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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:

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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

The Basic Urban Model

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

    • 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


    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

    • These steps lead to:

    • 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

    • Thus the answer is

      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


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