Urban Economics – An Introduction

Marco Salvi Real Estate Finance, Uni ZH, 1.3.2011

Our greatest invention - cities

Urban economics – what’s that?

Property prices, land uses, density, mobility and migration – how do these phenomena interact in cities?

Different sciences have different answers. Urban economics as the economic approach to cities.

Urban areas are viewed as arising from the rational location choices of individuals responding to incentives.

A motivating example: who makes money with your latte?

•  A “Latte” at Starbucks‘ on Bahnhofstrasse costs CHF 6 – a lot of money for some coffee and a little milk.

•  Where does the money go?

•  Here are some possible answers…

Honorio, farmer

Luís, Barista

Howard, CEO

?

Who makes money with your latte?

The willingness to pay for a Latte on Bahnhofstrasse is higher than the cost of production (coffee, milk, wages). That‘s why it is an attractive spot for a café.

Starbucks has to compete with bars, banks and boutiques to rent a spot on Bahnhofstasse. Each spot goes to the highest bidder.

The rent on a given spot is bid up until it is higher than the willingness to pay of the second highest bidder.

  The advantages of a location are reflected in its land price or rent. They accrue to the owner of the land. If the advantages increase, this will benefit the owner of the land, because it holds the factor of production which is in fixed supply.

Residual land value theory

•  A landowner sells land to an investor. The investor develops the plot as a condominium and sells the apartments. Which of the two parties receives the difference between the sale price and the construction costs? Do they share?

Property price

Construction costs

Investor

Land price?

Profit?

Land price Land- owner

•  On a competitive land market, the price of land will be bid up until the total dissipation of excess profit.

  Land price as a residual.

•  The developer will try to maximize the value of the property; in doing so she maximizes the land value ("highest and best use" of the land).

Spatial equilibrium approach

In a spatial equilibrium there is no free lunch to be gained from changing location.

Land prices adjust and/or firms and people move until there is no gain to be made from further changes.

Assuming a linear utility function, utility from

Income + Amenities – Housing Costs – Transportation Costs

is equalized across locations.

At the margin, consumers are indifferent between locations.

An application: city rankings

City rankings periodically report the “best” places to live in Switzerland. Some “winners” of past editions: Zug, Zürich, Feusisberg etc.

Suppose the spatial equilibrium holds. What are its implications for city rankings?

If your tastes are not too specific (i.e. if you qualify as a marginal consumer), you can spare yourself the reading…

Alternatively, consider Credit Suisse’s “Financial Attractiveness Ranking”, which ranks cities according to the metric (income being fix in all cities):

What does this ranking measure? Where is it likely to be higher?

The Monocentric Model

Application of the spatial equilibrium concept to land uses in a city.

Answers the questions: Where will city households choose to live, given their income and characteristics? How does the gradient of land prices arise?

Alonso-Mills-Muth [1964] (David Ricardo [1821], Johann Heinrich von Thünen [1826]).

A theory of urban form

Main model assumptions 1.  Production takes place exclusively in the Central Business District (CBD).

2.  The city is located on a featureless plain. Distance d to CBD is the only characteristic of a location.

3.  All city residents commute to the CBD with transport costs T proportional to the distance of the commute, i.e. T(d) = k*d.

4.  All dwellings (housing units) are identical, with surface q.

5.  Households have identical preferences and income. They consume only two goods: housing (q) and a composite good z (“consumption”). The utility levels are linear in these goods ( U(q,z) = q + z ). They earn an income Y.

(Assumptions 4 and 5 will be eased later)

Zürich as a monocentric city

Employment density in Zurich (in built areas) as a function of the distance from the CBD

CBD is identified with Zurich main station.

High concentration of employment in the city center.

Land opportunity cost ra / m2

Spatial equilibrium in Circle City We are looking for the equilibrium rent of a housing unit R(d) at a given location d. At this rent no household as an incentive to change location. No house is vacant.

CBD

City boundary

b

Commuting costs T= k*d

Location d

Lot size q Construction costs c Housing rent R(d)

Property prices at the city boundary

How much does the developer has to pay for land at the city boundary? What is the price of a standard dwelling located at the boundary.

The land price at the city boundary (=marginal land!) is equal to the land opportunity cost, i.e. the value of land in the alternative use, e.g. agriculture.

If the alternative use is not very valuable (e.g. a desert city), land price at the boundary will be virtually zero, reflecting the lack of scarcity of land.

Now, let’s move onward towards the city center, i.e. towards more valuable locations. What happens to the land price?

Location rent

A location near the CBD is preferred to a more distant location because of the lower commuting costs (i.e. higher consumption levels) associated with central locations.

Households being identical, they all prefer to live near the CBD. Is this a spatial equilibrium?

Rents at the city center will increase until the utility levels of all households are equalized, i.e. until utility out of consumption and housing is the same across locations.

Housing rent and land rent will increase 1-to-1 with the transportation cost (cost of commuting). More central location will command a location rent, reflecting the scarcity of central places.

Housing rent and commuting costs

In the monocentric model there is a direct relationship between communting costs and housing rent. Given our assuptions, it can be shown

Y – T(d) = R(d)·q + z

Net income Total expenditures

Suppose d increases by one unit. What change of R leaves a household with the same utility level (= constant Y)?

