Throw out the standard urban economics model

The workhorse model of urban economics is the Alonso-Muth-Mills (AMM) model of the mono-centric city (the modern treatment is attributable to Jan Bruckner). This model is basically the representative agent optimal-control model of neoclassical economics. It is modified with additional functions that account for the cost of commuting to a city centre from different distances and allows capital, K, to be optimally geographically dispersed as well.

Sweet right?

The only problem is this. When you convert the model to English you realise it has little basis in reality. The only real pattern that is consistent with the model is that higher buildings are near the city centre. But I could come up with a million other models that are consistent with that pattern.

One of the main flaws in the AMM model is that there is no possibility for development of sites within the city into new buildings. Every site is already used at its optimal level. There are no vacant sites or sites with old buildings ready for knock-down and reuse. There is no development industry. There are no landowners.

Also because of the comparative-static nature of how the model is used, every time there is a marginal change in any of the parameters of the model — a new person moves to the city, the rental price of the second best land use increases, or the efficiency of construction methods change — the whole city is wiped clean of homes and buildings. The single social planner who controls everything in the city then dictates that the whole city will be rebuilt with a new optimal allocation of housing and commercial buildings under new conditions, and this whole new stock of buildings rebuilt in an instant to that new specification.

Don’t believe me? Here are comments from eminent urban economist David Pines from back 1987 making the same point.

The static approach in the Alonso-Mills-Muth model is useless in explaining many stylized facts regarding the urban structure and its evolution through time. In the static analysis... land is continuously utilized within the city boundaries and the city boundaries are continuously extended with income and population size.
...
The reason for the failure of the static model in explaining these ‘irregularities’ is that the housing stock is assumed to be perfectly malleable, which, of course, is highly unrealistic.

Perfectly malleable. That’s the crux here. Behinds this term hides the complete nonsense I just described about the constant rebuilding of the entire city.

This is a massive problem for anyone wishing to apply economics to urban planning. Because in the AMM model any constraint on land use — be it a natural feature such as a lake, river, or mountain, or a regulatory constraint in the form of height limits, floor area restrictions (FAR), or greenbelts — increases prices by forcing the malleable capital stock of homes and buildings to spread further from the city centre.

But this simply cannot be true outside of the model. There are so many contradictions between the model and reality that its conclusions cannot be taken seriously. For example, the existence of a development industry that takes sites that are vacant or in low-value uses and invests in new buildings isn’t captured in the model. There is no such mechanism because there is no vacant or under-utilised land. Every piece of space already has a building at the perfect economically-optimal height for that location.

I created the below image to show the common real-life elements of real cities that can’t exist in the standard AMM model. Let me explain.

The horizontal axis represents the distance from the centre of the town. Imagine taking a slice of the city along the roadside as you drive outwards from the city centre. You will see the density of buildings fall, which are represented here in dark grey. What you see in the real world is just the dark grey. 

The world of the AMM model is represented by the blue line, showing the optimal development density at each point along the road at a given time. In the city centre, where rents are highest, it is optimal to build higher buildings. Higher rents justify the investment in taller buildings. But then as you go further from the city centre, the rents at each location can only justify a smaller building on each site. I call this the “site economic frontier” because for each individual site at a given point in time, it is the economic limit of development.

In the AMM model, the whole city is full to the blue line. Always and everywhere. So you can begin to see the problem. There are substantial gaps between this model outcome and the reality of the grey buildings.

Moving along, the dashed orange line represents town planning constraints. Near the city centre I have shown how a height limit will create a gap between the site economic frontier and the “planned frontier”, or planning limit. I have also shown how such gaps are created by site-specific controls such as heritage protection (meaning you can’t demolish the building and then build to the site’s economic frontier). And I have shown how city boundaries like greenbelts or urban footprints create a similar gap.

The blue shading is therefore the economic-planned frontier gap. In the AMM model this is a problem, because before introducing such a gap the city is full to the brim, with buildings always built in every location to the economic frontier, so it results in a net loss of dwellings and buildings, even after accounting for feedback into higher prices and a higher economic frontier in other areas.

Yet in the real-world view, introducing such a gap changes nothing. Buildings are not demolished and rebuilt in different locations. Landowners in certain locations are simply limited by a “regulatory geography” rather than the “economic geography”, neither of which the city as a whole is anywhere near.

