ECN741: Urban Economics

1. ECN741: Urban Economics Household Heterogeneity

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

3. 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:

4. 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}!

5. 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.

6. Household Heterogeneity

7. 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*.

8. 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 fromu* to for the people in the other class.

9. 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.

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

11. Household Heterogeneity Implications for Neighborhood Change • This graph is 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.

12. Household Heterogeneity Neighborhood Change These neighborhoods shift from high-to low-income

13. Household Heterogeneity Normal Sorting • The natural question to ask is whether rich or poor people live closer to the CBD. • A situation in which poor people live closer is often called “normal” sorting—meaning what is expected, not what is good! • Under what circumstances does normal sorting arise?

14. 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.

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

16. 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.

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

18. 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 t0 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!

19. Household Heterogeneity Normal Sorting, 7 • 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.

20. Household Heterogeneity Normal Sorting, 8 • Casual evidence indicates, however, that poor people tend to live in cities, not suburbs. • There is more formal evidence in Glaeser, Kahn, and Rappaport (JUE, 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?

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

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

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

24. Household Heterogeneity Sorting and Mode Choice • The resolution proposed by Glaeser et al. 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.

25. Household Heterogeneity Sorting and Mode Choice, 2 • One way to model this (which is not quite the way Glaeser et al. precede) is to make the following assumptions: • There are two modes, buses (B) and cars (C), each with costs that depend on Y. • The probability that a household type will choose B is a function of Y.

26. Household Heterogeneity Sorting and Mode Choice, 3 • With these assumptions, the elasticity condition becomes: • The third term is the standard case. • Higher income lowers pB and tB > tC. So the first term is negative. • But the second term is positive because the bus is slower (i.e. has higher time costs). • So normal sorting may be more likely than before, but only if the first term is larger (in absolute value) than the second.

27. Household Heterogeneity More General Treatment of Heterogeneity • Beckman (JET, 1969) suggests and Montesano (JET, 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.

28. Household Heterogeneity Bid Function Envelopes • Another approach, which I am working on, 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.

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

30. Household Heterogeneity Taking Envelopes Seriously • Before presenting my approach, I have to say that it is astonishing how many people are willing to ignore this point. • They 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.

31. Household Heterogeneity The Origin of Envelopes • An aside on the intellectual history of this stuff: • Remember that sorting is in von Thünen. • Alonso recognized that household heterogeneity would lead to sorting; he also 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 textbook interprets the coefficient of a distance variable as required compensation!

32. Household Heterogeneity Solving for the Envelope • In the paper I am working on, I am able to solve for the envelope with three main assumptions. • The derivation starts with something from an earlier class, namely, a bid function based on a constant elasticity demand function for H with a unitary price elasticity and income elasticity θ. The result Note: A is part of housing demand and Q is a constant.

33. Household Heterogeneity Solving for the Envelope, 2 • The second assumption is that t = tYY; which is implies (a) there are no operating costs and (b) tYis the same for all households. • Now tYdepends on the value of travel time as a fraction of the wage rate and on commuting speed. So holding this constant across households is a strong assumption—but it is needed for tractability. • The assumption can be weakened by switching to commuting time instead of distance; in this case only the value of travel time is constant across households and speed is built into the explanatory variable.

34. Household Heterogeneity Solving for the Envelope, 3 • The punchline here is that with these assumptions households are heterogeneous in two ways: their incomes, which affect t through tY, and the factors that determine their demand for H. • Thus, the slopes of bid functions can vary widely. • But households do not place a different value on their time as a fraction of their wage and they do not consider the operating costs of their travel (as when they ride a bus with a fixed fare).

35. Household Heterogeneity Solving for the Envelope, 4 • This assumption about t makes it possible to simplify the earlier equation for the bid-rent function. • The new version is:

36. Household Heterogeneity Solving for the Envelope, 5 • The ψterm contains all the information needed to determine the slope of a household type’s bid function, namely its income and other determinants ofH. • So if we, in effect, integrate it out, we will have the envelope. • My 3rd key assumption uses the theorem that people sort according to the slopes of their bid functions.

37. Household Heterogeneity Solving for the Envelope, 6 • 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-functions slope, ψ. • My approach is to approximate this relationship with the following equation, and then to estimate the parameters of this equation, namely, the σs:

38. Household Heterogeneity The Envelope • I can solve this for σ3 = ½, 1, or 2. • Because steeper slopes lead to more central locations, the value of σ2should be negative, which is testable. • If I bring in outside information on θ and tY (γand t in the following table and in the posted notes), then I can estimate the resulting forms with OLS. • The big empirical issue I am struggling with (and will report back to you on) is determining the best measure of commuting distance or time (u in the table).

39. Household Heterogeneity