Migration, Wages, and Corner Solutions

September 15, 2017
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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 lets 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.]

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