Thursday, 26 March 2015

Migration, amenities, and median multipliers

In the standard spatial urban economics model, the marginal person has to be indifferent between living in different cities. That doesn't mean that all cities have to be identical but rather that, from the perspective of the marginal resident, the downsides and upsides of different alternatives come to a similar balance. 

Because Detroit has fewer amenities than, say, San Francisco, its housing prices have to be lower than those in places with better amenities and stronger opportunities. If that weren't true, people would move out of Detroit until housing became sufficiently cheap that there were no longer reason to move from one to the other.

One reason for this is that houses are durable goods. Detroit built up a large stock of housing when the city's industry could support a much larger population. If people took their houses with them when they moved, housing costs wouldn't adjust downwards as much when there were substantial out-migration. 

Another reason can be time-to-build in high-amenity locations. Incoming migrants push up the price of the existing stock of housing; that provides a signal to developers to convert more houses into higher density uses and to expand on the fringes of town. High housing costs relative to incomes are then a disequilibrium phenomenon - they're something that happens in the interim until developers are able to get new housing on-stream. And you get a nice price gradient in the standard simple model where housing close to downtown amenities is very expensive to housing farther away, with the slope of the gradient depending on things like the ease of commuting and the desirability of the amenities. At the edges of the city, the cost of a house should be the cost of building it, plus the cost of providing basic infrastructure, plus the underlying base value of land in its next best alternative use.

So median house prices relative to incomes can tell us a few different things. High prices would be associated with strong amenity values. But if there are also very high prices at the city fringes, or if the very high prices persist for a long time, there's also something else going on - something related to building new houses and apartments. You don't get sharp drop-offs in land values at metropolitan urban limits, or apartment buildings built to only 10 stories when another 5 stories would cost less to build than the apartments' sale prices, unless something else is up.

What else can be up? Well, you could be in a place like Hong Kong or Singapore where there is little land available, and where much of the land that can be turned into high-rise apartments already has been. In places with hard physical constraints against further building, benefits of productivity increases or stronger agglomeration effects get capitalised into the price of existing land. Why? Spatial equilibrium: if people are just more productive in those places, that draws in more workers, which bids up the price of housing, which can't really draw in more housing supply, and then confers rents on the holders of existing land.

In other places, it's zoning constraints. Both the metropolitan urban limit and the city height limits seem binding: Arthur Grimes showed strong discontinuities in prices at the MUL, and that apartment buildings are being constrained by height restrictions. Yes, Auckland has physical constraints on land supply too, but there are plenty of places that could be upzoned (and haven't) and land at the boundaries that could be brought into use as suburbs were it not forbidden.

Peter Nunns really doesn't like Hugh Paveltich's median multiplier measure of housing affordability. Nunns says that because it ignores differences in amenities, it's worthless as a measure.

Maybe I'm an easier grader than Peter, but I do find good value in Hugh's measure. Cities that persistently have high median multiplier measures and no particular physical constraints on further building likely have issues in regulations around land supply. 

If Detroit implemented strong restrictions against upzoning or new suburbs, it wouldn't show up in a median multiplier measure because the regulations wouldn't be binding - there's a large stock of housing available relative to demand. A high median multiplier does not necessarily follow from bad regulations, nor does a low median multiplier necessarily follow from good regulations in places with binding physical constraints. But a persistently low median multiplier in a city with a growing population likely signals accommodative regulations and a persistently high one in a city without strong physical constraints likely signals regulatory issues.

The measure simultaneously tells us that places like Auckland have highly desirable amenities, and that they have pretty binding regulatory constraints against new building. 


  1. It's no wonder that people who believe in lots of planning laws dislike the median multiplier. The number clearly and succinctly shows the damage that all the planning laws they like have done.

    Perhaps there is a bit of an issue with median multiplier due to changes in interest rates though. High rates of interest would mean you'd probably wind up with a lower median multiplier.

  2. Only inasmuch as interest rates hit housing prices. Affordability measures more sensitive to interest rates tend to be things like "% of median income needed to service mortgage costs on median house".

