1. Public Finance Seminar Spring 2017, Professor Yinger
Bidding and Sorting

2. Allocative Efficiency and Equity
Bidding and Sorting
Class Outline
Bidding
Sorting
Allocative Efficiency and Equity

3. Three Related Questions
Bidding and Sorting
Three Related Questions
How do households select a community in which to live?
=The subject of this class.
How do communities select the levels of local public services and the property tax rate? (= Demand!)
Under what conditions is the community-choice process compatible with the local voting process? (=General Equilibrium)

4. Bidding and Sorting
Tiebout
Charles Tiebout argued in a famous 1956 JPE article that people shop for a community, just as they shop for other things.
This process came to be known as “shopping with one’s feet.”
The Tiebout model became very influential both because it identified an important type of behavior to be studied
and because it concluded that the community-choice process is efficient.
Although Tiebout’s model has neither a housing market nor a property tax, many people (not including me) still accept his efficiency claim, at least to a first approximation.

5. The Consensus Model, Assumptions
Bidding and Sorting
The Consensus Model, Assumptions
1. Households care about housing, public services, and other goods.
2. Households fall into distinct income/taste classes.
3. Households are mobile (i.e. can move costlessly across jurisdictions).
So an equilibrium cannot exist unless all people in a given income/taste class achieve the same level of satisfaction.
4. All households who live in a jurisdiction receive the same level of public services.

6. The Consensus Model, Assumptions, 2
Bidding and Sorting
The Consensus Model, Assumptions, 2
5. Residence in a jurisdiction is a precondition for the receipt of public services there (an assumption undermined by some education choice programs).
6. Public services are financed through a local property tax.
7. An urban area has many local jurisdictions, which have fixed boundaries and vary in their local public service quality and property tax rates.
8. Households are homeowners (or else they are renters and the property tax is shifted fully onto them through rents).

7. Bidding and Sorting
Bidding
These assumptions describe a housing market in which households compete for access to the most desirable locations, i.e., those with the best combinations of public services and property tax rates.
This competition takes the form of housing bids, with different bids for different types of household.

8. Bidding, 2 An analysis of bidding is based on 4 concepts:
Bidding and Sorting
Bidding, 2
An analysis of bidding is based on 4 concepts:
Housing is measured in units of housing services, H, which can be thought of as quality-adjusted square feet.
The associated housing price, labeled P, is what a household bids per unit of H per year.
The rent for a housing unit, R, equals PH. If the unit is an apartment, R = PH is equivalent to the annual rent.
If the unit is owner-occupied, R is not observed but is implicit. The value of a housing unit, V, is the amount someone would pay to own that unit; it equals the present value of the flow of net rental services associated with ownership or V = PH/r = R/r.

9. We will have to do some more work to figure out market prices.
Bidding and Sorting
The Bid Function
P is the amount a household is willing to pay per unit of H in a jurisdiction with service quality S and property tax rate t: P = P{S, t}.
A bid function is defined for all values of S and t, regardless of whether or not these values are the ones a household actually experiences.
We will have to do some more work to figure out market prices.

10. This logic is expressed in the following two figures.
Bidding and Sorting
The Bid Function, 2
This logic is expressed in the following two figures.
Figure 1 shows how bids (P) change as S changes (holding t constant).
Figure 2 shows how bids change as t changes (holding S constant)
These curves hint at the idea that housing prices reflect S and t but we are not quite there yet.

11. Bidding and Sorting
Figure 1: Housing Bids as a Function of Public Service Quality (Holding the Property Tax Rate Constant)

12. Bidding and Sorting
Figure 2: Housing Bids as a Function of the Property Tax Rate (Holding Public Service Quality Constant)

13. Bidding and Sorting
Sorting
If all household are alike, then Figures 1 and 2 describe market outcomes; in equilibrium housing prices will adjust to exactly compensate households who end up in low-S or high-t jurisdictions.
This compensation idea is a key to the intuition.
But obviously households are not all alike, and different types of households must sort across jurisdictions.

