Public Finance Seminar
Spring 2015, Professor Yinger
Bidding and Sorting
Bidding and Sorting
Class Outline

Bidding

Sorting

Allocative Efficiency and Equity
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)
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.
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.
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).
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.
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.
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.
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.
Bidding and Sorting
Figure 1: Housing Bids as a Function of Public Service
Quality (Holding the Property Tax Rate Constant)
Bidding and Sorting
Figure 2: Housing Bids as a Function of the Property
Tax Rate (Holding Public Service Quality Constant)
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.
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.
Bidding and Sorting
Figure 3: Consensus Bidding and Sorting
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!
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.
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 singlecrossing 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.
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!
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.
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.
Bidding and Sorting
Figure 4:The Demand for Local Public Services
D2 = MB2
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.
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.
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.
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.
Bidding and Sorting
Figure 5: Consensus Bidding and Sorting Net of Taxes
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.
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).
Bidding and Sorting
Figure 6: Hamilton Bidding and Sorting
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.
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.
Bidding and Sorting
Figure 7: Inefficiency Due to Heterogeneity
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!
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.
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).
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.
Bidding and Sorting
Subject
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Entry 1