1. ECN741: Urban Economics Household Heterogeneity Professor John Yinger, The Maxwell School, Syracuse University, 2016

2. Household Heterogeneity Class Outline  1. Simple models with more than one income-taste class  2. Normal sorting  3. Can an urban model predict where the poor live?  4. Bid-function envelopes with a general treatment of household heterogeneity

3. Household Heterogeneity Class Outline  1. Simple models with more than one income-taste class  2. Normal sorting  3. Can an urban model predict where the poor live?  4. Bid-function envelopes with a general treatment of household heterogeneity

4. Household Heterogeneity More than One Income-Taste Class  Consider a basic open urban model with two income- taste classes.  The bid-functions for the two classes are:

5. Household Heterogeneity Who Wins the Bidding Contest?  The key to putting these two bid functions into an urban model is to recognize that, in the long run, the seller cares about P, not about PH.  The value of H can be altered; houses can be split into apartments; apartments can be combined.  The seller wants to obtain the highest return for whatever level of H she provides.  So the winning household type at u has the highest P{u}!

6. Household Heterogeneity Who Wins the Bidding Contest?, 2  This leads to the concept of sorting: Households sort into different locations based on their bids.  In the following picture (which has household types 1 and 2 instead of A and B ), household type 1, which has a steeper bid function, lives inside u*.  And households type 2, which has a flatter bid function, lives outside u*.  A key principle: Sorting depends on bid-function slopes.

7. Household Heterogeneity

8. A Note on the Literature  The idea of sorting comes from von Th ü nen, and a formal model of sorting based on bid functions can be found in Alonso’s book.  The basic logic of a sorting based on the opportunity cost of time was provided by Becker (The Economic Journal, September 1965).  This type of sorting was also highlighted in the first edition of Mills’ urban textbook (Urban Economics, Scott-Foresman, 1972).

9. Household Heterogeneity Solving a Two-Class Model  In an urban model context, the household type with the steeper bid function wins the competition in the locations closer to the center.  With the standard single-crossing assumption, bid functions cross only once, so sorting is simplified.  This model has a new variable, u*, the boundary between the residential areas of the two classes; it is determined by setting the two bid functions equal at u*.

10. Household Heterogeneity Solving a Two-Class Model, 2  With an open model, the heights of the two bid functions are set by the utility levels in the system.  With a closed model, the heights have to be adjusted until there is enough room from 0 to u* for the fixed number of people in the “inner” class and enough from u* to for the people in the other class.

11. Household Heterogeneity Solving a Multi-Class Model  The logic of a two-class model can easily be generalized to as many classes as one wants.  Each new class adds a new boundary and a new boundary condition.  In principle, one could solve a model with rich young adults who don’t care about housing (high Y, low α), poor families (low Y, high α), and high-income families (high Y, high α).  But discrete household types make estimation difficult.

12. Household Heterogeneity Implications for H  Sorting depends on P but has implications for H.  If poor households win the competition for housing inside u*, then smaller units of housing will exist there.  In other words, poor households win the competition by accepting small units—and paying a lot for them per unit of H.

13. Household Heterogeneity Implications for Neighborhood Change  Bid function graphs are a great tool for starting to think about neighborhood change.  When the bid function of one group shifts, through immigration, for example, the location of u* will change.  Changes in u* require changes in the housing stock.  To use the earlier examples, by converting a house into apartments or combining apartments into a bigger unit.

14. Household Heterogeneity Neighborhood Change

15. Household Heterogeneity Implications for Neighborhood Change, 2  This graph oversimplifies, of course.  Neighborhood change is more likely where the housing stock can be converted relatively cheaply.  The shift in this graph squeezes the rich, so their bid function will shift upward and they will reclaim (as a matter of logic, not time) some of the area that is converted in this picture.  Ultimately, changes in density and inhabited area must be sufficient to provide enough housing for each group. Remember the logic of complete urban models!

16. Household Heterogeneity Class Outline  1. Simple models with more than one income-taste class  2. Normal sorting  3. Can an urban model predict where the poor live?  4. Bid-function envelopes with a general treatment of household heterogeneity

17. Household Heterogeneity Normal Sorting  The natural question to ask is whether rich or poor people live closer to the CBD.  This question, like so many others, is addressed in Alonso (although the treatment here differs somewhat from his).  A situation in which poor people live closer is often called “normal” sorting—meaning what is expected, not what is desirable!  Under what circumstances does normal sorting arise?