The existence of the light orange shading — the gap between the currently built city and the planned frontier — also cannot exist in the AMM model. There are no development opportunities. Even worse, there are no vacant land sites. This is an even bigger problem for the model.

I highlight this particular point by shading the gap between vacant sites and the planned frontier in darker orange because in the real world these are the most likely sites to be next developed. On the left of the diagram I also have a little curve that is supposed to show the probability of a site being developed as a function of its currently developed density or height. The smaller the current development, the more likely that site will be developed next, as there are lower costs in doing so in terms of demolition.

Overall then, we have a diagram that shows the major problems for the standard AMM model of urban economics. There can be no development industry in the AMM model because there is no planned-frontier gap. But even worse, the fact that reality doesn’t fit well to the model means that there must be some other mechanism determining the rate of investment in new housing and development. Something entirely ignored in this model. And even worse, entirely ignored in the current popular textbooks on urban economics.

I have been through this before. Vacant land is a perpetual real option to invest. The optimal timing of when to invest in a building — to exercise the development option — is when you expect that doing so maximises its value (read up on the Bellman equation if it takes your fancy). Otherwise, you wait. Because while today it might be optimal to build a five-storey building, in a couple of years it might be optimal to build a 12-storey building, providing even greater incomes. And you can’t do both.

This turns the standard model on its head. It means that because planning controls, such as height limits, take away this future option to build higher buildings, the value of waiting to build is lower, and the typical landowner will bring forward their investment, increasing the rate of new dwelling supply.

To me, the fact that the standard AMM model doesn't fit the data, and because we know land is best characterised as a real option, it must surely be time to throw out this model and update the textbooks.

Update: Read more about the option to delay development here.

The mysterious real interest rate of economic theory

The mysterious real interest rate – the one typically denoted as r in economic theory – does not have a real-life counterpart. This is a problem for economic theory. And it is a major problem for policymakers relying on monetary policy to boost economic activity.

While we think of the nominal interest rate minus inflation as getting close to the theoretical concept of real interest rates, changing this value in practice through central bank operations does not actually change the real return on capital and stimulate investment through that channel.

Why?

Because the price of capital is determined by the interest rate! We have known this for a long time. Joan Robinson wrote about the circularity of reasoning when we measure the quantity of capital by its price. She was ignored. As I expect to be.

For those who want to understand a little deeper, here are some more details. First, we take the standard economic view. In this view there is a thing called capital, K, that has a fixed cost (because it is a machine or building etc.), and each unit of K has an income-earning potential, net of depreciation, each period, which I call I. To buy each K people borrow money at the rate, r, meaning that as long as the ratio I/K > r it is profitable to invest in more capital, K.

So if my business can generate $100,000 in extra profit each from an extra machine, the business might see the value in spending $1,000,000 on that machine if they can borrow to pay for it at a 9% interest rate (costing $90,000 per year in interest), rather than an 11% interest rate (an annual interest cost of $110,000).

However here’s the circularity problem. The gains from a lower cost of new investment are made whether the investment is undertaken or not because they become capitalised in the value of the business immediately. That is because the value of the option to expand is always captured in the market value of the assets of the business.

What is this option I speak of? Where did it come from all of a sudden?

The way I snuck this into my definition of capital is part of the fundamental problem that permeates all the economic debates about capital. One group talks about capital as machines — independent robots, vehicles, machines and tools, who get to keep the returns from their existence. Yes, my bulldozer gets income from its efforts in this view, not the owner of the bulldozer. Because once you have an ownership structure overlaid, you have a system of property rights which contain real options for investment, and they have a value.

Think about land. Land is often referred to as capital, but it is nothing but a piece of paper offering a particular set of rights to a three-dimensional chunk of the universe. Land is an ownership right, not a physical object. See my mud map of economic concepts to help see what I mean here.

Once we have shifted to a view of capital of a system of property rights, some of which have physical machines attached to them — like a building attached to land rights, or a truck attached to various rights held by a trucking company — we can begin to see the circularity problem more clearly.

We now have a world were investors maximise the return on their property rights, not one where machines decide how to maximise the return on themselves.

