ECN741: Urban Economics The Basic Urban Model 2: Solutions.

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


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.


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


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


Housing Demand, 2

With a Cobb-Douglas utility function,

and

so

1U

Z Z

U

H H

1{ } .Z P u H


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


Housing Demand, 3

These conditions imply that

1 .Z Y tu

.Y tu

HP u


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 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


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


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 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 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


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


Differential Equation Approach, 3

The result:

or

11n 1n .P u Y t u

1 1P u e Y tu Y tu


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


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


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!


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


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


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


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


Determining the Outer Edge of the Urban Area

R(u)

_

R

CBD u u

-


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


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


A Complete Urban Model

So now we can pull equations together for the 6 markets

Housing Land Capital Transportation Labor Export Good


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


Land

Demand

Supply

[Ownership: Rents go to absentee landlords.]

.L u u

saP u H u

R uL u


The Capital Market

Demand

Supply: r is constant

1 sa P u H ur

K u


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.


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.


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


Locational Equilibrium

The bid function

The anchoring condition

1

1

*

kP u Y tu

U

.R u R


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 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.


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*.


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.


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.


Solving a Closed Model, 2

These steps lead to:

where

1 a

sH u DR u L u

11

aa

D Aa r



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



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 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 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 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 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


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


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 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!!


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


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

ECN741: Urban Economics The Basic Urban Model 2: Solutions.

Documents

Transcript of ECN741: Urban Economics The Basic Urban Model 2: Solutions.