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Nick - good post.

I'd like to push you to talk more about the welfare effects of relaxing planning restrictions. Here's your definition of strategic complementarity: ""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."

Strategic complementarity could arise through positional goods - so if you invest in peacock feathers, it increases my marginal utility of investing in peacock feathers, because I want to have more peacock feathers than you do.

Strategic complementarity could arise through complementarities in production - so if you post on WCI, it increases my marginal utility of posting , because regular posts attract traffic to the blog and so any given post gets more readers, feedback, influence etc.

Am I right in thinking that with strategic complementarity arising through positional goods, relaxing planning restrictions is a bad thing? But with complementarities in production, relaxing planning restrictions might be a good thing?

So which of the two stories do you think is the right one, and why?

I think this is basically correct. This will also increase total income in area, and unstated is an presumption that this will increase per capita income as well.

The other expectation is that prices will increase even more rapidly over time without this though that has to be squared with the last, so perhaps, increase per capita income, but by less than otherwise.

Frances: thanks!

"Am I right in thinking that with strategic complementarity arising through positional goods, relaxing planning restrictions is a bad thing? But with complementarities in production, relaxing planning restrictions might be a good thing?"

I think that's right. It could go either way. It depends on the exact nature of the strategic complementarities that create cities. All we know is that if others join, that raises my marginal utility from joining, which means my (Utility join - Utility not join) goes up. But there are many ways we can get the difference between two numbers to go up. Both numbers could go down, for example, with the second going down more than the first.

Those who control the narrative, being mostly cityfolk, tend to forget about the welfare of countryfolk. But this is a general equilibrium system. The rest of the country gets stripped of hospitals, for example.

Lord: "and unstated is an presumption that this will increase per capita income as well."

I'm not sure that's right. It might be that when others flock to the city, it leaves per capita income in the city unchanged, but reduces per capita income in the countryside. (Though we should perhaps talk about "welfare", rather than "income per capita".)

Nick: "Those who control the narrative, being mostly cityfolk, tend to forget about the welfare of countryfolk. But this is a general equilibrium system. The rest of the country gets stripped of hospitals, for example."

A couple of thoughts:

Do you think that you're seeing an absolute decrease in the standard of living in rural areas, an increase in the gap between standard of living in rural and urban areas, both - or neither? I think it would be hard to make the case that the standard of living in rural areas has decreased in absolute terms over the past 40 or 50 years given the amount of welfare-enhancing technological change. Yes, there are fewer hospitals - but would you really want to have your back operated on (say) in a small town hospital?

I was just in the Maritimes, and someone said to me: "you go to little towns like Antigonish, and they seem to be doing o.k., but what's happening is that people in the tiny little villages are moving away from those villages and into the little towns, and that offsets the loss of population from little towns to the bigger centres. But those little villages are gradually dying." There are lots of villages and towns and cities all over the world that have been abandoned at some point in time or other. People have chosen to move away, because they can achieve a better life elsewhere. Why should we, as economists, worry?

Politically, I like the conclusion that (some) increase in housing supply may increase demand by an even greater amount, pushing up already high rents and purchase prices. If I'm fighting against gentrification and displacement near the central core of a mega-region, the argument that added supply can increase rents and displacement is a powerful one.

But I think classical economics has a big problem with subjects like housing, and especially urban housing in the last 30 years. There are just too many individual actors with too many motivations. Too many ages. Too much mobility. Too many large differences among family wealth (provides easier access to housing for some), education (some high-growth jobs crowding out local residents with different skill sets), and a diffuse landlord class, with long-time mom and pops who own a few rental buildings to the large, public REITS that are motivated more by maximizing cash flow. Even if you could reasonably quantify the multitude of sub-groups, their behavior and preferences often change faster than they can be projected onto a model.

If someone is serious about understanding housing, especially urban housing, best to become very familiar with the tax code. Housing is the holy grail of both building individual wealth and receiving government tax subsidies to lower your tax bill. There's a very good reason landlording has exploded as an investment category over the last 40 years. The current tax system makes it one of the best investments. The wealth creation is often based on a relatively small amount of at-risk capital. With leverage, a mere 10% increase in the value of an apartment building equals a 50% ROI when a loan makes up 80% of the purchase price. In some parts of both US coasts RE is up over 100% in just 6 years. With a 20-30% downpayment and the rest financed, that's a 300-500% ROI. Most of us have not earned such high returns in our personal investment portfolios over the same period. The best part for speculators is that many (most) real estate loans are non-recourse, so if the property value sharply declines, it's the banks and government on the hook, not the landlords/speculators.

Besides the ability to grow personal wealth fast - using other people's money - the federal and state tax deductions are incredibly generous as well, including a phony depreciation deduction and (usually) full deduction of all mortgage costs, even if multi-millions are borrowed. Just make sure to buy during the up-swing cycle, because when the market turns - which it always does - you may lose your entire down payment if you're forced to sell. But if you can ride it out another 7-10 years, you should be fine.

I wish more economists would appreciate the observation that landlords have won the Economic War. The real issue for study should be, what will society do about it?

There is a trade off between making an existing building taller or building out farther away: density versus distance.

Fact 1: The returns to density are rapidly diminishing:
To go from a 1 story house to a 2 story house you lose space to stairs on both floors, need a stronger foundation, stronger walls, and a more expensive plumbing/HVAC system. It's a huge cost. But the cost of going from a 3 story house to a 4 story house is even worse, and the additional space devoted to elevators/stairs/HVAC must be provided on every floor in order for you to be able to add the next floor. So when you already have a lot of floors, the cost adding one more is worth it only if you are building the most luxurious of high end floors.