18. Household Heterogeneity Normal Sorting, 2  To determine the condition for normal sorting, differentiate the standard equilibrium condition with respect to Y, recognizing that t and H are functions of Y.

19. Household Heterogeneity Normal Sorting, 3  Normal sorting requires this derivative to be positive, which means that the bid function flattens with Y :

20. Household Heterogeneity Normal Sorting, 5  The left side is the elasticity of t with respect to Y ; the right side is the elasticity of H with respect to Y.  An increase in income raises t (higher opportunity cost of time) and therefore increases the compensation required in the form of P′.  But an increase in Y also raises H and therefore allows the compensation to be spread out over more units of H.  The net increase in P′ is positive if the former effect is smaller than the second.

21. Household Heterogeneity Normal Sorting, 6  Consider the case in a basic urban model. If Then t does not increase proportionally with Y ; that is

22. Household Heterogeneity Normal Sorting, 6  Moreover, with a Cobb-Douglas utility function, the income elasticity of demand for H equals 1.0.  So the condition for normal sorting holds by definition.  But t 0 might depend on Y, as well:  The rich might buy big cars that use more gas per mile  Or they might avoid old, inefficient cars and buy Priuses!

23. Household Heterogeneity Class Outline  1. Simple models with more than one income-taste class  2. Normal sorting  3. Can an urban model predict where the poor live?  4. Bid-function envelopes with a general treatment of household heterogeneity

24. Household Heterogeneity Is Normal Sorting the Norm?  More generally, the existing empirical evidence does not give a definitive answer concerning the values of these elasticities.  We saw studies that indicated income elasticities of demand for H in the range of 0.3 to 0.7.  Operating costs are often found to be only 15% of travel costs, so the above formula (with operating costs not a function of income) indicates that the income elasticity of t is 0.85.  These values indicate that normal sorting will not occur.

25. Household Heterogeneity Is Normal Sorting the Norm?, 2  Casual evidence indicates, however, that poor people tend to live in cities, not suburbs.  There is more formal evidence in Glaeser, Kahn, and Rappaport (henceforth GKR; JUE, January 2008), including the following two figures and table.  So the question is:  If the elasticity condition is not met, why do the poor live in cities?

26. Household Heterogeneity Source: Glaeser, Kahn, and Rappaport

27. Household Heterogeneity Source: Glaeser, Kahn, and Rappaport

28. Household Heterogeneity Source: Glaeser, Kahn, and Rappaport

29. Household Heterogeneity Sorting and Mode Choice  The resolution proposed by LeRoy and Sonstelie (JUE January 1983) and GKR is mode choice.  People who live in the city generally use a slow (=high- cost) mode, namely, public transit.  People who live in the suburbs generally use a fast (=low-cost) mode, namely, cars.  These mode choices are linked to the value of time— and hence to income.

30. Household Heterogeneity Sorting and Mode Choice, 2  These scholars link modes to speed and hence to the time cost of travel.  High-income people generally have a higher opportunity cost of travel because of their high income, but they may have a lower cost of travel if they travel faster.  What is the impact on sorting.  Start with a one-mode (=car) solution in which bid-function slopes (- t/H ) do not change with income, which implies no sorting.  Now add public transit and suppose high-income people commute in (fast) cars and low-income people commute on the (slow) bus.  This choice reflects the fact that a car is expensive, and only high-income people are willing to pay for one.  A bus has fixed costs, too, in to form of a monthly pass, but this cost is much lower (and is assumed not to vary with distance traveled).

31. Household Heterogeneity Sorting and Mode Choice, 3  With these assumptions, the bid function of low-income people gets steeper.  This changes in slope reflect the change in time cost brought on by the change in commuting speed.  The impact on the pattern of sorting depends on the nature of the transportation network.  If public transit is concentrated near the CBD (as seems likely since public transit needs a high population density to be feasible), low- income people, with their steep bid functions win the competition near the CBD and high-income people win in the suburbs (= normal sorting!).  If public transit is spread out, however, low-income people win the competition for housing near bus lines, and high-income people win the competition elsewhere ( ≠ normal sorting).

32. Household Heterogeneity Sorting and Mode Choice, 4  GKR miss this last result, by the way.  They say (p. 8) that “This assumes that public transportation is available everywhere; if public transportation was only accessible close to the city center, then this would further increase the tendency of the poor to centralize.”  In fact, the normal sorting result only holds if public transportation, which is what low-income people use, is centralized.  It is possible to have weak normal sorting with a semi- centralized public transportation, but that is not what GKR are talking about.