This means that anyone making a decision to invest in new machines must take into account the current value of their property rights as part of the cost of capital. Because the full opportunity cost of the investment in a machine is the next best alternative, which is to sell the property rights at market value. In the diagram below I try to capture the idea that all physical capital — buildings, machines and so forth — are attached to property rights, and that only if we look at the value of the whole can we get the true cost of new capital investment from the perspective of owners of property rights.

Let us now see the effect of decreasing interest rates in a world of property rights, and where the value of these rights is part of the cost of capital. We will take the simplest case of a piece of vacant land, where the full value of the property right is from the option to build a $1 million building on that land to earn a future income of $100,000 per year. Here only the building is part of physical capital in standard economic theory.

We will then see what effect a reduction in interest rates has on the cost of “property plus capital”, and therefore the incentive to invest for owners of property rights. The table below summarises.

	Before	After	Further
Interest rate	11%	9%	7%
Income from investment	$100,000	$100,000	$100,000
Capitalised value of income	$909,091	$1,111,111	$1,428,571
Cost of standard K	$1,000,000	$1,000,000	$1,000,000
Return on standard K	-9%	11%	43%
Value of property right	-$90,909.09	$111,111.11	$428,571.43
Cost property rights K	$909,091	$1,111,111	$1,428,571
Rate of return on property rights K	0%	0%	0%

Let me walk you through this. The interest rate is the real interest rate. Take it as the nominal interest rate in a zero inflation environment for simplicity. The income from investment is the annual income after the building is built. The value of that income is capitalised at the new interest rate to show the static value. Then we see that when the interest rate is reduced, the $1 million building gets a positive rate of return, and hence the change to the interest rate will provide the incentive to invest.

As a side note, the alternative way to see this is to simply assume that the cost of the building is borrowed at the interest rate, as I did earlier when discussing the standard view. In this case, the cost of capital is $110,000 per year before the interest rate fall, and $90,000 per year after the interest rate drop, shifting the investment from an unviable to viable way earn the $100,000 per year.

But, if we consider the value of the property right as well, we have a different picture. Here, the value of the property right is the residual after taking the investment return (capitalised value of income) and subtracting the physical investment cost (cost of investment). With interest rates of 9% in the 'Before' case, the value is negative, and there is clearly no return on capital (i.e. for property valuers out there, this building is not the highest and best use of the land). But even after the interest rate is dropped to 9%, the return on the combined “property plus capital” is zero, because the cost of capital now includes the opportunity cost of selling the property right at a positive price.

Even if we decrease interest rates further, say to 7%, the rate of return on “property plus capital” is still zero, as I show in the last column. Owners of property rights simply gain at the expense of those in society who do not own substantial property rights and will be future buyers of those rights.

Under this view, the investment effect of lower interest rates disappears. The reason is that the capital of economic theory, and hence the real interest rate of economic theory, cannot be detached from the reality of a system of property ownership rights.

I’m not the only one to say this either. Once you are in a world of property rights and real options, the key determinant of investment is not the real interest rate of standard theory. Here’s Raj Chetty showing that increasing interest rates from low levels can bring forward investment — the exact opposite of the standard view. In a world of property rights an real options, the key factor is not what to invest, but when to invest in order to maximise the rate of growth in the value of your property rights. Hence there is a huge role for speculation on the price of property rights, and a clear logic behind following the herd during asset cycles. Under these conditions, it is also the case the reducing interest rates reduces the cost of delaying investment, and may, in fact, slow rates of investment and economic activity!

Let me summarise. First, standard theory has machines earning incomes and ignores the system of property rights it attempts to model. Second, once you incorporate a system of property rights these right have values, and the value of these rights must be added to the cost of machines to calculate the economic (opportunity cost) of capital. Third, once you have done this, changing the nominal interest rate (or even nominal rate minus inflation) changes no investment incentives, as all property rights holders immediate gain the value, which becomes a cost of investment. Finally, other factors that affect the cost of delaying investment by owners of property rights probably have a larger effect on investment, and in fact, decreasing interest rates decreases the cost of delaying investment.

This is not to say that there may be some effect of monetary policy through other channels, such as decreasing interest costs of borrowers, allowing them to increase spending. But if this is the dominant effect, without an investment incentive, then loose monetary policy may primarily inflate asset prices and not economic activity. This prediction gels with the reality of the past decade.