For very tall buildings, over 50% of the footprint on every floor is not usable, being consumed by multiple elevators, HVAC and other supporting infrastructure, which is why only 0.0000001% of the world's population can afford to live in very tall buildings -- it's basically a wasteful status symbol, like owning a Lamborghini -- and very few can afford to live even in 3 or 4 story buildings. Almost everyone lives in a 1 or 2 story building because these are so much more efficient, in terms of materials required per unit of livable space.

Fact 2: the returns to distance are roughly constant in scale:
If you are willing to go x more miles out, you add x^2 more land, but to reach it you need x^2 more roads, so it's roughly constant.

Combine Fact 1 and Fact 2, and you see cities getting progressively less dense over time, as the technology of transportation outpaces the technology of construction. It's true that Manhattan today has taller buildings than it did 100 years ago, but it is less dense -- each unit added is larger. It is always cheaper to build another floor on a 1 story building 15 miles away from the center + invest more in transportation to the center than it is to go from a 10 story building to an 11 story building in the city center.

With this process, cities grow outward, even as they become less dense over time. Bottom line, in the old days when transportation technology was very poor, there might have been strong complemtarity, in today’s world it’s a weak complementarity.

When I was looking at apartments in San Francisco versus the East Bay, I was shocked to find that the difference in price was roughly equal to the difference in BART fares + travel time. E.g. if a BART ticket cost $4 per trip (one way), then 50 trips per month = $240. If the average person values their time at $10-$20/hr, and a trip takes 20 min, then that's about $170-$340 for time. So that would suppose that I'd be willing to pay about $400-$600 more to live right in San Francisco rather than in an apartment 20 minutes away (on BART). At the time I was looking, that was the delta in prices, and you could see the prices fall each mile you went out. This might be the only example of econ 101 stuff actually working that I've ever seen. Apple runs out of iPhones and they still don’t raise the price, but rents are pretty much what you’d expect them to be.

The above decreasing prices can be calculated in reverse -- e.g. take the construction costs in the unconstrained area and then add a premium for distance to get the rent charged in the city center. And this is the direction in which causality flows -- e.g. the monopolist doesn't charge based on his cost, he charges what the market will bear, whereas in the unconstrained areas, where land is cheap, the cost of construction determines the price of housing. So to calculate the price of the cities, start with housing outside the city and add premiums.

As an aside, it's the zoning laws in the surrounding suburbs that do the most damage to the cities, by short-circuiting the release valve offered by transportation.


Your numbers re usable floor space are right for commercial high rises.

I don't think they are right for residential high rises. Because people per square foot is so much lower, you don't need the lifts (elevators) nor the HVAC that an office building requires. Your useful square footage is at least 75%, maybe over 80%.

That's why very tall residential buildings make sense (HK, Sing, NYC, Sao Paolo etc). Plus of course people pay more for a better view (4th to 28th floor, north London, new build, +40% in price).

That 100 storey thing rising at Yonge-Bloor works in economic terms.

On houses in London you can add a 3rd storey (ie 2nd floor) for c. £200 psf = USD 300-350 say, the most expensive construction market in the country. Given there are not houses in London selling for less than £200 psf, this is a no brainer. The Victorians built 4-5 storey houses and 8 storey mansion blocks, and the economics have not materially deteriorated.

Re sprawl

Unclear to me 3rd World cities follow the pattern. Density seems to rise even as new suburbs are built (Mumbai, Shanghai, Cairo etc.). Population grows faster than area.

It would appear to me that Toronto (and Vancouver) density is actually rising of late. Yes household size is smaller, but the number of small central units being built is so large that that seems to offset the sprawl.

London of course doesn't sprawl (the Greenbelt: 1948 Town & Country Planning Act). There is a donut effect (a wave of satellite cities beyond the greenbelt-- Milton Keynes, Guildford, Basildon etc. So yes there is sprawl that way, but, again, population of central London has been rising since the early 1990s and population of London metropolis is back to 1938 high.

You are confusing density with total population. Density is always falling, in poor nations as well as rich ones.

Here is data:


Re: building up,

I'm not saying it never makes economic sense to build a taller building -- such buildings do exist! -- but rather that there are rapidly diminishing returns to this versus the constant returns to transportation. You don't see a lot of tall buildings in most cities. Also, you don't just "add another floor" to an existing building. You need a stronger foundation to support that additional weight, which also increases because of stronger supporting walls, more plumbing, more square footage to stairs/elevators, etc.

Very good post. I've been saying this for years. It seems almost inarguable that allowing higher density is more likely to raise prices than to lower them, because higher density locations are more valuable. We KNOW this is the case (at least over the range of densities we generally observed) because otherwise there would be no cities in the first place.

Now, it may be the case that that some point the congestion costs begin to rise faster than the benefits of density and the process reaches a "natural" stopping point. But it seems clear that most cities in rich countries are far short of that point.

This is obvious if we open our eyes. Where are housing prices highest? Almost always where densities are highest. Right there you know that downward sloping demand curves are not an important part of the story.

It's also true that increased housing density will lower housing costs where demand for the location is based on the presence there of an industry serving global/national rather than local markets, which has located there for some exogenous geographic reason. North Dakota can presumably reduce the runaway rents caused by the fracking boom by encouraging more housing construction. No large city fits this description. (Maybe Palo Alto does?)