ΔY = ΔT/Δd + ΔR/Δd·q = 0 ⇒ ΔR/Δd = - (ΔT/Δd)/q

The rent (per square meter) decreases by the amount of the change in commuting costs per square meter of housing.

Equilibrium housing rents

Equilibrium rent gradient across locations.

Households utility levels are independent of location.

Land rents runs parallel to the housing rent curve because housing units are identical.

What would happen if we allow for different housing units?

CBD b d

Housing rent R(d)

Agricultural rent (=opportunity cost) ra*q

Construction costs c

Location rent = k*(b-d) (=capitalized commuting costs)

R(d)=ra*q+c+k*(b-d)

An example

What is the land price at location d, given Avg. distance of commute: 15km, i.e. city radius: 15·√2 km Avg. commuting cost: 40 CHF/day · 220 days = CHF 8,800 per year

With linear commuting cost t = T/d = 8,800 CHF / 15 km = 586 CHF/km per year.

Each commuter is willing to pay 586 CHF per year to live 1 km closer to the CBD. Assuming an avg. dwelling size of 100 m2, the location rent will be CHF 5,86 per m2 per year (CHF 50 per month per dwelling)

Reality check: 100 m2 in Zürich (Kreis 1) according to ZKB/TA-Rating ca. CHF 3'850, in Horgen (ca. 15 km from city center) CHF 2'620. Difference is higher than implied by simplest monocentric model with linear utility and transport costs.

What is missing in the model?

Effect of a change in size/population

Comparative statics: How does a change of an exogenous variable impact the equilibrium rent?

Example: City size. Assuming a constant density, what is the effect of an increase in population, i.e. of an expansion of the city?

The larger the city, the higher the rents. Why?

Effect of a growing city on rents is similar.

d

Housing rent R(d)

ra*q

b b'

Larger city

Smaller city

Reality check: size and housing prices

Rents are higher in larger urban areas.

However, other factors do impact rents, as for example, environmental amenities (exogenous) and „man-made“ amenities (endogenous), as taxes, good urbanism etc.

Where do you expect incomes to be higher, in smaller or in larger cities – and why? (Hint: think about the implications of spatial equilibrium)

Quelle: ZKB, BfS (2008)

City is density Why does density vary within/across cities? Why does it change over time?

Is there an „optimal“ urban form – and so is there an „optimal“ density?

Use the monocentric model

From Bertaud, Alain and Stephen Malpezzi “The Spatial Distribution of Population in 48 World Cities”

Density is not constant across the city

In most cities we observe land and housing rents falling sharply when we move from the city center to the suburbs. Rent gradient is convex. Why?

Until now, we have assumed a constant housing density, i.e. a constant ratio between housing surface and lot size. That is, we have assumed a fixed ratio between capital and land in the production of housing.

We have also assumed a constant population density, i.e. a constant ratio between consumption (net of transportation costs) and housing.

This is not realistic! Households and builder will react to higher land rents in the city center. How?

Convex Location Rent Function

CBD b d

Land rent r(d)

Without substitution

Land rents with substitution

Land at the center of the city is scarce. Land can be used more productively.

„Better“ locations allow for a higher floor area ratio (FAR, ratio of housing floor area to land area), i.e. higher buildings at the city center.

Land prices increase more than proportionally because of the higher FAR.

€

r(d) = ra +k(b − d)q d( )

Demand- and Supply-side Substitution

On the supply side, developers will use more capital (concrete, elevators, etc) where land is more expensive.

On the demand side, assuming housing is a normal good, households will demand less housing, i.e. – for a given income, they will live in smaller dwellings where housing is expensive.

The extent of these substitution effects can be measured by the substitution elasticity. On the demand side it is (remarkably) equal to one, implying a constant share of housing expenditures.

On the supply side, a unit elasticity of substitution implies a constant share of land to the total value of the property. Evidence in Europe: El. < 1

Constant housing shares

The orange line shows the share of housing to household income in the Swiss regions.

The share is in a narrow band between 16% and 19%, largely independent from the location.

Same pattern is observed over time, consistent with unitary substitution elasticity, i.e. Cobb-Douglas preferences.

Reality Check: Population density in ZH

Location and building quality The Monocentric model implies a positive correlation between land prices and the „quantity“ of housing per unit of land.

Interpreting quality of housing as a quantity of housing services (better housing delivers more services), we expect a higher quality of housing (= more capital) on better locations.

We can thus explain some „wisdoms“ of real estate development –  Better to buy a lower quality building on an expensive location, than an

high quality building on a cheap location. –  Construction costs should be about twice the value of the land. –  When the city expands, returns from development are higher at the city

boundary than in the city center.

What else can we explain?

Based on the monocentric model (with varying density), what would you expect to the equilibrium land and housing prices at the city center and at the boundary

–  when transportation costs decrease?

–  when land outside the city is very productive?

–  when the population increases but the city is not allowed to expand?

–  when the city is located on a coast?

–  when we expect a growing city?

–  when the population increases but density is not allowed to rise at the city center?

Conclusions

Urban economics relies on the concept of spatial equilibrium [Glaeser, 2007]

We have touched upon some implications of this equilibrium for city rankings and for the choice of location of households in the simple case of monocentric cities.

We have not touched upon the question of why firms choose to locate in cities, i.e. why do cities have agglomeration returns – and why these returns vary. Topic of a huge literature in urban economics and regional science.