ECN741: Urban Economics The Basic Urban Model: Assumptions.

ECN741: Urban EconomicsThe 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


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 economics.

He also, by the way, invented general equilibrium analysis!


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.


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 model.


0 2 4 6 8 10 12 14 16 18 20

von Thünen's Model of Rents and Locations

Rent for Milk/Vegetables Rent for Wood Rent for GrainRent for Livestock

Distance from Central Market (miles)

Annual

Ren

t per

Acr

e

City Milk Wood Grain Livestock


Alonso

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.


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).


Urban Models

Urban models are designed to explain why urban areas look the way they do, with a focus on housing.

Where do people live? How far do they commute?

How much housing do they consume?

What is the price of housing?


Key Assumptions

We now turn to the key assumptions of a basic urban model.

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 introduced.


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


1. Housing Demand

Household utility functions depend on a composite consumption good, Z, and housing, H, and take the Cobb-Douglas form.

H is measured in units of housing services = quality adjusted square feet.

(1 )ln{ } ln{ },U Z H


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.

Housing is owned by absentee landlords.

Maintenance, rehabilitation, and conversion activities are ignored.

1 a aH K L


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 Thünen!).

There are no commuting arteries or street grids.

There is only one transportation mode (=car?)


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.


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.


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.


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.


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!


Preliminaries

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 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


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

where

*

1

{ } { }

1

M

yy

P u H P u H P u HV

i ii

* .

1 1M

ii

i


The Household Problem

The household problem is to

Maximize U{Z, H}

Subject to Y = Z + P{u}H + tu

The Lagrangian is

, ,U Z H Y Z P u H tu


The First-Order Conditions

The key first-order conditions are:

0,U

Z Z

0,U

P uH H

0.P u H tu


Implications

The first two conditions imply that

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:

/.

/ H

U HMB P u

U Z

P u H t


Location Choice

$

u

u*

t

-Pʹ{u}H


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 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 Locational Equilibrium Condition

The locational equilibrium condition is written:

Because t and H are positive, the slope of P{u} must be negative.

Econ 101 tells us that H declines with P, so the slope must get flatter as u increases.

.t

P uH


The Bid Function for Housing(Price per Unit of Housing Services)

Slope = ΔP/Δu

= -t/H

P(u)

CBD u

ΔP

Δu


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 model.

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.


A Family of Bid Functions

P(u)

CBD u

LowerUtility

HigherUtility


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.

ECN741: Urban Economics The Basic Urban Model: Assumptions.

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Transcript of ECN741: Urban Economics The Basic Urban Model: Assumptions.