We should be clear that this is not an argument against allowing higher density. We absolutely should see zoning and other regulations changed to make it easier to allow housing (and mixed use) development at higher densities. But that's because the benefits that people are paying for are real. There are good reasons that people from poorer areas all over the world choose to pay vastly higher rents in places like New York City. We should allow more people to make that choice. But we shouldn't expect to get lower housing costs that way -- that will require some form of subsidy, direct or indirect.

The other direction this analysis points is that what's really wanted is policies to allow higher density development where densities and prices are currently low.

Alternate model: at any time, housing demand slopes normally, but higher density tends to shift the demand curve to the right.

Hi, Nick,

I think most microeconomists would say you are suggesting the policy change would shift both the supply and demand curves, not that the 'market demand curve slopes downward' (I think they would say it is definitional that the market demand curve is the summation of individual demand curves).

Perhaps the case you present is similar to the commonly discussed case of changed expectations about future prices, which is represented in micro as causing a shift in the demand curve. If so, it could be represented by microeconomists as a case of changed expectations about future quantity traded: The policy change leads everyone to think quantity traded in the city will be higher in the future than they had thought, and as a result their current demand for housing in the city shifts upward.

@Nick Rowe:

> 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.

Isn't there a stock/flow consistency problem here, or at least a short versus long-term dynamic?

People want to move to a city not because of the instantaneous number of houses for sale (the short-term supply), but because a lot of people already live there. The strategic complementarity comes in time-aggregated effects: the total number of (immortal) people who move to a city between T=0 and T=N increase the utility of moving to the city at T=N+1.

To put it another way, I don't think that any amount of new home construction in Timmins would turn it into Vancouver.

*in first paragraph, should be "not that the 'market demand curve slopes upward'" of course

(disclaimer: not an economist. I just wandered in off the street.)

You talk about small increases in supply increasing price, but isn't the response symmetric? If local government decreases supply (moratorium on building permits, condemn a few houses for a park) didn't they just simultaneously increase individual demand (I want to live by the new park) and decrease aggregate demand (Anytown isn't growing; don't move there), thus making housing more affordable?

The returns to density are rapidly diminishing

Maybe. But you can't just look at building costs. Lots of infrastructure is cheaper at higher densities. And if it turns out that even broadly measured costs rise with density, that just makes the point stronger. There has to be a reason why people pay those costs.

you see cities getting progressively less dense over time, as the technology of transportation outpaces the technology of construction.

Maybe. I suspect it depends on a lot on the domain you're comparing over. But I don't see how this is relevant to the point. Yes, of course more generous zoning rules (which we can think of as a kind of improvement in construction technology) will increase the density of cities. But that's not the question, the question is whether it will make them cheaper.

The above decreasing prices can be calculated in reverse -- e.g. take the construction costs in the unconstrained area and then add a premium for distance to get the rent charged in the city center.

I don't understand why people always look for these pure supply-side explanations. The question is what determines the premium and in particular if it can be treated as exogenous to the supply of urban housing (or office space etc.) If allowing higher densities in the

in the old days when transportation technology was very poor, there might have been strong complemtarity, in today’s world it’s a weak complementarity.

I don't see how this follows at all. The only thing that changes is the geographic scale at which the complementarity operates. And I'm not sure it's true even in that sense. Improved transportation also reduces the congestion costs of high densities.

This might be the only example of econ 101 stuff actually working that I've ever seen.

OK, but this is ketchup economics. The fact that the center-city premium is the same whether measured in rents or transportation costs doesn't tell us why that premium is what it is. Your comment seems to assume that the premium for central locations is independent of the supply of them, but the whole point is that that isn't true. Allow more development in the center makes it even more desirable and the premium will rise.

Alternate model: at any time, housing demand slopes normally, but higher density tends to shift the demand curve to the right.

Seems like a distinction without a difference. "Demand curves" aren't actual objects out in the world, they are jut ways of describing market behavior. If an increase in the quantity supplied increases the amount people are willing to pay, that's all that an upward sloping demand curve means.

JW, a demand curve is a representation of the relationship between price and quantity demanded. It slopes downward because there is a strong ceteris paribus assumption: 'What happens to quantity demanded, in a given context, if you vary only price?' Come then you and Nick to state, openly: 'Yes, but what if the expected quantity to be purchased ALSO changes?' Well, when we vary something other than price, that's when we talk of curves shifting, not of their slopes. A distinction without a difference? It makes quite a bit of difference whether you and Nick have discovered something that runs counter to a purported law of economics or not. Are we discrediting micro or just putting it to use, here? When you invoke technical concepts such as "demand curves" and their "slopes," forgive folks for presuming you are not here just to wave your hands.

On the empirical side, we've seen various cases of housing prices stalling when a market gets saturated with housing. Just recently there were articles on how Tokyo prices have stayed about the same, rising slower than population, vs. the soaring prices of popular US cities. Difference: a much more elastic supply market in Tokyo, with simplified zoning and an ability to build whatever you want within the zoning, without special permitting.


Also note that in perfectly orthodox micro, a rise in demand (curve shifts right) plus a reasonably elastic supply curve (45 degree slope, say) means an increase both in the equilibrium quantity supply and in the equilibrium price. No need for special theories.

In US coastal cities, supply is typically highly inelastic; we should expect increasing supply (shifting the curve, or making it more elastic) should decrease prices while increasing supply. It's possible, as I said, that higher densities might then shift the demand curve right again, raising prices again, but also leading to even *more* people being able to live there.

