1. Lecture 1: Bidding and Sorting: The Theory of Local Public Finance
John Yinger
The Maxwell School, Syracuse University
CESifo, June 2012

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

3. 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.
Introduction

4. 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.
von Thünen

5. 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.
von Thünen

6. A Stylized von Thünen Graph

7. 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 Consensus Model

8. But Tiebout’s model is simplistic. It has
The Role of Tiebout
This literature can be traced to a famous article by Charles Tiebout in the JPE in 1956.
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)
The Consensus Model

9. 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.
The Consensus Model

10. 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 Consensus Model

11. The Household Problem The household budget constraint
The household utility function
The Consensus Model

12. The Household Problem 2 The Lagrangian: The first-order conditions:
The Consensus Model

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

14. 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.
The Consensus Model

15. 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 PS and Pτ can be found using the envelope theorem. The results are the same!
The Consensus Model

16. 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.
The Consensus Model

17. 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:
The Consensus Model

18. 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 Consensus Model

19. 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.
The Consensus Model

20. 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).
The Consensus Model

21. 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 MBS 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.
Deriving a Bid Function

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

23. 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!
Deriving a Bid Function

24. 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.
Deriving a Bid Function

25. 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
Deriving a Bid Function

26. 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
Deriving a Bid Function

27. 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.
Deriving a Bid Function

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

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

35. 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

36. 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

37. Sorting 10
5. 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

38. 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

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

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

41. 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

42. Sorting 15
6. 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
Sorting

43. 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 1974.
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.
Preview