Bidding and Sorting: The Theory of Local Public Finance

ECN 741, Urban EconomicsProfessor Yinger

Lecture Outline

The Consensus Model of Local Public Finance

Deriving a Bid Function

Residential Sorting

Introduction

The U.S. Federal System

Constitutions and Politics◦ Broad outlines defined by constitutions◦ Details determined by politics

Units Defined by U.S. Constitution◦ The Federal Government◦ State Governments

Units Defined by State Constitutions◦ The State Government◦ Counties and (usually) Townships◦ Municipalities (Cities and Villages)◦ School Districts◦ Special Districts

The U.S. Federal System

The U.S. Federal System

County Township Municipality School District

The U.S. Federal System

1952 20070

20000

40000

60000

80000

100000

120000

140000

160000

180000

3052 303316807 19492

17202 16519

67355

13051

67355

13051

Number of Governments, by Type of Gov-ernment, 1952 and 2007

School DistrictsSpecial DistrictsTownshipsMunicipalitiesCounties

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

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 marketNo property tax (just an entry fee)No public goods (just publically provided private goods) or votingNo labor market (just dividend income)

The Consensus Model

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

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.

T mA V

T AmV V

The Consensus Model

The Household Problem

The household budget constraint

The household utility function

1 (1 *)

Y Z PH V

Z PH Z PHr

{ , , }U Z H S

The Consensus Model

The Household Problem 2

The Lagrangian:

The first-order conditions:

{ , , }

{ , } (1 *)

U Z H S

Y Z P S t H

(1 *) 0S SU P H

0ZU

(1 *) 0PHP Hr

The Consensus Model

The First-Order Conditions The 1st and 2nd conditions imply:

The 3rd condition simplifies to:

/(1 *) (1 *)S Z S

SU U MBPH H

/( ) (1 *)

P P rPr

The Consensus Model

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.

This is, of course, the issue that arises in a basic urban model.

The Consensus Model

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!

0

Maximize (1 *)

Subject to { , , }

Y ZPH

U Z H S U

The Consensus Model

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.

ln{ { }} ln{ }P r C

1( )

PP r

The Consensus Model

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:

ˆ{ , } { } when 0P S P S

ˆ ˆ{ } { }{ , }( ) (1 *)rP S P SP Sr

The Consensus Model

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).

ˆ

(1 *)S

SPP

The Consensus Model

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.

ˆ{ , } { } { } { }P S H X P S H XVr r

The Consensus Model

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

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

ˆ{ }P S

(1 *)S

SMBP

H

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.

SS K Y W

ˆ(1 *)H HH K Y P K Y P

Deriving a Bid Function

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

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

1/

SS

SW MBK Y

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

1/1/

1/ ( / )ˆ ˆ ,S

S H

SP P SK K Y

11/ ( / ) .S HK K Y

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

1 2( ) ( )ˆ{ }P S C S

( ) 1 if 0 and ln{ } if 0XX X

1 211 and

Finding a Functional Form 7

This equation is called a “bid function.”

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.

It is analogous to the bid functions in a basic urban model—it indicates how much a household would pay at different locations (=levels of S) holding utility constant.

Deriving a Bid Function

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: 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

Sorting 2 Sorting in this context is the separation of different

household types into different jurisdictions.

As in an urban model, 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

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

ˆ{ }P S

ˆ{ }P S

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):

Sorting

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.

Sorting

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

Sorting 7 2. Some jurisdictions may be very

homogeneous, such as a jurisdiction between the intersections in the following figure.

Sorting

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

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.

1( )

PP r

Sorting

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

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.

2

ˆ 1 0S S SP MB MB HY Y H H Y

Sorting

Sorting 12 After some rearranging, we find that

Normal sorting occurs if the income elasticity of MB exceeds the income elasticity of H.

2

ˆ 10 if

or

S S S

S

S

P MB MB HY Y H H Y

MB Y H YY MB Y Y

Sorting

Sorting 13 The constant elasticity form for S implies

that

Hence, the slope, , will increase with Y so long as:

SMB YY MB

ˆ /SP Y

Sorting

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

Sorting 15 6. This analysis of bidding and sorting applies to any

public service or amenity that is linked to a location.

Examples include:

The perceived quality of local elementary schools;Distance from a pollution source;Access to parks or other neighborhood amenities.

As we will see, this framework also links nicely with the largely empirical literature on so-called hedonic regressions.

Sorting

Sorting 16 7. Finally, the logic of bidding and sorting does not apply

only to the highly decentralized federal system in the U.S.

It also applies to any country in which

A location-based public service or neighborhood amenity varies across locations,

Housing markets are competitive and households can decide where to live, and

Access (or the cost of access) to the service or amenity depends on residential location.

Sorting

Preview In the next set of classes, we 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