ECN741: Urban Economics The Basic Urban Model: 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 price of land is a derived land.  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 H S {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, :

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

The Basic Urban Model Anchoring P{u}  Now using the relationship between R{u} and P{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}, H S {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 area 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 H S {u}.  For H{u}, use the demand function, but put in P{u}=R{u} a /C.  For H S {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 H S {u} t o 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 c 1 = Y, c 2 = -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