Public Finance Seminar
Spring 2015, Professor Yinger
Demand for Public Services: The
Median Voter and Other
Approaches
Demand for Public Services
Class Outline

Household Demand for Public Services

The Median Voter Model

Estimating Household Demand
Demand for Public Services
The Starting Point

An household’s demand for local public services,
like its demand for private goods, depends on its
income, the price of the services, the price of
alternatives, and its preferences.

But with 2 big twists:
◦ There is no market price.
◦ The demand for public services can be expressed in
several different ways:
Demand for Public Services
Lack of a Market Price

Most public services are funded by taxes, not
prices.

Hence, the “price” is defined as the cost of
an additional unit of service

And this price depends on the tax system.
 Tax Price = How much would the individual pay
if taxes were raised enough to provide one more
unit of the service to everyone in the jurisdiction.
 A more formal definition will be derived later.
Demand for Public Services
How Is Demand Revealed?

The demand for services can be expressed:
◦ Through voting (today’s class)
◦ Through bidding for housing and choice of a
community (the subject of later classes)
◦ Through the purchase of private substitutes, such
as private schools, security guards, or access to a
gated community (not covered in any class)
Demand for Public Services
Household Demand for Public Services

A Household’s Budget Constraint

Income (Y) must be spent on housing (H
with price P), property taxes (tV=tPH/r) and
other stuff (Z with price 1):
Y  Z  PH  tV
PH
 Z  PH  t
r
t

 Z  PH 1  
r

Demand for Public Services
The Community Budget Constraint

In a community, sending per household (E) to
achieve the desired service level (S) must equal
property tax revenue per household (t
multiplied by average V ).
E{S}  tV  A

We will skip state aid and other local revenue
sources for now.
Demand for Public Services
Solving for Tax Price, 1

Solve the community budget constraint for t:
E{S}  A
t
V

Substitute into the household budget
constraint:
V 
Y  Z  PH  ( E{S}  A)  
V 
Demand for Public Services
Solving for Tax Price, 2

Tax price is the cost of one more unit of S, i.e.,
the derivative of the household budget
constraint with respect to S, or,
Tax Price = TP =
 E   V 
V 

    MC  
V 
 S   V 

where MC is the resource cost of another unit
of S, and the ratio of V to average V is the tax
share.

Note: If E is total spending, the tax share is V
divided by total V—a true share.
Demand for Public Services
Estimating Household Demand

With TP defined, we can write down a
household demand function:
S  S{Y , TP, Other Prices, Preference Variables}

The problem: How to estimate this function?
◦ One answer: through surveys.
Demand for Public Services
Survey Studies of Household Demand

Approach 1: Surveys of voting on a referendum.
◦ The demand function defines a latent variable, which
can be studied with a discrete-choice model, with Y
and TP as explanatory variables.
◦ This approach also can be applied to a survey of
preferences for increasing, decreasing, or not
changing spending.
Demand for Public Services
Survey Studies of Household Demand, 2

Approach 2: Surveys of spending preferences:
“How much would you like to spend?”
◦ Use a multiplicative form with desired spending
(= (S)(AC)) as the dependent variable (assuming
AC=MC):
S  AY  TP  
V


AY MC 
V



E  (S )( AC ) 

1  V

AY MC

V





Demand for Public Services
The Median Voter Theorem

Although household voting is not observed,
the outcomes of voting in a community are
easy to observe—on referenda or in the
form of spending or service levels.

The median voter model provides a way to
estimate a demand model at the community
level—where the data are!
Demand for Public Services
Bergstrom and Goodman

This famous paper (AER 1973) starts with an obvious
point (the voter in the middle of the demand
distribution is always on the winning side)

It then adds assumptions about the structure of
demand and taxes (that demand depends on Y and
TP, that there is a property tax, and that the demand
for H is a function of Y)

And shows that the voting outcome in a community
is determined by the voter with the median Y and
median TP.
Demand for Public Services
Bergstrom and Goodman, 2

In symbols:
S  S{YMedian , TPMedian , X }  S YMedian , MC  VMedian  , X 

 V



This was revolutionary because it specified the
demand for S using data just on median Y and
median TP, which are readily observed.

Scholars can proceed “as if” voting outcomes
depend only on the demand of this abstract
median voter.
Demand for Public Services
Problems with Median Voter Models

1. Logical problems
◦ If demand is not one-dimensional and preferences do
not take certain forms, the public choice mechanism
may not be well defined. This is Arrow’s
Impossibility Theorem: it is impossible to write
down a general model of public choice for complex
decisions.
◦ Example: private schools. Some people with a high
demand for public services under some circumstances
(no private alternative) may have a low demand under
others (a good private school nearby).
Demand for Public Services
Problems with Median Voter Models, 2

2. Institutional problems
◦ The median voter model says institutions are neutral.
Politicians and bureaucrats have no impact on
observed spending or service quality (except perhaps
through inefficiency—more later). Also, results are
assumed not to be skewed by non-participation. This
may not be true.
◦ Example: renters. The tax price idea applies only to
owners. But it is very hard to find a significant renter
variable in a median voter model.
Demand for Public Services
Problems with Median Voter Models, 3

3. Tiebout bias
◦ More on this later.
◦ Basically, this is a form of selection bias in which
people with low incomes but high demand for
services based on unobserved factors end up in
jurisdictions with high-quality services.
Demand for Public Services
The Budget Constraints