Frances: *Over time*, standard of living has been going up in both urban and rural areas. But it's not obvious to me how the growth of cities has affected welfare in either urban or rural areas. (And at the margin, standard of living must be the same in both, or people would have moved.)

rsj: that's an interesting train of thought. Now I'm thinking about scaling up a 2D circle vs scaling up a 3D hemisphere (with wastage due to roads/elevators/same thing). Wouldn't 3D always beat 2D, assuming symmetry? Except for being able to see the sky, of course. And gravity. Battery hens don't live in single floor apartments.

JW: Thanks!

Yes. Unless a couple of million Canadians really are willing to pay stupid prices to live in Toronto simply because they love Toronto's natural geography so much! AFAIK, there's a positive correlation between house prices (or rents) and city population size. And unless it's beautiful natural geography causing the demand curve to shift along an upward-sloping supply curve, there's gotta be some sort of strong strategic complementarity at work.

But, if you look at my first diagram, in the absence of controls, the only Nash equilibria with strong strategic complementarity is the unstable one. So we would be at a stable equilibrium, with weak complementarity. With controls, we don't know for sure whether the current equilibrium has strong or weak complementarity. Could go either way, depending on how tight the controls are.

"We should be clear that this is not an argument against allowing higher density."

Agreed. But strategic complementarity is also not *necessarily* an argument *for* allowing higher density (see my first post linked in this post). I *think* it could go either way.

Damien and JBriggeman: I tend to side with JW on slopes vs shifts of demand curves in this case. You *could* think about it either way. But demand curves are a lot more useful if they don't automatically shift every time the supply curve shifts. (Ditto for supply curves.) But *maybe*, you might find it useful to distinguish between a Short Run and Long Run demand curve in this case. (There's a similar argument in macro over whether the AD curve really is a demand curve in the normal sense.) Strictly speaking, the demand curve I'm talking about here (like the Keynesian AD curve, as opposed to the AE curve) is a "semi-equilibrium locus" -- combinations of {P,Q} such that Q=Qd(P,Q). Quantity demanded depends on quantity, as well as on price, just like in Clower's interpretation of Keynes.

Nick: "And at the margin, standard of living must be the same in both, or people would have moved."

No, because there's a limit to the number of people who can find houses in the city: "The government places a quota S on the supply of housing in the city." Or are you thinking about a city much warmer than Ottawa, so people don't need housing to live there?

Frances: Well, I think I see what you mean, but you can always buy or rent a house and swap places with someone already living in the city. The price of houses must adjust until the marginal person is indifferent.

Wouldn't 3D always beat 2D, assuming symmetry?

Yes, I think so! But over *time* you will see a diffusion process, where the steep areas get less steep as the mass spreads out. That is what has been happening since housing first became a thing. In the beginning we had to be cramped into small spaces, because the communication, transportation, and defense infrastructures made living outside the castle walls dangerous, and we were crammed into those tiny places. Then over time, as people become wealthier and transportation/communication improves, it becomes possible to get a bigger place farther out.

I remember looking at the old buildings in San Francisco built in 1915. They were all what we would call studios, and whole families lived in those studios. Basically everyone in a single room, with a fold up bed. You see old pictures from that era. Then, people started getting one bedroom apartments as the landlords combined two studios into a single unit. Then people wanted a second room for their kids. Then they wanted each kid to have their own room. Then they wanted more than one bathroom. Curiously enough, as landlords combined units and built new housing that has large units, density decreased! And that process has been happening for a very long time.

What you see is

1) space/unit increasing steadily over time
2) density decreasing steadily over time in city centers.

This is just an inconvenient fact for density proponents, who like to pretend that zoning laws are "preventing" density and that if we got rid of them, density would go up. They keep searching for this hidden demand for smaller units, and it's just not there.

The fact is, people like having a living room. Sure, they want to be in a hip urban setting, but you can leave Manhattan and find that in Brooklyn, and then you can leave Brooklyn and find a lively place to live in New Jersey, etc. As transportation and communication makes it possible for them to move out, they move out. That causes cities to grow horizontally and it causes density of city centers to decrease, not increase.

I would really recommend looking at the Lincoln Institute study I pointed to earlier. The Lincoln Institute should be your goto source for anything related to land use, as they are excellent, and did a lot of great work in looking at historical populations of census tracts in the U.S. and even comparable metrics globally. It's an amazingly researched monumental 150 page study, fully dedicated to historical changes in density around the world.

Here are some historical density evolutions around the world from that study:

View post on imgur.com

And here are their key findings:

To conclude this section, we review its key findings:

• Average built-up area densities in cities in land-rich developed countries in the
1990s were roughly half those of cities in other developed countries, and the latter
were, in turn, roughly half those of cities in developing countries.

• More specifically, the average built-up area density in cities in land-rich
developed countries (the U.S., Canada, Australia and the Russian Federation) was
34±7 persons per hectare (p/ha) in 1990 and 28±5 p/ha in 2000.

• The average built-up area density in cities in other developed countries (Europe
and Japan) was 86±9 p/ha in 1990 and 70±8 p/ha in 2000.

• The average built-up area density in cities in developing countries was 174±14
p/ha in 1990 and 135±11 p/ha in 2000.

• Multiple regression models can explain some three-quarters of the variation in
average built-up area density and do not appear to suffer from omitted variable
bias. They show that:
• Cities in countries with higher incomes—because of a variety of causes such as
higher land consumption, higher car ownership, and lower household sizes—have
significantly lower densities.

• The average built-up area densities in more populated cities are significantly
higher than those found in smaller cities.

