Lecture Outline Introduction to Series von Thünen The Consensus Model of Local Public Finance Deriving a Bid Function Residential Sorting

Series Overview This is the first of 3 lectures on local public finance, hedonics, and fiscal federalism I have three objectives Review recent developments in the theory of local public finance, in hedonics, and in associated empirical methods. Introduce you to a new technique I have developed for estimating key parameters and testing key hypotheses that come out of this research. Convince you that these ideas and methods are relevant for Germany, even though it is not nearly as decentralized as the United States.

von Thünen The main topics in my lecture all trace back to the German economist, Johann Heinrich von Thünen, who lived from 1783 to 1850.

The von Thünen Model von Thünen combined his practical experience running an estate and his training in economics to build a model of rural land use around a central market. His model introduces the concept of land bids that vary by location and of the sorting of competing activities into different locations. Everything I say about bidding and sorting descends from his model.

A Stylized von Thünen Graph

Local Public Finance The literature on local public finance in a federal system is built around three questions: 1. How do housing markets allocate households to jurisdictions? = Bidding and sorting! 2. How do jurisdictions make decisions about the level of local public services and taxes? 3. Under what circumstances are the answers to the first two questions compatible?

The Role of Tiebout This literature can be traced to a famous article by Charles Tiebout in the JPE in Tiebout said people reveal their preferences for public services by selecting a community (thereby solving Samuelson’s free-rider problem). Tiebout said this choice is like any market choice so the outcome is efficient. But Tiebout’s model is simplistic. It has No housing market No property tax (just an entry fee) No public goods (just publically provided private goods) or voting No labor market (just dividend income)

Key Assumptions Today I focus on a post-Tiebout consensus model for the first question based on 5 assumptions: 1. Household utility depends on a composite good ( Z ), housing ( H ), and public services ( S ). 2. Households differ in income, Y, and preferences, but fall into homogeneous income-taste classes. 3. Households are mobile, so utility is constant within a class. 4. All households in a jurisdiction receive the same S (and a household must live in a jurisdiction to receive its services). 5. A metropolitan area has many local jurisdictions with fixed boundaries and varying levels of S.

Additional Assumptions Most models use 2 more assumptions: 6. Local public services are financed with a property tax with assessed value ( A ) equal to market value ( V ). Let m be the legal tax rate and τ the effective rate, then tax payment, T, is and 7. All households are homeowners or households are renters and the property tax is fully shifted onto them.

The Household Problem  The household budget constraint  The household utility function

The Household Problem 2  The Lagrangian:  The first-order conditions:

The First-Order Conditions The 1 st and 2 nd conditions imply : The 3 rd condition simplifies to:

The Market Interpretation These conditions indicate the value of S and τ that a household will select. But all households cannot select the same S and τ ! Thus, these conditions must hold at all observed values of S and τ, that is, in all communities. As in an urban model, this is called, of course, locational equilibrium. No household has an incentive to move because lower housing prices exactly compensate them for relatively low values of S or relatively high values of τ.

Alternative Approach  Solve the budget constraint for P ; find the most a household is willing to pay for H at a given utility level  Now P S and P τ can be found using the envelope theorem. The results are the same!

Bidding for Property Tax Rates These two conditions are differential equations. The tax-rate equation can be written as This is an exact differential equation which can be solved by integrating both sides to get: where C is a constant of integration.

Property Tax Rates 2 We can solve for C by introducing the notion of a before-tax bid, sometimes called the bid “net of taxes” and indicated with a “hat”: Substituting this condition into the above (after exponentiating) yields:

Property Tax Rates 3 Note for future reference that we can differentiate this result with respect to S, which gives This result makes it easy to switch back an forth from before-tax to after-tax bid-function slopes (with respect to S ).

The House Value Equation To test this theory, we want to estimate an equation of the following form: The dependent variable is house value, V, or it could be apartment rent. The key explanatory variables are measures of public services, S, property tax rates, τ, and housing characteristics, X.

Capitalization In this equation, the impact of τ on V is called “property tax capitalization.” The impact of S on V is called “public service capitalization.” These terms reflect the fact that these concepts involve the translation of an annual flow ( τ or S ) into an asset or capital value ( V ).

Finding a Functional Form This house value equation cannot be estimated without a form for. To derive a form we must solve the above differential equation for S: To solve this equation, we obviously need expressions for MB S and H. These expressions require assumptions about the form of the utility function (which implies a demand function) or about the form of the demand function directly.

