Urban Models

ECN741: Urban Economics
The Basic Urban
Model: Assumptions
The Basic Urban Model
Outline of the Class
 Origins of Urban Economics
 Key Assumptions of a Basic Urban Model
 The Basic Household Maximization Problem
The Basic Urban Model
von Thünen
 Urban economics was invented (sort of) by a German
agricultural economist, Johann Heinrich von Thünen,
who lived from 1783 to 1850.
 More specifically, von Thünen invented the concepts of
bidding and sorting, which form the basis for urban
 He also, by the way, invented general equilibrium
The Basic Urban Model
von Thünen’s Model
 von Thünen modeled the location of agricultural
activities around a central market place.
 Each activity had a maximum amount it was willing to
pay for land at each location—its land bid.
 The winning activity at a given location was the one
that bid the most there. This leads to the sorting of
activities across locations.
The Basic Urban Model
von Thünen’s Model, Continued
 The key to the model is transportation costs.
 Some firms produced heavy or perishable products, so
they would not pay much for land far from the center;
transportation costs would eat up all their profits at
distant locations.
 But some other factors, based on von Thünen’s
practical knowledge of agriculture, also came into the
The Basic Urban Model
Annual Rent per Acre
von Thünen's Model of Rents and
Rent for Milk/Vegetables
Rent for Wood
Rent for Grain
Rent for Livestock
Distance from Central Market (miles)
The Basic Urban Model
 In 1964, William Alonso published Location and Land
Use, based on his dissertation in regional planning.
 This amazing book applied the von Thünen logic to the
location of households in an urban area.
 People all work in a central work site.
 Each type of household bids on land in every location.
 Household types sorted into different locations based on
their bids, and types with high commuting costs locate
nearer to the center.
The Basic Urban Model
Mills and Muth
 Edwin Mills (at Princeton and my thesis adviser) and
Richard Muth (at Chicago) extended Alonso to consider
housing at about the same time.
 Mills: A 1967 publication (May AER), which cites a Muth
working paper; a 1972 book (Studies in the Structure of
the Urban Economy).
 Muth: A 1969 book (Cities and Housing).
The Basic Urban Model
Urban Models
 Urban models are designed to explain why urban
areas look the way they do, with a focus on
 Where do people live? How far do they commute?
 How much housing do they consume?
 What is the price of housing?
The Basic Urban Model
Key Assumptions
 We now turn to the key assumptions of a basic urban
 These assumptions lead to a simple, complete urban
model that describes urban residential structure
 After we explore a model with these assumptions, this
class will investigate how urban residential structure
changes when more general assumptions are
The Basic Urban Model
Key Assumptions
1. Housing Demand
2. Housing Supply
3. The Transportation Network
4. Why Location Matters
5. Types of Households
6. The Labor Market
7. Household Mobility
8. Local Governments
The Basic Urban Model
1. Housing Demand
 Household utility functions depend on a composite
consumption good, Z, and housing, H, and take the
Cobb-Douglas form.
U  (1   )ln{Z }   ln{H },
 H is measured in units of housing services = quality
adjusted square feet.
The Basic Urban Model
2. Housing Supply
 Housing services are produced with land, L, and
capital, K, according to a Cobb-Douglas production
function with constant returns to scale.
H K
1 a a
 Housing is owned by absentee landlords.
 Maintenance, rehabilitation, and conversion activities
are ignored.
The Basic Urban Model
3. The Transportation Network
 Households commute between place of residence and
place of work in a straight line at a constant
transportation cost per mile, t, using a single
transportation mode.
 The urban area is located on a featureless plain (=von
 There are no commuting arteries or street grids.
 There is only one transportation mode (=car?)
The Basic Urban Model
4. Why Location Matters
 Distance to work is the only locational characteristic
households care about.
 This assumption rules out neighborhood amenities,
such as
School quality
Access to shopping or recreation
Air quality
The characteristics of their neighbors.
