ECN741: Urban EconomicsThe Basic Urban Model 2: Solutions

The Basic Urban Model

Motivation for Urban Models

Urban models are built on the following simple sentence:

People care about where they live because they must commute to work.

This sentence contains elements of 6 markets:

Housing Land Capital Transportation Labor Export good

The Basic Urban Model

Motivation for Urban Models, 2

So now we are going to write down equations for these 6 markets.

It is difficult to solve a general equilibrium model with 6 markets.

That is why we rely on the strong assumptions discussed in the previous class.

Moreover, the best way to understand a complex system is to write down a simple version and then try to make it more general.

That is what we will do in this course.

The Basic Urban Model

Housing Demand

A household maximizes

Subject to

where

1 ln ln .U Z H

.Y Z P u H tu

0 yt t t Y

The Basic Urban Model

Housing Demand, 2

Recall from the last class that the Lagrangian for this problem is:

And the first-order conditions for Z and H imply that

, ,U Z H Y Z P u H tu

/.

/ H

U HMB P u

U Z

The Basic Urban Model

Housing Demand, 2

With a Cobb-Douglas utility function,

and

so

1U

Z Z

U

H H

1{ } .Z P u H

The Basic Urban Model

Housing Demand, 2

Now add the first-order condition with respect to λ:

Combining results:

0.Y Z P u H tu

1{ } 0.Y P u H P u H tu

The Basic Urban Model

Housing Demand, 3

These conditions imply that

1 .Z Y tu

.Y tu

HP u

The Basic Urban Model

Deriving a Bid Function

A bid function, P{u}, can be derived in two different ways:

The indirect utility function approach, pioneered by Robert Solow

The differential equation approach, in Alonso, Muth, Mills.

The best approach depends on the context!

The Basic Urban Model

The Indirect Utility Function Approach

Substitute the demands for H and Z into the exponential form for the utility function:

where

-k Y tuU

P u=

11k

The Basic Urban Model

An Aside: Transformation of Utility Functions

Any positive monotonic transformation of a utility function is also a utility function that represents the same preferences.

Thus, demand functions are not affected by a positive monotonic transformation of the utility function.

For example, the following three utility functions yield the same demands:

1 2

1

ˆ ln{ } ln{ }

(1 ) ln{ } ln{ }

U Z H

U Z H

U Z H

The Basic Urban Model

Indirect Utility Function Approach, 2

All household receive the same utility level, U*, so

or

The height of the bid function, γ, obviously depends on the utility level, U*.

1

1

*

kP u Y tu

U

1P u Y tu

The Basic Urban Model

The Locational Equilibrium Condition

Remember from last class: The price of housing adjusts so that, no matter where someone lives, savings in housing costs from moving one mile further out exactly offsets the increased commuting costs.

The savings in housing costs is:

The increase in commuting costs is just t.

P u H

The Basic Urban Model

The Differential Equation Approach

Thus, the locational equilibrium condition is:

Now substitute in the demand for housing to obtain the differential equation:

or .tP u P u t

P uY tu P u Y tu

tP u

H

The Basic Urban Model

Differential Equation Approach, 2

This is an exact differential equation. It has the function, P{u} on one side and the argument, u, on the other.

It can be solved simply by integrating both sides.

The key integral is: 1n

g udu g u

g u

The Basic Urban Model

Differential Equation Approach, 3

The result:

or

11n 1n .P u Y t u

1 1P u e Y tu Y tu

The Basic Urban Model

Housing Supply

The housing production function is assumed to take the Cobb-Douglas form:

where the “S” subscript indicates aggregate supply at location u, K is capital and L is land.

Because this is a long-run model, the role of labor in housing production is ignored.

1 a a

sH u AK u L u

The Basic Urban Model

Input Demand

Profit-maximizing forms set the value of the marginal product of each input equal to its price:

saP u H u

R uL u

1constantsa P u H u

rK u

The Basic Urban Model

Note on Land Prices

Note that the demand for land is a derived demand.

