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

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
Figure 1.5. Von Thünen's Model of Rents
Annual Rent per Acre
and Locations
0
2
City
Milk
4
Wood
6
Rent for Milk/Vegetables
8
Grain
10
Rent for Wood
12
14
Livestock
Rent for Grain
16
18
20
Rent for Livestock
Distance from Central Market (miles)
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 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)
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
T  mA   V
and
 
T
 A
 m 
V
V 
 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
Y  Z  PH   V
 
 Z  PH 1    Z  PH (1   *)
r


The household utility function
U {Z , H , S}
The Household Problem 2

The Lagrangian:
 U {Z , H , S }
  Y   Z  P{S , t}H (1   *)  

The first-order conditions:
U S    PS H (1   *)   0
UZ    0
PH 

  P H (1   *) 
0
r 

The First-Order Conditions

The 1st and 2nd conditions imply:
US / UZ
MBS
PS 

H (1   *) H (1   *)

The 3rd condition simplifies to:
P
P/r
P  

(r   ) (1   *)
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
Y Z
Maximize P 
H (1   *)
Subject to U {Z , H , S}  U 0

Now PS and Pτ 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
P
1

P
(r   )
 This is an exact differential equation which can be
solved by integrating both sides to get:
ln{P{ }}   ln{r   }  C
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”:
P{S , }  Pˆ{S} when   0

Substituting this condition into the above
(after exponentiating) yields:
rPˆ{S }
Pˆ{S }
P{S , } 

(r   ) (1   *)
Property Tax Rates 3

Note for future reference that we can
differentiate this result with respect to S, which
gives
PˆS
PS 
(1   *)

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:
P{S , }H { X } Pˆ{S }H { X }
V 

r
r 
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
ˆ{S} . To derive a form we
without a form for P
must solve the above differential equation for S:
MBS
PS 
H (1   *)


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.
Finding a Functional Form 2

One possibility is to use constant elasticity
forms:

S  K SY W
H  KHY


 P(1   *) 


ˆ
 KHY P

where the Ks 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:
1/ 
 S 
W 
 
 K SY 

 MBS
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:
PˆS Pˆ 
S
 KS 
1/ 
1/ 
  S 1/  ,
K H Y ( /  ) 
where

   KS 
1/ 
KHY
( /  ) 

1
.
Finding a Functional Form 6

The solution to this differential equation is:
( 1 )
( 2 )
Pˆ{S }
 C  S
where C is a constant of integration, the
parentheses indicate a Box-Cox form, or,
X
( )
and


X 1

if   0 and  ln{ X } if   0
1  1  and 2 
1 

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 Pˆ{S} is on the vertical axis and S is on
the horizontal axis.
Each household type has its own bid function;
that is, its own Pˆ{S}.
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,
P
1

P
(r   )

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 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 11

Normal sorting occurs if the slope of
household bid functions increases with Y, that
is, if
PˆS MBS 1 MBS H


0
2
Y
Y H
H Y

This condition is assumed in my JPE picture.
Sorting 12

After some rearranging, we find that
PˆS
MBS 1 MBS H
 0 if

Y
Y H
H 2 Y
or
MBS Y
H Y


Y MBS Y Y

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
MBS Y


Y MB


Hence, the slope, PˆS / Y , 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 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 locationbased 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 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.