The Basic Urban Model
5. Types of Households
 All households are alike.
 All households have the same income, family structure,
job location, and utility function!!!
 We obviously need to weaken this assumption—and
eventually we will.
The Basic Urban Model
6. The Labor Market
 Income is fixed and all households have a single worker
with a job in the central business district (CBD).
 This assumption greatly limits labor market analysis;
there is no explicit export good and the implicit
demand curve for labor is horizontal.
The Basic Urban Model
7. Household Mobility
 Households are assumed to be perfectly mobile within
an urban area.
 If a household has an opportunity to improve its utility,
it will take it!
 This assumption implies that all (identical!)
households will achieve the same level of utility and is
crucial to the logic of urban models.
The Basic Urban Model
8. No Local Governments
 The basic urban model ignores local governments.
 There are no local government services and no
property taxes.
 This is obviously and important omissions, and we will
spend the last third of the class trying to fix it!
The Basic Urban Model
 Now to set up the basic household problem that is the
core of an urban model, we consider
 The price of housing
 Owning versus renting
The Basic Urban Model
The Price of Housing
 The annual price per unit of housing services is P.
 This price depends on household location, measured in
miles from the CBD, u; that is, P = P{u}.
 Apartment rent is P{u}H.
 The derivative of P with respect to u, is negative:
Pʹ{u} = dP{u}/du < 0
The Basic Urban Model
Owner-Occupied Housing
 This model can also be applied to owners.
 The value of a house, V, is the present value of the
rental benefits from owning it. Let i be a household’s
real discount rate and M be the expected lifetime of a
house in years. Then
P u H
y 1
1  i 
V 
i 
P{u}H P{u}H
1  1  i 
The Basic Urban Model
The Household Problem
 The household problem is to
U{Z, H}
Subject to
Y = Z + P{u}H + tu
 The Lagrangian is
 U Z , H    Y  Z  P u H  tu  ,
The Basic Urban Model
The First-Order Conditions
 The key first-order conditions are:
   0,
Z Z
  P u  0,
H H
   P u H  t   0.
The Basic Urban Model
 The first two conditions imply that
U / H
 MBH  P u .
U / Z
 The third condition indicates that a household moves
away from the CBD until it finds the u* at which the
savings in housing costs from moving 1 mile further out
equals the associated increased commuting cost:
 P u H  t
The Basic Urban Model
Location Choice
The Basic Urban Model
A Puzzle
 Can you see a problem with this result?
 Every household is alike, so every household picks the
same u*.
 How can we have a city in which everyone lives the
same distance from the CBD?
 We can’t!
The Basic Urban Model
The Urban Model Twist
 The key to solving this puzzle is to recognize that the
form of P{u} is determined by the market.
 Hence, the equation for a household’s location choice
can be interpreted as a equilibrium condition for P{u},
that is, as a locational equilibrium condition.
 If P{u} meets this condition, then every household will
be satisfied no matter where it lives and no household
will have an incentive to move.
The Basic Urban Model
The Locational Equilibrium Condition
 The locational equilibrium condition is written:
P u  .
 Because t and H are positive, the slope of P{u} must be
 Econ 101 tells us that H declines with P, so the slope
must get flatter as u increases.
The Basic Urban Model
The Bid Function for Housing
(Price per Unit of Housing Services)
Slope = ΔP/Δu
= -t/H
The Basic Urban Model
Features of Equilibrium
 1. Note that the model does not say where any given
household will live.
 Location choices are now idiosyncratic and outside the
 2. Note also, that the locational equilibrium condition
defines a family of bid functions.
 Higher bid functions correspond to a higher cost of
living—and a lower utility level.
The Basic Urban Model
A Family of Bid Functions
The Basic Urban Model
Looking Ahead
 Next time we will see how this household problem fits
into equations for the other markets in an urban area.
 The result is a general equilibrium model of urban
residential structure.

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