Migration, Wages, and Corner Solutions

I'm trying to get my head straight on something. Macro farmboy lost in Urban Economics again. Read at your own risk.

If immigration always increases real wages (or well-being), do we end up in a "corner solution", where everyone bunches together in one location leaving other locations empty? If so, that's a reductio ad absurdam, because we do not observe everyone living as close together as is physically possible, like sardines in a can.

Start with the simplest model. Everyone is identical, they only care about wages, and there are no moving costs. Divide the country up into two regions (East and West), and assume an initially arbitrary distribution of the population between the two regions. Suppose that wages are higher in the West than in the East, so people move from East to West. What does the equilibrium look like?

If we assume Diminishing Returns (think of an agricultural economy where land and labour produce food, so wages are a decreasing function of the labour/land ratio) we will (usually) get an interior solution -- an equilibrium where some live in the East and some live in the West. Because East to West migration causes West wages to fall and East wages to rise, until they are equal and migration stops.

If instead we assume Increasing Returns, so wages are an increasing function of population, we get a corner solution. People migrate West, which causes West wages to rise, East Wages to fall, which increases the incentive to migrate West. The whole population moves to one of the two regions. (There is a third equilibrium where wages are exactly equal between the two regions, so nobody moves, so wages stay equal, but we only get to that equilibrium if we start there by sheer fluke, plus it's unstable, so I will ignore it.)

Now divide the country into three regions. Everyone moves to one of the three regions.

Now divide the country into four regions. Everyone moves to one of the four regions.

You can see where this is going.

Divide the country into an infinite number of regions. Everyone moves to one infinitesimally small region.

Which is not what we observe.

Plus, it's daft. People wouldn't do that, even if they could. So don't start talking about zoning laws, which are just a red herring. Zoning laws could be good or bad, depending on externalities, and if you assume they are bad you are begging the question this post is trying to address. Plus zoning laws are what they are, and you don't have a magic wand to get rid of them. Plus, zoning laws can only make an area of land effectively smaller than it otherwise would be; eliminating zoning can't make an infinitesimally small area have a finite size.

Introducing moving costs into the model might prevent it hitting a corner solution, if those moving costs are big enough. But if some people have small enough moving costs, those people will still all crowd together into an infinitesimally small space. And some people do have small (or even negative) moving costs (they like to travel) at some point in their lives. So it's still daft.

Introducing a variety of preferences for location (physical geography and climate) into the model might prevent it hitting a corner solution, if those preferences are strong enough. But if some people have small enough preferences for location, those people will still all crowd together into an infinitesimally small space. And some people don't care much about the trivial difference in climate between the point with the greatest population density and the suburb next door. So it's still daft.

If we assume that real wages (or well-being, taking account of housing costs, dislike of crowding, etc.) are always a positive function of the number of people living in a given area, we end up with daft conclusions. So that assumption must be a daft assumption. Reductio ad absurdam.

But the opposite assumption, that returns are always decreasing, isn't totally sensible either. Because we do observe people bunching into villages, towns, and cities, rather than being spread out fairly evenly across a country.

A better assumption is that there are (at least) two inflection points in the relationship between wages and population density. We start out with decreasing returns, then increasing returns, then eventually hit decreasing returns again. So let's run with that assumption, and forget about all the other unhelpful complications like moving costs and preferences for climate.

If I have got my head clear on this (and I'm not sure I have) we get an equilibrium with two sorts of regions in the country: all regions in the country have the same real wage (adjusting for housing costs, dislike of crowding, etc.); all regions have locally diminishing returns (otherwise the equilibrium is unstable); but some regions have low population density and other regions have high population density. There are no regions with (locally) increasing returns, because that is an unstable equilibrium. (I would need to complicate the model with more inflection points to get mid-density regions.)

Now here comes the important bit, and the tricky bit, that I can't get my head around. What happens if there is immigration into the whole country? Does the wage (adjusted for housing costs, crowding, etc.) rise, or fall, or stay the same?

