Worthwhile Canadian Initiative

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.

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.

Upward-sloping Demand curves for Labour and Housing

I have a simple thought-experiment ("model") that helps me think about the relationship between: migration; planning restrictions; wages; and house prices (or rents).

Assume all workers are identical, all houses are identical, and it's strictly one worker lives in one house. The government of a small city/state controls both immigration and the number of houses that are built. Because the city/state is small, relative to the world, it has no effect on the rest of the world. There's a perfectly elastic supply of labour from the rest of the world at a real wage w*.

There is some sort of economies of scale or agglomeration effect that causes labour productivity in the city/state to be an increasing function of the number of workers who live there. And self-employed workers earn the Average Product of Labour (there are no other inputs).

There are four types of equilibrium: 1. Where the migration constraint is binding because the government allows in fewer workers than it allows houses to be built; 2. Where the housing constraint is binding because the government allows fewer houses to be built than it allows workers to enter; 3. A kife-edge equilibrium where the government allows exactly the same number of houses and workers; 4. An equilibrium where the government allows free entry and free building and the limited land of the city/state is the binding constraint.

Here is what the four types of equilibrium look like:

1. House prices and rents are the same as in the rest of the world (houses can be imported from abroad and erected on the city/state) but wages are higher. Small relaxations on entry (not big enough to make the building constraint binding) cause wages to rise (because productivity rises due to agglomeration effects) but do not affect rents.
2. House prices and rents are higher than in the rest of the world, and so are wages. But the wage differential exactly equals the rent differential, so that real wages (adjusted for the higher cost of housing) are exactly the same as in the rest of the world (otherwise it can't be an equilibrium because more workers would try to enter, pushing up house prices and rents). Small relaxations on building cause house prices and rents to rise and wages to rise but real wages to stay the same because both rise by the same amount.
3. An indeterminate mix of 1 and 2 above.
4. Exactly like 2 above, except that it is landowners who capture the benefits from the high rents, rather than the government auctioning off the building permits (or the government's friends who get given the new building permits for free). Real wages (adjusted for the higher cost of housing) are exactly the same as in the rest of the world.

(I would like to relax the simplifying assumption of one worker one house, but it's too hard for me.)

This post is for Open Borders YIMBYs. Your stories about whether standard Econ 101 supply and demand apply to labour and to housing have got to be consistent.

Statistics Canada's historical housing cost data is wrong

In the early 1960s, Canadian economic historian Marvin McInnis started digging through the Dominion Bureau of Statistics archives, looking for city-level information on rental prices. While there, he discovered something strange and disturbing: 

A prominent theme of my career has been to reveal anomalies in what has been put forward as evidence. One instance is known only to me. In 1961 I was doing research for my doctoral thesis at what was then called the Dominion Bureau of Statistics. I wanted to know spatial price level differences across Canada. The data were collected for several major cities across the country but only changes over time for those cities were published as the CPI. I wanted geographical differences. I was on good terms with Simon Goldberg, the Assistant Dominion Statistician and he said , OK just go into the archive where the original reports from the cities were stored and dig out what you need. So I did.

What I discovered is that the clerks who recorded the data were under an instruction that, since the CPI was to represent prices paid by better off working class families, to edit out any rental figures what were above a designated threshold. By the end of the 1950s they were throwing out more than half of the reported rents. There had been a serious downward bias in a price that got fairly heavy weight in the CPI. I reported that to Simon but I don't think any alteration was ever made. It was water under the bridge. - Marvin McInnis, personal communication, November 23, 2016

We Are Adding Less to Housing Supply

Housing prices particularly in places like Toronto and Vancouver are still a big issue and what is driving them is the subject of debate. There is Josh Gordon’s recent policy paper, which places the main emphasis on demand side factors and there is the recent story raising alarm on Toronto’s “housing bubble”. There are of course demand and supply side factors and the supply of new housing coming onto the market is a factor. So today’s post is going to do two things: first, add to the debate on housing supply as a driver of housing prices in Canada and second, celebrate Canada’s 150th by adding a long term time series on housing starts going all the way back to Confederation.

