Household Heterogeneity

Household Heterogeneity
ECN741: Urban Economics Household Heterogeneity Professor John Yinger, The Maxwell School, Syracuse University, 2018

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

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:

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}!

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.

A Note on the Literature The idea of sorting comes from von Thünen, 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).

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*.

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.

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.

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.

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.

Neighborhood Change

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!

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?

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.

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

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.

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

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!

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.

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?

Source: Glaeser, Kahn, and Rappaport

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.

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).

Sorting and Mode Choice, 3 With these assumptions, the bid function of low-income people gets steeper. This change in slope reflects 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).

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.

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.

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.

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

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.

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.

More General Treatment of Heterogeneity There are two approaches in the literature to the issue of household heterogeneity. First, 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.

More General Treatment of Heterogeneity, 2 Second, two scholars have defined household classes based on observable traits and estimated separate bid functions for each. Wheaton (JUE 1977; AER 1977) defines household “strata” based on size, age, and income class, substitutes the budget constraint into a utility function, and solves for net income as a function of utility level and housing traits—which is what he estimates. Income varies within a class so utility is assumed to be a function of income (a strong, untestable assumption!). This means that income is on both sides of the regression, which violates the assumptions of OLS.

More General Treatment of Heterogeneity, 3 Yinger (JRS 1979) solves for a bid function with a Cobb-Douglas utility function, defines household types using rings around the CBD, and estimates a separate bid function for each ring. Defining household types using rings is very restrictive. Income varies within a ring, of course, which contradicts the assumption that the people in the ring belong to the same household type. Some of Yinger’s regressions address this by assuming (implausibly) that the average ring income indicates permanent income for the class. Yinger’s approach also puts price on the left side and income on the right; as shown in later classes, this raises an endogeneity problem because the choice of a location by an income class involves the “choice” of a housing price

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: mmuting/Ch_3.4.pdf .

Bid-Rent Functions and Their Envelope Envelope Bid Functions

Taking Envelopes Seriously Before presenting the (new) 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.

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!

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.

The Mathematics of Envelopes, 2

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.

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.

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.

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, tm; (2) their income, Y, which affects t through tY; 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).

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

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.

Solving for the Envelope, Insert

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.

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 σs, can be estimated:

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 gα 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 gα and hence to the envelope solution.

The Envelope Yinger solves this for σ3 = ½, 1, or 2 (= square-root, linear, or quadratic ψ function). Because steeper slopes lead to more central locations, the value of σ2 should be negative, which is testable. With outside information on γ and tY 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.

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.

Normal Sorting, 2 Solve the equilibrium function for ψ. Since the σs are estimated or (in the case of σ3) 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.

Household Heterogeneity

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