14. Bidding and Sorting
Sorting and Slopes
The key idea in the consensus model is that households sort according to the steepness of their bid functions.
The slope is ∂P/∂S; a steeper slope corresponds to a greater willingness to pay (WTP) for an increment in S.
Housing sellers prefer to sell to the highest bidder, so a steeper slope wins where S is higher.

15. Bidding and Sorting
Sorting and Slopes, 2
Note that in the long run sellers prefer the highest P, not the highest PH.
H can be adjusted by adding on to a housing unit or dividing it into smaller units.
The seller wants the highest return per unit of H, which is given by P.

16. Figure 3: Consensus Bidding and Sorting

17. Bidding and Sorting
The Price Envelope
In Figure 3, the observed market price function is the envelope of the underlying bid functions, say PE.
This function shows how S and t show up in housing prices.
This phenomenon is known as capitalization, because it has to do with the impact of annual flows S and t ) on the value of a capital asset (a house).
As we will see, many studies estimate capitalization!

18. Sorting and Bid-Function Heights
Bidding and Sorting
Sorting and Bid-Function Heights
One subtle point in Figure 3 is that the heights of the bid functions adjust so that each type of household has enough room.
To get the logic right, start with a point at which the bid functions of two groups intersect—i.e., their heights are the same.
From this starting point, it is obvious that the group with the steeper bid function bids more where S is higher.
It is impossible to shift the heights around so that (a) both groups live somewhere and (b) the group with the flatter slope lives where S is higher.

19. The Single-Crossing Condition
Bidding and Sorting
The Single-Crossing Condition
It is logically possible for household type A to have a steeper bid function than household type B at one value of S and for a household type B to have a steeper bid function at a different value of S.
In this case it may be impossible to find an equilibrium.
Almost all scholars rule out this case with the so-called single- crossing condition, which is that if household type A has a steeper bid function than household type B at one value of S, it also has a steeper bid function at any other value of S.
This assumption is equivalent to a regularity condition on the underlying utility functions.

20. Sorting and Empirical Research
Bidding and Sorting
Sorting and Empirical Research
Figure 3 has 2 key implications for empirical work.
First, any estimate of the impact of S on P (or on R or V) blends willingness to pay (movement along a bid function) and sorting (shifting across bid functions).
In Figure 3, for example, (P3 – P1) does not measure any group’s WTP for S3 versus S1.
It is very difficult to untangle these two effects!

21. Sorting and Empirical Research, 2
Bidding and Sorting
Sorting and Empirical Research, 2
Second, the market relationship between S and P, that is, the envelope, is very unlikely to be linear (or even log-linear).
Because sorting is based on bid-function slopes, the slope of the winning bidder (the one we observe) goes up as the level of S increases.
The next slide gives a simple example with linear bid functions (which correspond to a constant MWTP, which is equivalent to a horizontal demand curve).

22. Figure 3A: A Nonlinear Envelope
Bidding and Sorting
Figure 3A: A Nonlinear Envelope P S

23. Bidding and Sorting
Normal Sorting
Under normal circumstances, higher-income households will pay more for an increment in S. See Figure 4, where MB = MWTP (and a 2 subscript indicates high-income).
So high-income households will normally win the competition in high-S jurisdictions.
Low-income households win the competition in low-S jurisdictions because they are willing to accept low H, e.g. by doubling up, and hence can bid a lot per unit of H.

24. Figure 4: The Demand for Local Public Services
Bidding and Sorting
Figure 4: The Demand for Local Public Services
D2 = MB2

25. Bidding and Sorting
Normal Sorting, 2
The formal condition for normal sorting is that the ratio of the income and elasticities of demand for S must exceed the income elasticity of demand for H .
Note that α matters because a household’s willingness to pay for S (and hence its bid) is spread out over the number of units of H it buys.

26. Sorting and Inequality
Bidding and Sorting
Sorting and Inequality
Sorting results in a system of local governments in which some jurisdictions are much wealthier than others.
Moreover, as discussed earlier in the class, the cost of public services is often linked to poverty.
The overall result is a federal system in which both resources and costs are unevenly distributed—and in which higher-income people receive higher levels of S.
The key normative tradeoff: community choice versus equality of opportunity.