Supply Constraints and Ontario Housing Prices

A key feature of housing markets in Canada over the last decade is the sustained increase in prices particularly in larger urban centers such as Vancouver and Toronto. Data from Canada Mortgage and Housing (CMHC) on average MLS housing prices for Ontario as a whole shows that between 1990 and 2015 the increase was from $171,979 to $465,441 – a rise of 171 percent. In Toronto, the average MLS housing price was $254,890 in 1990 and $622,046 in 2015 and is expected to range from $$694,000 to $742,000 in 2017.

Do (local) housing demand curves slope up?

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.

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.

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.

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

I can think of two plausible answers:

1. Natural geography matters. Countries are not flat featureless plains. Some places are more desirable than others to live and work. For example, you will tend to see cities located at natural harbours.
2. Strong Strategic Complementarity. Let me explain what I mean by that.

Secular stagnation, liquidity, and rent/price ratios

No answers here, only questions.

By "secular stagnation" I mean "declining equilibrium real interest rates".

Most explanations of secular stagnation say it is caused by a rising desire to save and/or a falling investment demand. Call this the "Saving/Investment Hypothesis".

But there are lots of different real interest rates. For example, the real interest rate on Bank of Canada currency is around minus 2%. (That currency pays 0% nominal interest, and the Bank of Canada targets 2% inflation). But people are willing to hold currency, despite that, because it is very liquid. But all assets differ in their liquidity. More liquid assets will have a lower real yield than less liquid assets. And if an asset becomes more liquid over time, its real yield will fall over time.

David Beckworth, JP Koning, and now Steve Williamson, say that falling real interest rates (on bonds and/or stocks) might be due to rising liquidity, (of bonds and/or stocks). Call this the "Liquidity Hypothesis".

One set of data, and two competing hypotheses to explain that data. So we need more data, to try to distinguish between them.

1. Would looking at data on rent/price ratios, for houses, or farmland, help us resolve this question? Because my random Googling suggests that rent/price ratios have generally been falling (prices rising faster than rents), and this is a global trend over the last couple of decades that is consistent with falling equilibrium real interest rates.

House prices and inflation targeting

"Should house prices be included in the CPI?" is not a good question to ask.

The best answer to that question is another question:

"Why do you want to know?" Or, "It depends; what are you planning to use the CPI for?"

Instead, it would be better to ask the question in a different way -- one that already answers that second question. Like:

"Should house prices be included in the price index that the Bank of Canada uses to define 'inflation' when it targets inflation?" (They aren't really included now, except very indirectly over the longer term -- long story (pdf).)

CMHC reserves revisited

There are two questions that Canadian taxpayers need to ask about the Canadian Mortgage and Housing Corporation:

1. Are CMHC fees for mortgage insurance high enough?

2. Does CMHC have a big enough reserve against potential losses?

Canadian Housing Prices: Some More Data to Ponder

Well, given the continuing pronouncements that Canada’s housing market is overvalued, I thought I would follow up my November post with another take on the data.  Among the many suggestions received on that post was to extend the data back further and to look at price/rent ratios. This time, I decided to gather CMHC data on average annual MLS residential prices and average annual rents for a two-bedroom apartment for the period 1992 to 2013 to estimate price/earnings ratios for a number of major Canadian CMAs. This provides a nice two decade look at housing prices in Canada.

Canadian Housing Markets: Bubbles if Necessary but Not Necessarily a Bubble

The debate over whether or not Canada’s housing market is a bubble that is going to crash heated up again this month. Earlier this month, the Financial Times warned that the Canadian housing market was “perched precariously at its peak”. Bank of Canada governor Stephen Poloz said this week that Canada’s housing market was not characterized by a bubble and expected housing prices to stabilize.  Of course, this debate assumes the existence of a “Canadian housing market.”

 

Random thoughts on house prices

First random thought: a person who was very ignorant about future house prices would nevertheless be almost certain that real (inflation-adjusted) house prices will be lower than today at some time in the future.