In residential use, the price of land is determined by the price of housing.

Land at a given location has value because someone is willing to pay for housing there.

It is not correct to say that someone has to pay a lot for housing because the price of land is high!

The Basic Urban Model

Solving for R{u}

Now solve the input market conditions for K{u} and L{u} and plug the results into the production function:

1

1

1

1.

aa

s ss

asaa

a P u H u aP u H uH u A

r R u

P u H uaAa

r R u

The Basic Urban Model

Solving for R{u}, 2

Now HS{u} obviously cancels and we can solve for:

or

where

aR u CP u

1 aR u CP u

11

aa a

C Aar

The Basic Urban Model

Solving for R{u}, 3

Combining this result with the earlier result for P{u}:

This function obviously has the same shape as P{u}, but with more curvature.

111

1*

aa

a

R u C Y tu

Y tu

The Basic Urban Model

Anchoring R{u}

Recall that we have derived families of bid functions, P{u} and R{u}.

The easiest way to “anchor” them, that is, to pick a member of the family, is by introducing the agricultural rental rate, , and the outer edge of the urban area, :

.R u R

Ru

The Basic Urban Model

Determining the Outer Edge of the Urban Area

R(u)

_

R

CBD u u

-

The Basic Urban Model

Anchoring R{u}, 2

This “outer-edge” condition can be substituted into the above expression for R{u} to obtain:

With this constant, we find that

*

11 aa

R

C Y tu

1 a

Y tuR u R

Y tu

The Basic Urban Model

Anchoring P{u}

Now using the relationship between R{u} and P{u},

where the “opportunity cost of housing” is

1

Y tuP u P

Y tu

aR

PC

The Basic Urban Model

A Complete Urban Model

So now we can pull equations together for the 6 markets

Housing Land Capital Transportation Labor Export Good

The Basic Urban Model

Housing

Demand

Supply

D = S

where N{u} is the number of households living at location u.

{ } { } { }SN u H u H u

Y tuH

P u

1 a a

sH u AK u L u

The Basic Urban Model

Land

Demand

Supply

[Ownership: Rents go to absentee landlords.]

.L u u

saP u H u

R uL u

The Basic Urban Model

The Capital Market

Demand

Supply: r is constant

1 sa P u H ur

K u

The Basic Urban Model

The Transportation Market

T{u} = tu

Commuting cost per mile, t, does not depend on

▫ Direction▫ Mode▫ Road Capacity▫ Number of Commuters

Results in circular iso-cost lines—and a circular city.

The Basic Urban Model

Labor and Goods Markets

All jobs are in the CBD (with no unemployment)

Wage and hours worked are constant, producing income Y.

This is consistent with perfectly elastic demand for workers—derived from export-good production.

Each household has one worker.

The Basic Urban Model

Labor and Goods Markets, 2

N{u} is the number of households living a location u.

The total number of jobs is N.

So

_

0

.u

N u du N

The Basic Urban Model

Locational Equilibrium

The bid function

The anchoring condition

1

1

*

kP u Y tu

U

.R u R

The Basic Urban Model

The Complete Model

The complete model contains 10 unknowns:

H{u}, HS{u}, L{u}, K{u}, N{u}, P{u}, R{u}, N, , and U*

It also contains 9 equations:

(1) Housing demand, (2) housing supply, (3) housing S=D, (4) capital demand, (5) land demand, (6) land supply, (7) labor adding-up condition, (8) bid function, (9) anchoring condition.

u

The Basic Urban Model

The Complete Model, 2

Note that 7 of the 10 variables in the model are actually functions of u.

An urban model is designed to determine the residential spatial structure of an urban area, so the solutions vary over space.

In the basic model there is, of course, only one spatial dimension, u, but we will later consider more complex models.

The Basic Urban Model

Open and Closed Models

It is not generally possible to solve a model with 9 equations and 10 unknowns.

