Understanding Urban Economics: Building the Basic Model

1. ECN741: Urban Economics The Basic Urban Model 2: Solutions Professor John Yinger, The Maxwell School, Syracuse University, 2019

2. The Basic Urban Model Class Outline • 1. Motivation for Urban Models • 2. Housing Demand • 3. Deriving a Bid Function • 4. Housing Supply • 5. Anchoring Bid Functions • 6. A Complete Urban Model • 7. Solving Open and Closed Models

3. The Basic Urban Model Class Outline • 1. Motivation for Urban Models • 2. Housing Demand • 3. Deriving a Bid Function • 4. Housing Supply • 5. Anchoring Bid Functions • 6. A Complete Urban Model • 7. Solving Open and Closed Models

4. 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

5. 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 the previous class. • 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 throughout this course.

6. The Basic Urban Model Class Outline • 1. Motivation for Urban Models • 2. Housing Demand • 3. Deriving a Bid Function • 4. Housing Supply • 5. Anchoring Bid Functions • 6. A Complete Urban Model • 7. Solving Open and Closed Models

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

8. 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

9. The Basic Urban Model Housing Demand, 3 • With a Cobb-Douglas utility function, and so

10. The Basic Urban Model Housing Demand, 4 • Now add the first-order condition with respect to λ: • Combining results:

11. The Basic Urban Model Housing Demand, 5 • These conditions imply that • These are standard Cobb-Douglas results, except that P varies with u and the income term is net of commuting costs to u.

12. The Basic Urban Model Class Outline • 1. Motivation for Urban Models • 2. Housing Demand • 3. Deriving a Bid Function • 4. Housing Supply • 5. Anchoring Bid Functions • 6. A Complete Urban Model • 7. Solving Open and Closed Models

13. 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 (Swedish J. of Econ., March 1972). • The differential equation approach, in Alonso, Muth, Mills. • The best approach depends on the context!

14. 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

15. The Basic Urban Model An Aside: Transformation of Utility Functions • Any positive monotonic transformation of a utility function is also a utility function that represents the same preferences. • Thus, demand functions are not affected by a positive monotonic transformation of the utility function. • For example, the following three utility functions yield the same demands:

16. 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*.

17. 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.

18. 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:

19. 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:

20. The Basic Urban Model Differential Equation Approach, 3 • The result: or • Not surprisingly, both approaches yield the same answer!

21. The Basic Urban Model Class Outline • 1. Motivation for Urban Models • 2. Housing Demand • 3. Deriving a Bid Function • 4. Housing Supply • 5. Anchoring Bid Functions • 6. A Complete Urban Model • 7. Solving Open and Closed Models

22. 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, both of which vary with location. • Because this is a long-run model, the role of labor in housing construction is ignored.

23. The Basic Urban Model Input Demand • Profit-maximizing forms set the value of the marginal product of each input equal to its price: • Note that r is the area’s capital rental rate, and R{u} is rent per unit of land per unit of time at a location u miles from the CBD.

24. 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!

25. 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:

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

27. 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, because it has 2 forms of substitution—in production and in utility.

28. The Basic Urban Model Class Outline • 1. Motivation for Urban Models • 2. Housing Demand • 3. Deriving a Bid Function • 4. Housing Supply • 5. Anchoring Bid Functions • 6. A Complete Urban Model • 7. Solving Open and Closed Models

29. 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, :

30. R{u} Determining the Outer Edge of the Urban Area The Basic Urban Model u CBD _ R _ u

31. 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

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

33. The Basic Urban Model Class Outline • 1. Motivation for Urban Models • 2. Housing Demand • 3. Deriving a Bid Function • 4. Housing Supply • 5. Anchoring Bid Functions • 6. A Complete Urban Model • 7. Solving Open and Closed Models

34. 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

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

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

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

38. The Basic Urban Model The Transportation Market • T{u} = tu = (t0 + tYY)u • Commuting cost per mile, t, does not depend on • Direction • Mode • Road Capacity • Number of Commuters. • These assumptions imply circular iso-commuting-cost lines—and a circular city.

39. 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.

40. 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

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

42. 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.

43. 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.

44. The Basic Urban Model Class Outline • 1. Motivation for Urban Models • 2. Housing Demand • 3. Deriving a Bid Function • 4. Housing Supply • 5. Anchoring Bid Functions • 6. A Complete Urban Model • 7. Solving Open and Closed Models

45. 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*.

46. 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.

47. The Basic Urban Model Solving a Closed Model • The trick to solving the model is to go through N{u}=HS{u}/H{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.

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

49. 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:

50. 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 N to indicate that it is fixed.