If we want a null hypothesis about house prices, I think the best null hypothesis would be that the log of real house prices follows a martingale. That means they are equally likely to rise (say) 10% next year as they are to fall 10% next year. And equally likely to rise 20% next year as fall 20% next year. Etc. For any year, regardless of anything. I think that is what someone who admitted total ignorance would assume.

Is this a bank bailout by the CMHC? Quotas vs tariffs.

I learn from the CBC that the Canadian Mortgage and Housing Corporation has imposed a quota of $350 million per month per individual bank (or other lender) on the amount of mortgage-backed securities it will guarantee. Presumably the CMHC did this because the Federal government wanted to put a cap of $85 billion on the total amount of new mortgage-backed securities the CMHC (and thus, indirectly, the government and the taxpayer) would guarantee in 2013.

This raises two questions:

1. Was the Federal government right to want to impose a total annual cap of $85 billion?

2. Was the CMHC right to impose a monthly quota of $350 million on individual banks as a way of implementing the total cap of $85 billion?

I want to set aside the first question and focus on the second question.

Montreal: An Island of Calm?

The latest release on the value of building permits for Canada's CMAs by Statistics Canada provides an interesting perspective on a slowing economy.  The numbers show that there has been a downward trend in the total value of permits since late 2012.

Worrying about the Canadian housing market, Part 3: Risks of default

Concerns about the housing market are not really about what happens to the market for houses per se; its role as a driving force for the recovery was done long ago. The real concern is the risk is a US-style financial crisis. I think it's a valid concern, but I don't think it's very likely.

UWO's Jim Magee has written a couple of very persuasive - that is, persuasive to me - commentary pieces on the topic:

• "Why didn't Canada's housing market go bust?" (Federal Reserve Bank of Cleveland, 2009)
• "Not here? Housing market policy and the risk of a housing bust" (CD Howe Institute e-brief, 2010)

Worrying about the Canadian housing market, Part 2: The effects of higher leverage

In a previous post on the housing market, I noted that we were unlikely to see the sorts of interest rate increases that would generate increases in mortgage payments of the size we saw in the early 1980s (up to 60%). And it's a good thing too, because most new homeowners are not as able to absorb those sorts of increases the way new homeowners were back then.

This difference is that the economics of mortgage debt in a low-inflation environment are quite different from those of the high-inflation world of the early 1980s.

Worrying about the Canadian housing market, Part 1: Higher interest rates

Like Nick, I've been thinking and worrying about the Canadian housing market, but not blogging about it. I've decided that it's worth putting some points out for discussion on the topic, if only because it's something we should be talking more about. I've broken it down to three separate posts: the second is here and the third is here.

The Bank of Canada has been warning homeowners that low mortgage interest rates are not here to stay, and homeowners' financial planning should be based on higher rates in the future. This concern is also the reason why the Department of Finance has been tightening conditions on mortgages. We don't want to see a wave of houses being dumped on the market because highly-leveraged home owners were unable to renew their mortgages. Or worse: a wave of defaults as people find they can no longer make their payments after renewing at a higher rate.

Below the fold, I'll try to sketch out how changing mortgage interest rates have affected mortgage payments in the past, as well as the sorts of changes people should be preparing for.

Things I think about and worry about but don't blog about

I mean macroeconomic things, not just cars and canoeing and stuff.

This is another "selection bias in blogging" post. Probably a more important selection bias than the one I talked about in my previous post.

If you thought that my blog posts are about all the things in macroeconomics I think are important, you would be wrong. Yes, my thinking something is important is one criterion, but it's not the only one. There is no point in me blogging about something I think is important if I also think I don't have anything useful to say about it. Because other people know more than me. Because other people are already saying anything I could say. Because I don't have any new thoughts I haven't already said. Because I can't make up my mind on the subject so don't know what to think. Etc.

The Lucasian map is not the Hayekian territory

In defence of Lucas '72.

Take any macroeconomic model of a market economy with inefficient aggregate fluctuations. In fact, take any economic model where something bad might happen.

Assume that model is literally true.

The people in that model are idiots.