Since there is locally decreasing returns in all regions, you might think that the wage must fall. But that's not a valid argument, because it forgets that some regions might flip from low population density to high population density. The effect of immigration is to create more cities. It is not obvious (to me) whether wages rise or fall, or what it depends on. (The question of what happens to the real incomes of people who own land that new cities are built on but live in the countryside off their rental income is quite different. They gain.)

[And if we make the assumption (common in non-US macro) that the country in question is "small", which by definition means it has no effect on prices in the rest of the world, there is a simple answer. If that country alone opens its borders, then either: its wages (adjusted for housing costs, crowding etc.) fall to the rest of the world's level; or we get a corner solution where everyone moves there.]

Comments

[Tiago]

"And if we make the assumption (common in non-US macro) that the country in question is "small", which by definition means it has no effect on prices in the rest of the world, there is a simple answer. If that country alone opens its borders, then either: its wages (adjusted for housing costs, crowding etc.) fall to the rest of the world's level; or we get a corner solution where everyone moves there."

I think this is only correct if you assume homogeneous productivity among workers, which, in this particular case, is not a realistic assumption. Suppose the small country is a very high wage country. Living costs there will be high, even if there is very little regulation. The most visible reason is housing - building up is more expensive. Taxes would be high, at least in absolute terms. A low productivity worker might not find it advantageous to move there, even if he is allowed to do it.

[Nick Rowe]

Tiago: I'm always talking about *real* wages, which means adjusted for the cost of living, including housing.

Now it's true that people will differ in the percentage of their wages they will spend on housing (for a given price of housing). And those that spend less (like couples with no kids, or people who don't mind crowding as much) will be more likely to move to the high density locations. So we get a corner solution only in the sense that everyone like that moves to the high wage country. But where do the rest of the original inhabitants go? Do they all emigrate? Because they are worse off.

[Jacques René Giguère]

Nick: "But where do the rest of the original inhabitants go?" short answer on a busy afternoon: They stay in Ohio and vote Trump.

[Matthew Young]

There is a subdistrict in Hong Kong that, for legal reasons, has uncontrolled packing, and it is very close to sardines flopping, in its economy.

It is a highly dense swarm of small shops that do odd manufacturing and service jobs for the large corporations in Hong Kong. So, the question you bring up is great, and there is a measurable scale effect available to almost answer it, some real sphere packing going on.

[Nick Rowe]

Jacques Rene: I think Stephen did a post recently, with data showing net emigration from Toronto of those born in Canada. That would be an example.

Matthew: I think I remember reading about extreme crowding for the urban poor in England a couple of centuries back. The poor in Hong Kong are presumably richer than them, but their relative price of housing, compared to other goods, will also be much higher. So if they spent all their wages on housing, they might even be poorer.

[Roger Sparks]

I've been thinking about this subject as time allows. Here are my current thoughts on the matter.

Wages are secondary to some other criteria. One obvious criteria is the object(s) that constitute wage-paid. It is easy to see that pay in yen would not be the same as pay in dollars. A second criteria could be found in a region that allowed borrowing, which might have higher wages than a region that shunned borrowing.

The results of labor (for which wages are paid) should also make a difference. If the results of labor is portable products, portability of the product should serve to balance wages between regions, complementing migration (as a method of labor (wage) balancing). If the results of labor is immovable products, migration of labor becomes the only balancing mechanism available to balance wages (excepting taxes). Skilled labor could be a third category of labor, finding improved expression when the population increases.

Having considered these wage influencing factors, I think a corner-solution would be a political choice. A government that printed (or borrowed) money year-after-year and spent consistently in one region (ignoring some regions) would tend to the corner solution.