Finding a Functional Form 2 One possibility is to use constant elasticity forms: where the K s indicate vectors of demand determinants other than income and price, and W is the price of another unit of S.

Finding a Functional Form 3 These forms are appealing for three reasons: 1. They have been successfully used in many empirical studies. ◦ Duncombe/Yinger (ITPF 2011), community demand for education ◦ Zabel (JHE 2004), demand for housing  2. They can be derived from a utility function. ◦ The derivation assumes a composite good (=an “incomplete demand system”), zero cross-price elasticities, and modest restrictions on income elasticities [LaFrance (JAE 1986)].  3. They are tractable!

Finding a Functional Form 4 Note that the demand function for S can be inverted to yield: This is, of course, the form in which it appears in earlier derivations.

Finding a Functional Form 5 Now substituting the inverse demand function for S and the demand function for H into the differential equation yields: where

Finding a Functional Form 6 The solution to this differential equation is: where C is a constant of integration, the parentheses indicate a Box-Cox form, or, and

Finding a Functional Form 7 This equation is called a “bid function.” It is, of course, a descendant of the bid functions derived by von Thünen. It indicates how much a given type of household would pay for a unit of H in a location with a given level of S.

Sorting It is tempting to stop here—to plug this form into the house value equation and estimate. As we will see, many studies proceed, incorrectly, in exactly this manner. But we have left out something important, namely, von Thünen’s other key invention: sorting. To put it another way, we have not recognized that households are heterogeneous and compete with each other for entry into desirable locations.

Sorting 2 Sorting in this context is the separation of different household types into different jurisdictions. The key conceptual step to analyze sorting is to focus on P, the price per unit of H, not on V, the total bid. In the long run, the amount of H can be altered to fit a household’s preferences. A seller wants to make as much as possible on each unit of H that it supplies.

Sorting 3 This framing leads to a standard picture in which is on the vertical axis and S is on the horizontal axis. Each household type has its own bid function; that is, its own. The household that wins the competition for housing in a given jurisdiction is the one that bids the most there.

Sorting 4 I did not invent this picture but was an early user. Here’s the version in my 1982 JPE article (where I use E instead of S ): P(E, t *)

Sorting 5 The logic of this picture leads to several key theorems. 1. Household types with steeper bid function end up in higher- S jurisdictions. Group 2 lives in jurisdictions with this range of S.

Sorting 6 This theorem depends on a “single crossing” assumption, namely, that if a household type’s bid function is steeper at on value of S, it is also steeper at other values of S. This is a type of regularity condition on utility functions.

Sorting 7 2. Some jurisdictions may be very homogeneous, such as a jurisdiction between the intersections in the following figure.

Sorting 8 3. But other jurisdictions may be very heterogeneous, namely, those at bid-function intersections, which could (in another figure) involve more than two household types.

Sorting 9 4. Sorting does not depend on the property tax rate. As shown above, Nothing on the right side depends on Y (or any other household trait); starting from a given P, the percentage change in P with respect to τ is the same regardless of Y.

Sorting In contrast, income, Y, (or any other demand trait) can affect sorting. Because τ does not affect sorting, we can focus on before-tax bids. We will also focus on what is called “normal sorting,” defined to be sorting in which S increases with Y.

Sorting 11 Normal sorting occurs if the slope of household bid functions increases with Y, that is, if This condition is assumed in my JPE picture.

Sorting 12 After some rearranging, we find that Normal sorting occurs if the income elasticity of MB exceeds the income elasticity of H.

Sorting 13 The constant elasticity form for S implies that Hence, the slope,, will increase with Y so long as:

Sorting 14 The available evidence suggests that θ and μ are approximately equal in absolute value and that γ ≤ 0.7. It is reasonable to suppose, therefore, that this condition usually holds. Competition, not zoning, explains why high- Y people live in high- S jurisdictions.

Sorting Finally, the logic of bidding and sorting does not apply only to the highly decentralized federal system in the U.S. I also applies to any situation in which a location- based public service or neighborhood amenity varies across locations and access (or the cost of access) depends on residential location. Examples include: The perceived quality of local elementary schools Distance from a pollution source Access to parks or museums or other urban amenities

Preview In the next lecture, I will bring in the complementary literature on housing hedonics, which builds on Rosen’s famous 1974 article in the JPE in The Rosen article provides some more theory to think about as well as the framework used by most empirical work on the capitalization of public service and neighborhood amenities into house values. I will also introduce a new approach to hedonics, that draws on the theory we have reviewed today.