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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, 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

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

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

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}, 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, :

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

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

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, 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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