The Median Voter’s Constraint
t

Y  Z  PH  tV  Z  PH  1  
r


The Community Constraint
C{S }
E
 tV  A
e

The Combined Constraint
C{S }  V 
V 
Y  A    Z  PH 
 
e V 
V 
Demand for Public Services
Components

Tax Price
Spending dC 1  V
TP 

e 
S
dS  V


1  V 
   MC  e  

V 
Augmented Income
V 
Y  Y  A 
V 
A
 This term leads to the Oates equivalence theorem:
$1 of aid weighted by tax share should have the same
impact on demand as $1 of income. More later.
Demand for Public Services
Constant Elasticity Demand

General Form


V  
V
S  K  Y  f A      MC  e 1 
V  
V






Linear Form to Estimate


 V 
V
ln{S}  K *   ln Y  f A      ln  MC  e 1 
 V 
V


 
 K *   ln Y 1 
 
 A  V
f  
 Y  V




 
1  V


ln
MC
e






 
V






 A  V  
 V 
 K *   ln Y    ln 1  f       ln  MC  e 1   
 Y  V  
 V 


 A  V 
V 
 K *   ln Y    f     1 ln MC  2 ln e  3 ln  
 Y  V 
V 
Demand for Public Services
The Big Problem: Endogeneity

Note that this equation includes MC, which
depends on the level of S

It also includes e, which may depend on MC
as well as on key explanatory variables, such
as Y and tax share.

A solution: Model MC and e.

Most studies ignore these problems!
Demand for Public Services
The Cost Function

A multiplicative form:
C{S}   S W N P




 implies that
C{S}
 1 
 
MC 
  S W N P
S
Demand for Public Services
Efficiency

Assume demand for services other than S
and monitoring incentives depend on
augmented income, TP, and other factors, M:



V  
 V 
e  k M  Y  f A      MC    
V  
V 


Demand for Public Services
The Full Expenditure Function

Also recall that E = C{S}/e. Substitute MC
and e into this expression to obtain
Ek
*
S
  ( 1)
W


N P

 1

  
V 
 M Y  f A  

V 


V  
  
V  

 which can be estimated in log form (with the above
approximation).
Demand for Public Services
The Demand Function

Now substitute MC and e into the demand
function and solve for S to get an estimating
equation:

V
S  K Y  f A
V

*
*
  *
    C  e
 

* 1
V

V



*
 where the expressions with asterisks can all be
identified with the expenditure results.

Note that the simultaneity problem is solved
with algebra, not econometrics.
Demand for Public Services
Common Error

Most studies ignore e.

But if e is a function of augmented income
and TP, then the coefficients of these
variables in a demand function reflect
efficiency effects as well as demand effects.

They do not just give demand elasticities!
Demand for Public Services
Massachusetts (Phuong/Yinger)
Property tax limits with overrides
 No independent school districts, so actions
may depend on costs of other services
 Observe 296 districts over 6 years
 Year dummies, but no fixed effects
 Service is measured with a state-defined
Student Performance Index

Demand for Public Services
Table 4. Demand Estimation Regression Results (2001-2006)
Dependent Variable: Log of Student Performance Index
Base
Without logged nonschool costs (𝑪𝑴 )
𝑪𝑴 interacted with
regional dummy (RD)
(1)
(2)
(3)
1.576
1.871
1.917
(2.47)**
(2.57)**
(2.58)**
0.082
0.076
0.075
(2.09)**
(1.96)*
(2.01)**
-0.265
-0.288
-0.287
(-4.05)***
(-3.87)***
(-3.75)***
-0.472
-0.513
-0.504
(-6.38)***
(-6.33)***
(-6.08)***
1.548
1.547
1.705
(4.01)***
(3.87)***
(3.67)***
Income and price variables
Chapter 70 aid component of adjusted income
Log of median income
Log of tax share
Log of cost index
Log of efficiency index
Log of non-school costs
-0.020
-0.034
(-1.82)*
(-1.92)*
Demand for Public Services
Other variables
Regional districts (RD) (= 1 for RD and = 0 otherwise)
-0.068
0.009
-0.079
(-1.98)**
(1.02)
(-2.03)**
-0.249
𝐶𝑀 ×RD
(-2.48)**
Percent of college graduates
0.003
0.004
0.003
(3.63)***
(3.60)***
(3.75)***
0.000
0.000
0.000
(0.02)
(0.07)
(0.18)
-0.001
-0.001
-0.001
(-2.33)**
(-2.30)**
(-2.13)**
0.008
0.007
0.007
(1.48)
(1.37)
(1.22)
Year dummies (2002, 2003, 2004, 2005, 2006)
Yes
Yes
Yes
Constant
Yes
Yes
Yes
Number of observations
1776
1776
1776
Percent of senior citizens
Percent of low-income students in comparison districts
Percent of special ed students in comparison districts
Demand for Public Services
Tax Price with Parcel Tax

The budget constraints
Y  Z  tV  P
C{S }
E
 tV  A  NP
e

Solve for P and substitute
tV  A
C{S }
Y
 Z  tV 
N
eN
Spending dC 1 MC
TP 


S
dS eN eN
Demand for Public Services
California Estimates (D/Y 2011)

About 900 school districts in two years
(2003-04 and 2004-05)

Service is measured by an index (API) of
several tests in several grades developed for
the California school accountability system.

No fixed effects, but clustered errors.
Demand for Public Services
Demand Results from California
Demand for Public Services
California, 2
Demand for Public Services
California, 3

These variables are instruments in the cost
equation.