27. Sorting and Inequality, 2
Bidding and Sorting
Sorting and Inequality, 2
Some evidence comes from plotting the market value of all neighborhood amenities and local public services against income (more later on the estimating method) based on my 2015 JUE article:

28. Heterogeneity Within a Jurisdiction
Bidding and Sorting
Heterogeneity Within a Jurisdiction
Sorting implies a tendency for jurisdictions to be homogeneous by income.
But this homogeneity is not complete:
Other factors influence demand.
Heterogeneous communities arise where bid functions cross. See Figure 3. Many household types could live in a large jurisdiction.

29. Sorting and Tax Rates Figure 3 holds t constant.
Bidding and Sorting
Sorting and Tax Rates
Figure 3 holds t constant.
In fact, t does not affect sorting.
Any household is willing to pay $1 to avoid $1 of property taxes (or to bid 1% more in a location where t is 1% lower).
Sorting only involves S.
Nevertheless, sorting is sometimes shown with net bids that reflect both S and t. See Figure 5.

30. Figure 5: Consensus Bidding and Sorting Net of Taxes

31. Bidding and Sorting
The Hamilton Approach
Bruce Hamilton (Urban Studies 1975) developed a model with three additional assumptions:
Housing supply is elastic (presumably by movement in jurisdiction’s boundaries).
Zoning is set at exactly the optimal level of housing for the residents of a jurisdiction.
Government services are produced at a constant cost per household.

32. Bidding and Sorting
Hamilton, 2
The striking implication of the Hamilton assumptions is that capitalization disappears. See Figure 6.
Everyone ends up in their most-preferred jurisdiction, so nobody bids up the price in any other jurisdiction.
This prediction (no capitalization) is rejected by all the evidence (to be covered in future classes).

33. Figure 6: Hamilton Bidding and Sorting

34. Is a Federal System Efficient?
Bidding and Sorting
Is a Federal System Efficient?
Tiebout (with no housing or property tax) said picking a community is like shopping for a shirt and is therefore efficient (in the allocative sense).
This result is replicated with a housing market and a property tax under the Hamilton assumptions.
These assumptions are extreme and the model is rejected by the evidence.

35. Sources of Inefficiency
Bidding and Sorting
Sources of Inefficiency
1. Heterogeneity
Heterogeneity leads to inefficient levels of local public services.
Outcomes are determined by the median voter; voters with different preferences experience “dead-weight losses” compared with having their own jurisdiction.
See Figure 7.

36. Figure 7: Inefficiency Due to Heterogeneity
Bidding and Sorting
Figure 7: Inefficiency Due to Heterogeneity

37. Heterogeneity in Services?
Bidding and Sorting
Heterogeneity in Services?
The consensus model assumes that all households in a jurisdiction receive the same services.
This is not true; higher-income neighborhoods have better police protection and probably better elementary schools, for example.
This eliminates some of the inefficiency in Figure 7—but adds to inequality.
This topic is poorly understood; research is needed!

38. Sources of Inefficiency
Bidding and Sorting
Sources of Inefficiency
2. The Property Tax
Producers of housing base their decisions on P, whereas buyers respond to P(1 + t/r), (which, as we will see, is constant across jurisdictions).
This introduces a so-called “tax wedge” between producer and consumer decisions, which leads to under-consumption of housing.

39. Hamilton and Inefficiency
Bidding and Sorting
Hamilton and Inefficiency
These two sources of inefficiency disappear in Hamilton’s model:
Boundaries adjust so that all jurisdictions are homogeneous (despite extensive evidence that boundaries do not change for this reason).
Zoning is set so that housing consumption is optimal (despite the incentives of residents to manipulate zoning).

40. Conclusions on Efficiency
Bidding and Sorting
Conclusions on Efficiency
The Hamilton model shows how extreme the assumptions must be to generate an efficient outcome.
Nevertheless, most scholars defend our federal system as efficient,
Instead of identifying policies, such as intergovernmental aid programs, to improve its efficiency.

41. Bidding and Sorting