33. Household Heterogeneity Sorting and Mode Choice, 5  It is also worth pointing out that the GKR approach is incomplete; it describes an equilibrium in which high- income people drive cars, but it does not show how bidding leads to this equilibrium.  As discussed in the class on transportation costs, a formal model of mode-choice was provided by Anas and Moses (JUE, April 1979).  Moreover, income-based sorting models were discussed earlier in this class.  But no scholar has put the two together.  So here is a highly simplified version of how such a model might work.

34. Household Heterogeneity Sorting and Mode Choice, 6  The introduction of a slow, centralized public transit system has several impacts on bid functions:  It leads to dual bid functions for each household type—one for car travel and one for transit; the one for transit is steeper because it involve slower travel.  It shifts the bid function for transit to a higher intercept than the one for car travel because the income loss is much greater for a car and the two bid functions must correspond to the same utility level. This shift is larger for low-income households, who have trouble affording a car.  It leads to a transit bid function that stops at the outer edge of the transit system, not the outer edge of the area.

35. Household Heterogeneity Sorting and Mode Choice, 7 Larger Difference in Bid Functions for Low- Income Households

36. Household Heterogeneity Sorting and Mode Choice, 8  Start with the two high-income bid functions.  Some high-income households would live centrally and pick transit.  But the high-income bid-functions are above the low- income bid-functions everywhere.  So we must raise the low-income bid function until they win the competition for housing somewhere.  Because of the greater gap between the two low-income bid functions, an upward shift in the two low-income curves (remember they are linked by utility) leads to higher low-income bids in the central area based on transit.

37. Household Heterogeneity Sorting and Mode Choice, 9  The outer edge of the transit system ( u *) might not correspond to the outer edge of the area inhabited by low- income households.  If there is not enough room for low-income households inside u *, low-income people will compete against each other and drive up the price of housing, resulting in a price gap at u *. But low- income people will not switch to cars.  Some high-income households might live inside u * but still drive. High-income people will not switch to transit.  An area-wide public transit system would lead to sorting near transit stations and not lead to normal sorting.

38. Household Heterogeneity Class Outline  1. Simple models with more than one income-taste class  2. Normal sorting  3. Can an urban model predict where the poor live?  4. Bid-function envelopes with a general treatment of household heterogeneity

39. Household Heterogeneity More General Treatment of Heterogeneity  Beckman (JET, June 1969) suggests and Montesano (JET, April 1972) derives a bid function based on the assumption that household income has a Pareto distribution.  Unfortunately, however, this approach can only be solved for one special case and even then leads to a very complex housing price function.

40. Household Heterogeneity Bid Function Envelopes  Another approach is to take advantage of the fact that the observed relationship between housing prices and distance is the mathematical envelope of the underlying bid functions.  We cannot observe the bid functions for a given household type. We can only observe the highest bid at each distance.  See Yinger, “The Price of Access to Jobs: Bid Function Envelopes for Commuting Costs,” Working Paper, Syracuse University, June 2013, Available at: http://faculty.maxwell.syr.edu/jyinger/ebooks/housing_co mmuting/Ch_3.4.pdf. http://faculty.maxwell.syr.edu/jyinger/ebooks/housing_co mmuting/Ch_3.4.pdf

41. Household Heterogeneity Bid Functions Envelope Bid-Rent Functions and Their Envelope

42. Household Heterogeneity Taking Envelopes Seriously  Before presenting the Yinger approach, it is worth pointing out that most empirical studies of commuting and housing prices have simply ignored household heterogeneity.  These studies include distance from a worksite in a regression with the log of house value as the dependent variable and then interpret the coefficient as a measure of the compensation required as a household moves farther from the worksite.  But any such estimate combines two factors:  the degree of compensation that is needed for a household at each location  and the change in bids that arises as households sort—that is, as one type of household replaces another.

43. Household Heterogeneity

44. The Origin of Envelopes  An aside on the intellectual history of this stuff:  As discussed earlier, sorting appears in von Thünen (1826!) with an implicit envelope.  Alonso (1964) recognized that household heterogeneity would lead to sorting; he also recognizes that market prices are an “envelope” and introduces the single-crossing condition.  Gary Becker, in his famous paper on the allocation of time (EJ, 1965), has a paragraph with a clear statement about sorting by income using commuting time, not distance.  Muth and Mills clearly discuss sorting by income, but the Mills/Hamilton textbook interprets the coefficient of a distance variable as required compensation!