• Cities in countries with ample arable lands per capita have significantly lower
average built-up area densities.

• Cities with no geographical constraints on their expansion in all directions have
significantly lower average built-up area densities.

• Cities in countries with high levels of income inequality have significantly lower
average built-up area densities.

• But cities with a large share of their population in informal settlements have
significantly higher average built-up area densities.

Nick - o.k., I see what you mean - the welfare of the marginal individual is the same in both. I always think of "standard of living" as being an average concept, not a marginal one.

All else being equal, people do like more space (until there's so much that it gets to be a pain to climate control or clean, but that's not a tight limit.) But all else isn't equal, and someone facing a $2000 rent might like to pay half the rent for half the space. But... that's commonly illegal.


"Meanwhile, in Kawasaki City Japan, there’s a proposal to increase the minimum apartment size from 18 sq m to 25 sq m. In New York, there’s pressure to lower the minimum size of new apartments to less than the present 400 sq ft (37 sq m)."

Are you willing to live in a 30 m2 apartment for less money, or even a 15 m2 microapartment? Too bad, NYC won't allow them to be built. What my friend was living in in Tokyo (15-20 m2 by my estimate) is illegal in NYC, not to mention other US cities (such as San Francisco, which originally invented minimum housing sizes as an anti-Chinese measure, just as later zoning is often an anti-poor and implicitly anti-black measure.) That's a problem.

And for all those international correlations, don't forget that within countries, dense cities are the expensive places to live. NYC and San Francisco are high density and super expensive and people still want to move in -- which is why they're super expensive.

And yet both NY and San Francisco are less dense today than they were in 1990, which in turn were less dense than in 1960, which were less dense than in 1930, etc.

Density continues to decrease in city centers, as it has been for 100 years, and as it will continue over the next 100 years, even though all these cities have different zoning laws and regulations. It doesn't matter whether you are talking about Mexico City, or Tel Aviv, or Moscow, or Tokyo, or Philadelphia, or Boston, or Jeddah. Density has been falling and continues to fall by about 1% per year across the globe.

In 20 years, it will even be cheaper and faster to commute to city centers than it is today, and in 20 years, communication will continue to be cheaper and more efficient than it is today, and each such improvement tilts the balance towards getting that cheaper/bigger place out of the city center. To compete the center creates even bigger spaces to keep and density falls there.

So density continues its 100 year slide, despite the fact that people do still pay to live in cities (as they pay to live in suburbs), and despite the fact that zoning laws do annoy some people on the margins (preventing both the combining of units and subdividing of units). Still density will continue to fall as this is driven by long term technological change and the human desire to live in larger units.

NYC has 8.4 million people today and had 7.3 million in 1990. If you divide by ten you get almost the same numbers for San Francisco. How is that getting less dense? Your claim of ever falling density is false.


As for positive demand curves, I don't like them because they make bad predictions. Consider two straight lines of positive slope. Moving the more elastic line to the right predicts falling equilibrium quantity, which makes no sense here whether that line is supply or demand. It could for some prestige good like diamonds where showing off how much you spent was entirely the point, so increased supply would lead to lower desirability, but we wouldn't expect building more housing to lead to fewer people.

Actually, the more I play with them, the less sense two positive lines make.

rsj: now my mind is thinking: we need to distinguish between:

2D density (people per square meter)

3D density (people per cubic meter)

As people get richer, and demand more housing, we would expect 3D density to fall. But it's not obvious (to me) whether 2D density would rise or fall. I read you as saying 2D density tends to fall.

I remembered another case of odd demand: asset bubbles, where as price goes up more people want to buy it. And housing often bubbles. OTOH that's more a response to change in price than to the price itself, and at any given moment I'd prefer to buy shares at a lower price rather than higher.

Compared to immigrant tenements, both 3d density (people per living space) and 2d (people per land) have gone down. But recently 2d density has gone back up, and it would go up even faster if zoning allowed it. And legal barriers to higher 3d are common too.


Yes! This is exactly my argument. The reason 2D density falls is that the costs on the z-axis go up much faster than the costs on the x-y axis. Building up has rapidly diminishing returns to z, but transportation has constant returns in the distance from center. Transportation (and communication) technology advances much faster than construction technology.

Without increased supply, price will increase with income. (Slightly faster than income in the case of SF (Eric Fischer, Steve Randy Waldman<). Increased supply can moderate this somewhat. Increased supply, being new, will enter near the high end but will temporarily lessen price pressure on existing stock while drawing in the more affluent. Investment won't be made unless the risk and tax adjusted return at least matches alternatives so prices can't be expected to fall, only increases temporarily moderated. Increased growth and population should result in greater productivity and specialization while increased affluence results in demand for more living space. Increased supply can allow more space as well as more 2d density, while lack of it results in lower density, but only centers can support the cost. Zoning is local and there are always other localities, so the most it does is shift development. Commuting and congestion fluctuate with energy costs and the economy. While marginal welfare is close, there are still some movements and trends though less than in the past.

Here is some more data on "2d" density. All density is 2D density in any conversation about land usage.

Change in population density for urbanized areas: 1900->2000

y2000 density/y1900 density - 1

Algiers: -68%
Bangkok: -55%
Beijing: -53%
Buenos Aires: -44%
Cairo: -68%
Chicago: -67%
Istanbul: -64%
Johannesburg: -19%
London: -80%
Los Angeles: -32%
Manilla: -58%
Mexico City: -77%
Moscow: -69%
Paris: -58%
Santiago: -46%
Sao Paolo: -32%
Shanghai: -72%
Sydney: -85%
Teheran: -68%
Tokyo: -49%
Warsaw: -90%

densities of top 30 cities, average person/hectare in urbanized areas:

1900: 285
1920: 238
1940: 200
1960: 165
1980: 129
2000: 102

You can see lots more data in the Lincoln Institute study.

