ECN741: Urban Economics

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

2. 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

3. 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.

4. The Basic Urban Model Housing Demand • A household maximizes • Subject to • where

5. 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

6. The Basic Urban Model Housing Demand, 2 • With a Cobb-Douglas utility function, and so

7. The Basic Urban Model Housing Demand, 2 • Now add the first-order condition with respect to λ: • Combining results:

8. The Basic Urban Model Housing Demand, 3 • These conditions imply that

9. 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!

10. 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

11. 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*.

12. 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.

13. 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:

14. 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:

15. The Basic Urban Model Differential Equation Approach, 3 • The result: or

16. 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.

17. The Basic Urban Model Input Demand • Profit-maximizing forms set the value of the marginal product of each input equal to its price:

18. 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!

19. 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:

20. The Basic Urban Model Solving for R{u}, 2 • Now HS{u} obviously cancels and we can solve for: or where

21. 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.

22. 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, :

23. Determining the Outer Edge of the Urban Area The Basic Urban Model R(u) _ R • CBD u u -

24. 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

25. The Basic Urban Model Anchoring P{u} • Now using the relationship between R{u} andP{u}, where the “opportunity cost of housing” is

26. 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

27. The Basic Urban Model Housing • Demand • Supply • D = S where N{u} is the number of households living at location u.

28. The Basic Urban Model Land • Demand • Supply • [Ownership: Rents go to absentee landlords.]

29. The Basic Urban Model The Capital Market • Demand • Supply: r is constant

30. 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.

31. 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.

32. 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

33. The Basic Urban Model Locational Equilibrium • The bid function • The anchoring condition

34. 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.

35. 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.

36. 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*.

37. 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.

38. 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.

39. The Basic Urban Model Solving a Closed Model, 2 • These steps lead to: • where

40. 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:

41. 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.

42. The Basic Urban Model The Integral • Here’s the integral we need: where c1 = Y, c2 = -t, and n = [(1/aα)-1].

43. 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 .

44. 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).

45. 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.

46. 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.

47. 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.

48. 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!!

49. 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.

50. 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