So urban economists have two choices:

Open Models:

▫ Assume U* is fixed and solve for N.

Closed Models:

▫ Assume N is fixed and solve for U*.

The Basic Urban Model

Open and Closed Models, 2

Open models implicitly assume that an urban area is in a system of areas and that people are mobile across areas.

Household mobility ensures that U* is constant in the system of areas (just as within-area mobility holds U* fixed within an area).

Closed models implicitly assume either

(1) that population is fixed and across-area mobility is impossible,

or (2) that any changes being analyzed affect all urban areas equally, so that nobody is given an incentive to change areas.

The Basic Urban Model

Solving a Closed Model

The trick to solving the model is to go through N{u}.

Start with the housing S=D and plug in expressions for H{u} and HS{u}.

For H{u}, use the demand function, but put in P{u}=R{u}a/C.

For HS{u}, plug K{u} (from its demand function) and the above expression for P{u} into the housing production function.

The Basic Urban Model

Solving a Closed Model, 2

These steps lead to:

where

1 a

sH u DR u L u

11

aa

D Aa r

The Basic Urban Model

Solving a Closed Model, 3

Now plug in the supply function for L{u} and the “anchored” form for R{u} into the above. Then the ratio of HS{u} to H{u} is:

1 1

1

a

a

R u Y tuN u

a Y tu

The Basic Urban Model

Solving a Closed Model, 4

Substituting this expression for N{u} into the “adding up” condition gives us the integral:

Note: I put a bar on the N to indicate that it is fixed.

1 1

10

au

a

R u Y tudu N

a Y tu

The Basic Urban Model

The Integral

Here’s the integral we need:

where c1 = Y, c2 = -t, and n = [(1/aα)-1].

2

1 2 1 22

2

1

1 22

1

1 2

1

n n

n

u c c u du c c uc n n

uc c u

c n

The Basic Urban Model

The Integral, 2

Thus the answer is

where b = 1/aα and the right side must be evaluated at 0 and .

1

1 20 0

,1

ub bu

a

Y tu u Y tuR bN u du

tbY tu t b b

u

The Basic Urban Model

The Integral, 3

Evaluating this expression and setting it equal to yields:

A key problem:

This equation is so nonlinear that one cannot solve for (the variable) as a function of (the parameter).

N

1

11

b

b

R Y Y tuN u

t t bt b Y tu

uN

The Basic Urban Model

The Problem with Closed Models

One feature of closed models is convenient:

The utility level is not needed to find anything else.

But another feature makes life quite difficult:

As just noted, the population integral cannot be explicitly solved for .

This fact (and even more complexity in fancier models) leads many urban economists to use simulation methods.

u

The Basic Urban Model

Solving an Open Model

The equations of open and closed models are all the same.

However, one equation plays a much bigger role in an open model, namely, the key locational equilibrium condition, because U* is now a parameter (hence the “bar”), not a variable. *

a

k Y tu k Y tuU

P RC

The Basic Urban Model

Solving an Open Model, 2

This equation can be solved for as a function of parameters of the model.

This makes life a lot easier! This expression can be plugged into the solution to the integral to get N, which is now a variable.

u

* *aP U R U

Y Yk C ku

t t

The Basic Urban Model

The Problem with Open Models

Open models are much easier to solve than are closed models.

The problem is that they address a much narrower question, namely what happens when there is an event in one urban area but not in any other.

Be careful to pick the model that answers the question you want to answer—not the model that is easier to solve!!

The Basic Urban Model

Density Functions

A key urban variable is population density, which can be written D{u} = N{u}/ L{u}.

Our earlier results therefore imply that:

This function has almost the same shape as R{u} and, as we will see, has been estimated by many studies.

R u

D ua Y tu

The Basic Urban Model

Building Height

The model also predicts a skyline for housing, as measured by building height—a prediction upheld by observation!

One measure of building height is the capital/land ratio, or K{u}/L{u}, which can be shown to be

where

K u

qR uL u

11

aA a

qC r