Someone mentioned London, which had people/hectare of 211 in 1900, dropping to 83 by 1950, and 44 in 2000.

Hoping that people/hectare goes up in urbanized areas is a bit like hoping that incomes decline.

You're on the wrong side of history, and in both cases for the same reason. You expect incomes to increase because of technological innovation, and the same innovation makes land farther out relatively more attractive than it was before.

if we are in the Star Trek future where you can teleport anywhere you want instantly, there is no benefit (other than vanity) to living in any particular place, as you can beam yourself to your favorite locales the moment you step out of bed. So who cares where your bed is? In this infinite transportation world, all land is equally priced, because there is no advantage to monopolizing space when transportation costs instantaneous and of zero cost.

In the other extreme when the moment you step out of your neighborhood you are attacked by highwaymen and have to slog on foot along muddy roads, everyone is bunched together and the landlord can take virtually all of your income. This is Ricardo's nightmare scenario where population growth and scarce land leads to permanently falling incomes after rent. Density was increasing in many cities in the 19th century, prior to the transportation revolution.

Ricardo never saw the era of mass rail travel, otherwise he might have had a different view of the future.

So as we trend towards more innovation in which technology and produced goods become substitutes for privately monopolized non-produced goods such as space, we get to a better place -- with less density and more transportation in which the power of the urban landlords is diminished over time.

But even if you disagree that falling densities are beneficial, they are nevertheless inevitable.

None of this contradicts your uncle Leon who loves living in a studio, or the fact that apartments in Paris continue to get more expensive over time.

Cities whose populations, and presumably densities, have increased between 1990 and 2013, source Google's handy stats:
San Francisco (1971-2013)
San Jose
San Diego
Saint Paul
Cambridge, MA
Salt Lake City


(2001 to 2013)
Hong Kong


There are, of course, many cities that have shrunk in population and become less dense. We call many of them dying cities, or at least in trouble.

Yes, densities have shrunk compared to 1900, when we were much poorer, and people lived jammed into apartments, and cars had barely been invented yet, let alone massively subsidized to take over the world. The long crime wave from 1960 to 1995 drove people out of the cities, too, not to mention (in the US) white flight from school integration. That trend has reversed; people want to live in cities again. Millennials don't like to drive as much as older generations, even factoring in their worse job prospects. Walkable, bikeable, and transit neighborhoods appeal to more and more people.

Most of the cities whose population has been going up would be going up even more if zoning allowed it.


No, you are confusing population with density. Cities grow! Urbanized areas increase horizontally while becoming less dense in each hectare. That's kinda the whole point of a diffusion process. Imagine a pile of mud slowly spreading out.

Walkable, bikeable, and transit neighborhoods appeal to more and more people.

Small towns can be walkable and "special". Any sized town can be livable. London has great night life even though it's density has dropped by half since 1950. I bet the night life improved over that time.

The fact that cities are becoming "nicer" gives you *no* information about density. because above those nice, walkable streets, landlords are merging studios into 1 bedrooms and making the city less dense, and that process doesn't make the streets below any less fun or the people walking those streets less interesting to be around.

I know that I'm advocating in vain here for a discussion based and data and reason rather than cultural preferences and wish-fulfillment, but the data about density is as clear and obvious as the data about technology or incomes. As a rule, incomes go up and density goes down.

Here's a nice article about Manhattan. They even have a video to convince the breathless skeptics that yes, density does decline even in "special" cities in which tall buildings are constantly built.

"[...] Shlomo Angel, a senior research scholar at NYU Stern’s Urbanization Project, set out to do just that for Manhattan between 1800 and 2010.

It is a story of a century-long intensification of people, jobs, and houses on the island, followed by an equally long — and equally dramatic — decongestion. In the second half of the 19th century, Manhattan’s population growth far outstripped the rate of new home construction. In 1910, Manhattan reached a peak population of 2.2 million, from which it has never since rebounded, even after modest growth in the past three decades. Angel’s research found that today, Manhattan’s population density is down a surprising 40% from 1910. "

Construction won't help materially but it can reduce the rate at which density falls and keep them from decaying. Transportation tech is episodic, so innovation can widen reachable areas, but these areas add population, fill up, and increase congestion until a new innovation is invented and implemented. There will still be cities because people need places to gather and destinations to go to and this is what cities will be, hubs for interaction.

Yes, I think there will be always be cities because transportation will always have a cost in terms of time and money. E.g. we wont ever reach the Star Trek teleporter scenario.

In terms of reducing the rate at which decay occurs, yes anything that makes new construction cheaper, particularly infill construction, will do that. We should make construction much cheaper than it is. We should also loosen zoning restrictions, particularly in the suburbs surrounding big cities. And we should tax the hell out of landlords.

But also keep in mind that if we get rid of zoning all together, the result in already densy areas will be many more units combined into a single unit than micro apartments, on net. In terms of risk/reward, you get more bang for the buck as a developer with luxury housing than with affordable housing. You can charge more per square foot and require fewer materials.