[cm]

I recently moved away from the "Silicon Valley" (more specifically south SF Bay Area between Palo Alto/Mountain View and San Jose). My complaints included increasingly long periods of high summer temperatures, drought, and water restrictions as well as crowding and worsening traffic (on more and more occasions I encountered trivial inconveniences such as difficulty finding parking at stores or public institutions outside walking distance, or going familiar places taking more and more time and observing inconsiderate driving and road rage incidents - small stuff, but together with the other issues it adds up to a diminished quality of life); but the main trigger was escalating housing cost. I had to move out of my long time residence, and found that area rents had gone up to almost 2X of what I had been paying; and I wasn't quite prepared to hand a whole paycheck minus food/gas/utilities and moderate savings allowance to a landlord (and I'm not talking about a mansion, but a SFH with one-car garage and one room used as a home office).

I heard many similar complaints from coworkers, and I know of quite a few acquaintances who made similar moves in the past year, or are planning to. These people as well as myself arrange remote/working from home deals or move to a "less hot" area where there is a corporate satellite office - others move farther away from the office to areas in the same metro with lower cost (or same after increases) housing cost, and then complain about long commutes.

This is all in an area where public transit is "OK" but not very good coverage wise (and IMO cannot be for everybody) so pretty much everybody commutes by car (in some cases by "shared economy" car rides - i.e. still car). Transit will not improve things overall - if you have a transit line you don't have to drive but can read, listen to music, etc., but the commute will take longer due to more transit stops (even if you are not in congested traffic). That's just how transit works - but social expectations are set by what "most people" (can) do.

I have found similar patterns in metropolitan area situations, also in Europe.

[Nick Rowe]

Roger: yes, if all goods (and services) can be transported anywhere at zero transportation cost, then migration doesn't matter. We can live and work anywhere, teletransport in our labour services, and teletransport in anything we want to consume. Migration is not needed, as long as we have free trade. And the big non-transportable consumption good is housing services (or the land on which it is built). And labour services may or may not be transportable ("can I work from home?")

cm: that sounds much like what I had at the back of my mind. Housing costs are the big one. But the one important thing I left out of my model is commuting between home and work. It gets costlier with distance, and also with crowding (though with an inflection, because there's no roads and no buses where there's no people).

I don't think I could handle living anywhere near SF nowadays. It was OK in 1976. Southern England is pretty impossible too. So is Toronto. Ottawa is barely manageable. Think I might head even further up into the hills when I retire fully.

[cm]

Nick: You can arrange a convenient deal (working from home etc. and being in the main office only a few days a week/month or even more occasionally if at all - also depending on whether it is in driving distance or flying/hotel stay are in the picture) if you have money/high-ish income, which strongly correlate with better working conditions and employers/principals being willing to make more accommodations. It doesn't work for most people, almost by definition (physical workplace attendance required either by the nature of the work or social supervision i.e. management not trusting workers to work if not under the whip or at least "peer surveillance").

As a subset of this, public sector or private companies catering to the public are very strongly correlated with physical presence either at the place of work or close by. From anecdotal evidence, my impression is that most employers in either the private or public sectors think nothing about contacting potential employees withing a 1-2 hours commute radius (at least) - based on recruiting calls my household received.

In the company where I work (mostly not public facing product company catering to global business customers, i.e. not public facing office work), I see the follwing patterns:

* people coming to the office very early (like 6 AM) and leaving in the early afternoon to beat the commute traffic
* people with kids coming to the office after 8 AM (dropping kids to kindergarten/school), and leaving when kids need to be picked up
* people coming in 1-2 hours after "regular" commute hours and leaving later accordingly
* the previous category includes a lot of people commuting to the office from farther away, traveling after the peak morning/evening commute; this is probably the category of people with the longest office presence as both morning and evening commute peaks extend for hours
* in addition to all that, many people "work from home" (at least checking emails) outside regular office hours - but the work pays well even without overtime

Another dimension in the whole thing is offshoring - there are employee populations in Americas/Europe/ME/PAC (and close time zones), and meeting schedules accommodating geographically distributed staff, which are at night or early morning for at least a subset of everybody.