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# ECN741: Urban Economics PowerPoint PPT Presentation

ECN741: Urban Economics. The 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:

ECN741: Urban Economics

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## ECN741: Urban Economics

The 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 previous classes.

• 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 later in this class.

### The Basic Urban Model

Housing Demand

• A household maximizes

• Subject to

• where

### 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

### The Basic Urban Model

Housing Demand, 2

• With a Cobb-Douglas utility function,

and

so

### The Basic Urban Model

Housing Demand, 2

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

• Combining results:

### The Basic Urban Model

Housing Demand, 3

• These conditions imply that

### 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

### 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*.

### 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.

### 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:

### 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:

### The Basic Urban Model

Differential Equation Approach, 3

• The result:

or

### 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.

### The Basic Urban Model

Input Demand

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

### 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:

### The Basic Urban Model

Solving for R{u}, 2

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

or

where

### 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.

### 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, :

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

### The Basic Urban Model

Anchoring P{u}

• Now using the relationship between R{u} andP{u},

where the “opportunity cost of housing” is

### 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.

### The Basic Urban Model

Land

• Demand

• Supply

• [Ownership: Rents go to absentee landlords.]

### The Basic Urban Model

The Capital Market

• Demand

• Supply: r is constant

### The Basic Urban Model

The Transportation Market

• T{u} = tu

• Commuting cost per mile, t, does not depend on

• Direction

• Mode

• 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

### The Basic Urban Model

Locational Equilibrium

• The bid function

• The anchoring condition

### 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.

### 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

• where

### 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:

### 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.

### The Basic Urban Model

The Integral

• Here’s the integral we need:

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

### The Basic Urban Model

The Integral, 2

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

and .

### 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).

### 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.

### 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.

### 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.

### 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.

### The Basic Urban Model

Building Height

• The model also predicts a skyline, 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