What a lot of people don't realize is that the zoning laws within already dense areas tend to promote density rather than reducing it, just as the zoning laws within the suburbs tend to restrict density, by making it difficult to build up. In San Francisco, there are barriers to merging units just as there are barriers to building micro units, but the demand for 2 and 3 bedrooms far outstrips the demand for micro studios. A lot of people leave the city because they need bigger housing once they have kids. So zoning laws keep the city artificially dense, even though the stymied micro developer is the one creating the moral panic.

Similarly in the suburbs, zoning restricts do thwart building up, but there is a natural barrier to building up, namely the cost of tearing down (and then rebuilding) what was already there. When you go up, you don't have the same marginal trade-offs as when you go out. In the absence of zoning laws, it would still be problematic to take a built up area and make it taller. It would not be problematic to take an existing floor and combine some of the units on that floor. The geometry is against the density proponents. Nevertheless in places like Palo Alto, the demand for housing is so insane that I'm sure we'd get more height just by removing zoning laws. Unfortunately there are very few places like Palo Alto in the U.S. (and no one cares about Canada :P).

I'm surprised that no one has brought up the obvious solution -- ban and or heavily tax transportation. That would really encourage density and serve to rollback the clock. People would be again crammed into small units because they wouldn't be able to live where land is cheaper while working in the city. Landlords would become much more powerful. Also, we can reduce the efficiency of communication. If we get rid of the internet, people wont be able to telecommute or easily communicate. They would need to meet face to face, so the value of location increases and density would also go up. Getting rid of telephones would help, too. I'm pretty sure if we roll back the technological advances in communication and transportation since the early 20th century, we'd achieve that era's level of density.

"No, you are confusing population with density. Cities grow! Urbanized areas increase horizontally while becoming less dense in each hectare"

You going to claim that e.g. San Francisco and NYC (or just Manhattan) have added land since 1990? We're not talking about the metro areas, here, we're talking about the formal city boundaries. Most have been hemmed in by suburbs for a lot longer than that, and annexation has gone out of style; they are *not* growing horizontally.

"I'm advocating in vain here for a discussion based and data"

No, you're misusing your data. Yes, density has gone down from 1900, or 1800. Trends change. Manhattan's population bottomed out in 1982 at 1.42 million, 34 years ago. It has since increased modestly but nearly constantly, to 1.62 million. Brooklyn and Queens have grown even more.

"the zoning laws within already dense areas tend to promote density"

Untrue. Maybe you're right about laws barring merging units, that's not something I've heard of much, and I have more experience with houses that were chopped up into multiple apartments. But there are also laws about maximum heights, minimum setbacks, minimum area per unit, maximum floor-area-ratios, zoning against multi-family housing, and required parking, all of which substantially inhibit density. Much of our denser cities are in fact illegal to build under their own current zoning codes; a recent study said 40% of Mantattan was non-compliant, and another found only around 20 conforming buildings in Somerville, MA, which has a population of about 80,000. And I think they weren't counting parking requirements, just building form.

"there is a natural barrier to building up, namely the cost of tearing down (and then rebuilding) what was already there"

That's not a big barrier, especially when turning one unit into six, as you might by taking a one-story house and turning it plus its backyard into a three story apartment (with similar wood-frame construction techniques as a house; no need to jump into the elevator-equipped cost tier.) You're reaching at straws.


There are a number of approaches here for your methodological question, but all of them boil down to scale.

For example, in the U.S. the Lincoln Institute looked at census tracts, classifying them as urbanized or non-urbanized, and then examined what happens over time to the density in the urbanized tracts. So in a city a census tract can still be viewed as non-developed if it's a park, for example, which allows cities to grow by paving over that park. If they pave over that park with single family homes, but the rest of the city is 2 story homes, then the urbanized areas in the city will have become simultaneously larger in aggregate population and less dense even though the official city boundary did not change.

It's a 150 page study, so go read up on it if you have good faith methodological questions.

If you don't like the Lincoln institute, then go to the NYU Stern’s Urbanization Project, a similar methodology was used. E.g. for Manhattan:

Only by 1951 was the island completely built out, so to account for years when significant swaths of Manhattan was uninhabited farmland, Angel and his team counted only the portion of Manhattan that was built up and occupied by residents at each census year. No open space or parkland is ever included in their averages.

They also found:

Angel’s research found a worldwide decline in metropolitan area density over the course of the 20th century, regardless of level of development.

So it really boils down to questions of scale. If you zoom in or out, you can sometimes cause the result to flip a little, but over time and over many areas, the trend is consistent. E.g. if you don't have granularity at the level of the census tract, you may need to artificially impose a grid a 1x1 mile grid. Then you declare that if the grid is more than x% paved over it is urbanized otherwise its not. Then again look at density in the urbanized grids. You will get slightly different answers based on your scale, but over larger periods of time and many grids the general trend is the same. FYI Manhattan, as an island, *did* increase in density slightly from 1980-2000, but that was nothing compared to the drop from 1900 to 1980. And if you average over many areas, you see a decline in density during every decade.

But you never want to use political boundaries such as city limits because they do change quite a bit as most cities are not hemmed in like SF and Manhattan. You want to use geographic boundaries instead. And you want the property of being urbanized or not to be an intrinsic characteristic of the plot of land, and not a characteristic of the topology.

A huge chunk of SF belonged to the Navy, for example, and as they turned that land over to the city it became open for development and urbanization. There are still large undeveloped areas everywhere that are constantly being developed and recruited for housing use.

This seems plausible as a model of the long-term consequences of allowing a small one-off increase in housing quota followed by preserving everything in zero-growth amber forever after. But that's not what anybody is proposing; a healthy city is a growing city.

