Space is expensive in cities, so why don’t we use it more efficiently?

Land is a scarce and expensive resource in Auckland, as the city’s strong economy and natural amenities (sunlight! beaches! bush!) mean that a lot of people want to live in a relatively small area. But we often insist upon acting like urban space is worth nothing – why else would we have so many underutilised parking lots around the place?

To an economist, this is perplexing. Econ 101 predicts that when one factor of production becomes expensive, firms and households will respond by substituting other inputs instead. This is easy and intuitive to grasp in practice. For example:

• If your local fish and chip shop puts up the price of snapper fillets, some people will choose to buy terakihi instead.
• If wages for checkout operators increase, supermarkets will consider installing self-checkout counters to save on staff costs.

We should expect the exact same thing to happen in the housing market. Broadly speaking, developers produce housing (H) using a mix of land (L) and capital (K), which we can loosely think of as the size of the building constructed on a site. So, for example, a ten-story apartment building will tend to have a quite high K/L ratio, while a detached house constructed on a large lot will have a low K/L ratio.

Warning: Arithmetic ahead. Come back after three paragraphs if you don’t like that sort of thing.

If we assume (as economists so often do) that housing production follows a standard Cobb-Douglas production function, then total dwelling supply can be modelled as a function of land and capital inputs, where a is the input share of land:

We can use this equation (plus a little bit of simple calculus) to estimate the marginal rate of substitution between L and K. Or, in other words, the degree to which rising land prices will encourage us to build up to save on land. If we assume that P_L is the price of land and P_K is the price of capital, then the ratio of K to L is given by the following equation:

We can immediately observe a couple of crucial relationships from this equation. First, if the price of land increases (and the cost to build up doesn’t), we’d expect the K/L ratio to rise – in other words, we expect people to build taller buildings on more expensive land. Second, if the cost to build up decreases – for example, through a technological innovation such as steel-framed buildings or elevators – the K/L ratio should also rise. This explains the emergence of high-rise Manhattan in the early 20^th century. Third, the relationship between changes to prices and changes in the K/L ratio will hold true in both low-density and high-density areas, although changes will occur at different rates.

Armed with this economic framework, we can start to make sense of the way that various cities look.

Here’s New York. It doesn’t look like this because it’s full of people who, unlike Aucklanders or Texans, have a mysterious preference for tall buildings. It looks like this because land is expensive and people have responded in a rational way.

Here’s an aerial photograph of a suburb in Atlanta, Georgia, one of the world’s true hellholes. Once again, it doesn’t look like this because Georgians have some oddly-shaped utility function. It looks this way because land is cheap in Atlanta (and motorways are large).

And here’s a picture of a typical Paris boulevard that somebody has photoshopped an enormous woman into for unknown reasons. While I’m sure many Parisians would claim that they have a unique cultural preference for seven-story apartment blocks with cafes underneath, Paris actually looks this way because land is expensive and developers have responded accordingly.

With that in mind, how does Auckland stack up in terms of efficiently using its expensive land? Well, as it turns out we’re doing some smart things and some blitheringly idiotic things. Here’s a brief tour.

The Northern Busway: Really smart. Adding two lanes for buses has enabled the capacity-constrained Auckland Harbour Bridge to carry many more commuters than it otherwise would have been able to do. Today, 40-45% of the people crossing the bridge during rush hour are on buses. It’s the most revolutionary transport investment to hit the Shore since the Harbour Bridge’s completion.

Manukau Centre’s sea of carparks: Mind-bogglingly irrational. As the map shows, Manukau actually devotes more land to parking lots than to commercial uses. Whoever laid it out obviously hadn’t paid any attention to Auckland’s real estate prices.

City centre shared spaces: Bloody clever idea. Turning service lanes and carparks into spaces for businesses to expand and people to enjoy allows us to make much better use of space in the city.

Spaghetti Junction: A tortured trade-off. Demolishing a tenth of the city’s housing stock and abandoning much of the city centre to urban blight was undoubtedly an audacious gamble. The motorways move a lot of people, but we’re never going to reclaim the valuable, centrally located land that they occupy.

Vancouver’s Skytrain – a future option for Auckland? Now this is about as cunning as a fox who’s just been appointed Professor of Cunning at Oxford University. Vancouver built a space-efficient (and cost-effective) transport system that created an incentive to build more densely. A perfect example of the virtuous cycle in which better transport options encourage more efficient use of land.

Vancouver’s Skytrain also provides an impressive contrast to the effects of Spaghetti Junction on Auckland’s city centre, which raises the question – are we smart enough to start building like that, or are we going to carry on with the pretense that urban space is free?

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