45. Household Heterogeneity The Mathematics of Envelopes  An envelope is a function that satisfies 2 equations: where α is the parameter that varies with the family of curves and a subscript indicates a partial derivative.  A simple example is given in the following figure.

46. Household Heterogeneity The Mathematics of Envelopes, 2

47. Household Heterogeneity Solving for the Envelope  The Yinger working paper is able to solve for the functional form of the envelope with three main assumptions.  The first assumption concerns transportation costs.  He draws on a model with leisure time (e.g. DeSalvo, Journal of Regional Science, May 1985) to introduce the time cost of commuting as a fraction, λ, of the wage. Note that λ, but not the wage rate, is assumed to be constant across households.  He recognizes that “location” can reflect either distance, u, or time, v, and he uses the symbol m to mean either u or v.

48. Household Heterogeneity Solving for the Envelope, 2  The second assumption concerns the demand for housing.  He notes that operating costs are not fixed, but depend on household decisions (whether to take the bus, what type of car to buy) and treats them as part of Z in the budget constraint—not part of net income.  He assumes a constant elasticity demand function for H with income and price elasticities, γ and η.

49. Household Heterogeneity Solving for the Envelope, 3  With these assumptions, we can derive a bid function.  In an earlier class we derived a bid function based on a constant elasticity demand function for H with a unitary price elasticity, which simplifies the math but is not necessary. Using this approach here, we find that where C is a constant.

50. Household Heterogeneity Solving for the Envelope, 4  The punchline here is that with these assumptions households are heterogeneous in three ways: (1) their operating costs, which are included in total transportation costs per mile, t m ; (2) their income, Y, which affects t through t Y ; and (3) the other factors that determine their demand for H, such as age and family size, which are in α.  Thus, the slopes of bid functions can vary widely.  But households all place the same value on their time as a fraction of their wage and they do not consider the operating costs of travel when they decide how much H to buy (but do consider these costs when selecting a location).

51. Household Heterogeneity Solving for the Envelope, 5  Now by pulling out the constants, we can get an even simpler form for the bid function:  The new version is: where

52. Household Heterogeneity Solving for the Envelope, 6  The ψ term contains all the information needed to determine the steepness (=absolute value of slope) of a household type’s bid function, namely its operating costs, its income, and its determinants of H ( which are in α ) other than P and m.  Note that P and m affect the slopes of bid functions, but do not affect the slope of one household’s bid function relative to another’s.  When two bid functions cross, P and m are the same for both household types.

53. Household Heterogeneity Solving for the Envelope, 7  So, in effect, we want to integrate ψ out of the bid function to obtain the envelope.  Before we can do this, we need a third key assumption, which is based on the theorem that people sort according to the slopes of their bid functions.

54. Household Heterogeneity Solving for the Envelope, 8  More specifically, if people sort according to the slopes of their bid functions, then the market equilibrium must be characterized by a monotonic relationship between location, u, and bid-function steepness, ψ.  Yinger’s working paper shows that under many circumstances, this equilibrium can be approximated with the following equation, the parameters of which, namely, the σ s, can be estimated:

55. Household Heterogeneity Solving for the Envelope, 9  A mathematical envelope must satisfy:  In this case, the g function is a bid function written in implicit form, α = ψ, and g α is the change in the bid- function slope when α (= ψ) changes.  This α is just a symbol for a parameter, not the α in the demand function for H.  The assumption about the nature of the sorting equilibrium leads to an expression for g α and hence to the envelope solution.

56. Household Heterogeneity The Envelope  Yinger solves this for σ 3 = ½, 1, or 2 (= square-root, linear, or quadratic ψ function).  Because steeper slopes lead to more central locations, the value of σ 2 should be negative, which is testable.  With outside information on γ and t Y m, it is possible to estimate the resulting forms with OLS (assuming a unitary price elasticity).  The big empirical issue is determining the best measure of commuting distance or time.

57. Household Heterogeneity

58. Normal Sorting  One final point is that this approach leads to a test for normal sorting.  The theory says that transportation costs and housing demand both show up in ψ. Normal sorting implies a negative net elasticity of ψ with respect to income.  Once the model has been estimated, one can back out values for ψ and estimate this elasticity—i.e. test for normal sorting.

59. Household Heterogeneity Normal Sorting, 2  Solve the equilibrium function for ψ. Since the σs are estimated or (in the case of σ 3 ) assumed,  Estimate the log-linear quasi-demand equation (where C is a constant and α indicates other demand factors):  See if ∂ψ/∂Y (= coefficient of ln{ Y }) is negative.