In the short run, the increase in housing supply would cause prices to fall; everybody who could afford to live in the city at previous prices already did, and the new empty buildings aren't providing any complementarity/agglomeration benefits yet. As people move into the new buildings and start patronising local businesses, starting companies, writing blog posts about how great the city is, telling their friends to move there, etc, the desirability of moving to the city will increase, and prices start creeping up towards the higher level predicted by your model. However, before they get there the next batch of new housing should come on line, pushing housing prices down again. (In a free market developers would enter in response to rising profits so profits would never get too high; in a housing quota system it's important that the quota always increase fast enough to keep rents from rising too high.)

This cycle will continue until 1) everybody in the country who can possibly be persuaded to move to the city has done so, 2) marginal construction costs rise too close to market rents (but most economically-healthy first world cities have market rents well above even supertall construction costs), or 3) there's a recession, which will make it too hard for developers to get financing to build more housing, but will also drive down everybody's ability to pay so rents will stay low anyway.

I don't understand this type of free-floating ahistorical reasoning. There is also an incredible amount of wishful thinking here.

Once a city is built up, it's built up. It's extremely difficult to tear down existing units and replace them with taller units. This just doesn't happen, and wishing it to happen or thinking it must happen because it would lead to some greater good is to be mistaken about the basic laws of asset prices. Yes, the developed areas within cities cities are pretty much frozen because houses are long lived assets. Cities are "living" things because they grow out, recruiting new land. Alternately, mass fires or other calamities can be used to clear land for development. But apart from that, the built up areas don't change a whole lot.

It's simple economics: Say you are a developer and you have a choice between buying an empty lot and building X units on top of it, or buying an existing lot with a single family home (1 unit), tearing down that home, and building X units on top.

To buy the home from the family living in it, they would need to be compensated for 1) the loss of the land (in terms of its current market value), 2) the loss of the house (replacement cost), 3) the loss of the community they are living in. Many households in wealthy areas will refuse to sell unless they are given extreme windfalls, because the marginal utility of $ is low but the loss of their community is large.

Moreover the developer would need to earn a return on this payment because this is a risky, time consuming project. Say a return of 30% (generally a lower bound). What the developer gets, after spending money to tear down your house, is empty land. Assume the cost of tearing down a house is zero, and there are no zoning issues or permitting fees at all. For this equation to work out, the developer must be willing to pay a substantial premium over the current market price of the land. But in so doing, they just *raised the market price* of all the other small units next to the unit that was bought. E.g. because prices are set on the margin, the land in all the similar houses becomes priced as if it was developed even though only one representative house was developed. Therefore the developer -- or all developers collectively -- can only develop one house per neighborhood, at which point they will not have enough money to buy the other houses.

So in san francisco, 80% of the value of a property is the cost of the land, and only 20% is the cost of the structure, even though most of the houses are less than 3 stories tall. The "potential" for building something big is already capitalized into the price of the land even though only a handful of lots have been converted. So you can convert a handful of units in each neighborhood per generation, and that's about it.

For this reason, large scale re-development of already built up areas just doesn't happen without some element of coercion. It doesn't matter how much money the developer has to spend, there is never enough money to consistently pay much more than the market price for a large number of identical units of land. Even Nero had to start a fire.

And this coercion -- e.g. seizing thousands of homes, compensating the families for less than what they could have gotten by holding out, and then throwing them out of their homes is very unpopular politically.

So it's not going to happen.

It doesn't matter that it "should" happen, developed areas just don't change without a large amount of state coercion. Development within a city happens by some units slowly deteriorating due to age, or infill development (landfill, golf courses and parks being paved over), etc.

What you do see, is as houses start to fall apart, they are sold as teardowns and rebuilt. But that happens randomly across a city. It doesn't allow for any type of new planning for a neighborhood, and it happens *extremely slowly*. A house can easily last 100 years, and many last longer. Even wood houses, if they are built of solid old growth wood like the redwoods in San Francisco, are going to remain standing for a couple of hundred years.

So let's please get back to reality and stop pretending we can redesign built up areas without mass evictions of incumbent homeowners. The room for design is only on the margins -- as the city grows -- and the time for city planning is when the city is being built up, not when it's already built up. There is very little scope for significant changes once the area is already built up.

A special case that is becoming more common as urban development continues?

But, the argument falls apart when you think at the margin. If another 50,000 housing units magically appear out of nowhere in downtown Toronto, this wil reduce, not increase purchasing prices.

The effects you refer to would only occur in the long run. I haven't got a clue how you'd tie this together mathematically though.

Posted by: rsj | August 22, 2016 at 09:35 PM "There is very little scope for significant changes once the area is already built up"

Sorry, but you are totally wrong, Sydney is the obvious counter example, but also places like Singapore.

Of course, there is some government involvement, but if anything government makes it more difficult to build by putting regulatory obstacles in the way... still the developers go for it, and they don't have to pay much more than market price, they are very good at being patient and paying just on market price. They build in every way imaginable, by retrofitting existing warehouse space, by razing and building a bigger structure, by building up through existing structures, building above railway stations and leaving the original station underneath... all of the above and more. You see that profit motive makes people extremely creative. Astounding!!

You only have to find some archival photos, most of the skyscrapers in Sydney are less than 20 years old.

There's been no fire I assure you, just a lot of new population and massive investment spending. Small numbers of homes have been seized by government, but never for development of towers, they will do it for roads and sometimes for rail. However, a lot of the rail is underground now, and a few of the roads are going that way too.

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