  3. Hi Eric - nice discussion of the spatial equilibrium model! One thing that I've noticed, in my reading of the literature, is that nobody's really put together a model that integrates both amenities and planning constraints as determinants of prices. (Let alone financial factors.) It seems like the only way to make urban economics models tractable is to make simplifying assumptions about some components of the housing market in order to analyse another component.

    My concern about Demographia's analysis is that they don't consider all of the possible explanations for differences in the level of the median multiple. Failing to do so is likely to mean over-estimating the welfare gains from relaxing planning regs. The overall effect of doing so may still be positive, but the existence of persistent differences in amenity levels means that it's unrealistic to expect that house prices would be 60-75% lower in the absence of any rules. (As Demographia implies when they state that every city should have a median multiple under 3.)

  4. I have a very hard time imagining price drops of that magnitude under any conceivable change in Auckland planning regs. Worse, that others might fear such drops might motivate opposition to useful relaxation of the rules on some margins: if the government were worried, for example, that getting the land use rules right would result in a halving of household wealth, well, they're just not going to do it.

  5. That's why I've always been a bit skeptical about "housing affordability" comments from politicians. Surely they wouldn't deliberately choose to reduce the value of their constituents' assets!

    That's one of the reasons I'm happy renting (and investing the savings in a diversified basket of equities): it ensures that my personal incentives are aligned...

  6. Peter and Eric there is a mysterious law that accurately predicts the size of cities in a country or economic region. Zipf''s law in most places has held true for the last century. The biggest city is double the size of the 2nd biggest city. Triple the size of the 3rd biggest city and so on.

    "by kiwimm (from

    Interesting article and comments.

    Based on Zipf's Law, we should see (City, Expected, Actual)
    - Auckland 1.42M 1.42M
    - Christchurch 0.71M 0.37M
    - Wellington 0.47M 0.35M
    - Hamilton 0.36M 0.22M
    - Tauranga 0.28M 0.13M
    - Dunedin 0.20M 0.12M

    Clearly Auckland is overly large for the country by this measure and to increase productivity, we need to grow the remaining cities."

    Kiwimm got his analysis slightly wrong Greater Christchuch and Wellington are more like 0.47M and 0.45M and I would characterize some cities being overly small not Auckland being overly large!

    But his conclusion still seems valid in there is growth potential in cities like Christchurch, Hamilton, Tauranga and Dunedin.

    I wonder if Zipf's Law indicates there is a balance between agglomeration economic forces that encourages aggregation and dispersal forces that well encourages dispersal. That this doesn't just hold within a city as per the spatial equilibrium model but also between a countries cities.

  7. I assume by "planning laws" that you also include the many, many restrictions on my and your private property rights on what we can do with our own land.

    For example, minimum set backs, minimum lot size, minimum parking and maximum heights. All these planning laws both prevent housing being built in desirable areas plus unreasonably restrict my private property rights.

    Equally on the question of why not let the city sprawl, why not let it intensify?

    Cities like Houston and Atlanta have just as many planning rules that relate to restricting density, just not land use.

  8. I can't speak for Steve X, but I prefer easing back the restrictions on both density and on sprawl, then letting consumer choice and relative costs determine to what extent cities go up and out.

  9. I agree 100%. That is what annoys me about pointing to Houston and Atlanta as some illustration of consumer preference on housing. The planning laws there are massively weighted in favour of sprawling, auto dependent development while CBD fringe land has big restrictions on density.

    I believe a lot of people in Auckland (obviously not everyone) would choose smaller, denser housing close to the centre if it was a real option. Maybe apartments but more likely multiple storey, terraced housing with small outdoor areas - say on 200-300sqm sections.

    If we free the MUL and density rules and everyone floods out to low density fringe developments and medium density developments languish unsold, then I will be the first to say I am wrong.

    However, considering the speed that the few dense developments in established areas sell at already - including apartments - I suspect the spread would be much more balanced. Maybe even the 70/30 split the Unitary Plan was aiming for.

  10. "but more likely multiple storey, terraced housing with small outdoor areas - say on 200-300sqm sections."
    Count me in. But Wellington, rather than Auckland, pls.