoring agent heterogeneity to keep it simple, it might be easier to think of X and Y as the probability of owning a phone/living in the city.]

"Strategic Complementarity" means Uxy > 0. It means that if you were to do more of something, that would increase my marginal utility of doing more of the same thing. It means dX/dY > 0. It means my reaction function slopes up. It means there's positive feedback.

"Weak Strategic Complementarity" means that 0 < dX/dY < 1. My reaction slopes up, but the slope is less than the unit slope of the 45 degree line where X=Y.

Weak strategic complementarity creates a multiplier effect, that magnifies any exogenous forces. The simple Keynesian Cross macroeconomic model (see my old post) has weak strategic complementarity, because the Marginal Propensity to Consume is positive but less than one. The multiplier magnifies the effect of any exogenous shock to investment or government spending by an amount 1/(1-mpc).

Weak Strategic Complementarity is not strong enough to create cities by itself. All it can do is magnify (multiply) the effects of natural geography.

"Strong Strategic Complementarity" means that dX/dY > 1. My reaction function is steeper than the 45 degree line. If you do more of something, my response is to do even more of the same thing than you did.

With Strong Strategic Complementarity, over some range of the reaction function, we can easily get multiple equilibria. Like three equilibria: one where noone has a phone; a second where everyone has a phone; and an equilibrium somewhere in the middle where some people have phones and some don't.

It could look like this:

[Most economists don't like that middle Nash Equilibrium, because it's "unstable". Neo-Fisherians are an exception to "most economists".]

Strong Strategic Complementarity will create cities even on a flat featureless plain. Everybody wants to live near everyone else, wherever that happens to be. When I look at a map of some parts of Canada or the US, it seems fairly plausible. Even if there were some confluence of rivers that got the snowball rolling in the past, the pattern may persist and get stronger even after people stop floating logs and canoes down rivers. People follow jobs, and jobs follow people, because it's easier to live and work near where everyone else is living and working. It cuts down on transportation and communication costs. Right?

Now let's bring the price of housing into the picture.

If the price I have to pay for doing X goes up, that will shift my reaction function for doing X vertically down. Given everybody else's choice Y, I will want to do less X if the price of doing X goes up. My individual demand curve for X slopes down, like normal demand curves do. The higher the price of city housing, relative to country housing, the less I will want to live in the city.

Now let's bring quotas on housing into the picture.

The government places a quota S on the supply of housing in the city. But it's a transferable quota; I can buy (or rent) a city house from someone else. And the price of housing adjusts to clear the market, which means the price of housing adjusts, to move my reaction function up or down vertically, until it cuts the 45 degree line at a point where X=Y=S.

If there is Strong Strategic Complementarity, the equilibrium with quotas will look something like this:

[Note that the quota makes what would be an unstable equilibrium a stable equilibrium.]

Which illustrates the point of my post. Given strong strategic complementarity, a (small) relaxation of planning regulations (a small increase in S) means that the price of housing will rise, not fall, so my reaction function shifts down vertically and intersects the 45 degree line at a higher Y=S. Even though my individual demand curve for city housing slopes down, the market demand curve can slope up, and will slope up in the neighbourhood (sorry about that pun) of an equilibrium with strong strategic complementarity.

It's a bit like Say's Law ("supply creates its own demand"), only even more extreme. If you build delta S more housing, so delta S more people move to the city, even more than delta S more people will want to move to the city at the previous price of housing, so the equilibrium price of housing must rise. If you build 100, 150 will come.

The effect of a large change in the quota (or total elimination of restrictions) is left as a topic for further research. I think it could go either way.

This won't work for the demand curve for housing in the country as a whole (unless we are talking about a city-state with relatively open borders).

I'm not sure about the welfare effects of relaxing planning restrictions. Strategic complementarity is not the same as positive externalities (see my old post).

[Aside: we teach students to read demand curves from P to Q. Quantity demanded is a function of P. That's the Walrasian way, IIRC. But we could also read them from Q to P. Demand price (the maximum amount people would be just willing to pay, at the margin) is a function of Q. That's the Marshallian way, IIRC. The reason that science students get upset in ECON 1000, when we draw a demand curve with P on the Y axis and Q on the X axis, and then talk about Qd as a function of P, is because we use the Walrasian language with the Marshallian diagram. With upward-sloping demand curve, it's easier to use Marshallian language, and talk about Pd as a function of Q.]