Public Finance SeminarSpring 2013, 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 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):
1
Y Z PH tVPHZ PH trtZ PHr
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 ).
We will skip state aid and other local revenue sources for now.
{ }E S tV A
Demand for Public Services
Solving for Tax Price, 1
Solve the community budget constraint for t:
Substitute into the household budget constraint:
{ }E S AtV
( { } )V
VY Z PH E S A
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,
where MC is the resource cost of another unit of S, and the ratio of V to average V is the tax share.
Tax Price = = E V VTP MCS V V
Demand for Public Services
Estimating Household Demand
With TP defined, we can write down a household demand function:
The problem: How to estimate this
function?
◦One answer: through surveys.
{ , , Other Prices, Preference Variables}S S Y TP
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):
VS AY TP AY MCV
1( )( ) VE S AC AY MCV
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:
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.
{ , , } , ,MedianMedian Median Median
V
VS S Y TP X S Y MC X
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
◦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
The Community Constraint
The Combined Constraint
{ }C SE tV A
e
1t
Y Z PH tV Z PHr
{ }V C S VY A Z PH
V e V
Demand for Public Services
Components
Tax Price
Augmented Income
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.
1 1Spending dC V VTP e MC e
S dS V V
A VY Y A
V
Demand for Public Services
Constant Elasticity Demand
General Form
Linear Form to Estimate
1V VS K Y f A MC e
V V
* 1
* 1
* 1
*
ln{ } ln ln
ln 1 ln
ln ln 1 ln
ln
V VS K Y f A MC e
V V
A V VK Y f MC e
Y V V
A V VK Y f MC e
Y V V
A VK Y f
Y V
1 2 3ln ln lnV
MC eV
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:
implies that
{ }C S S W N P
1{ }C SMC 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 Ve 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
which can be estimated in log form (with the above approximation).
1* ( 1) V VE k S W N P M Y f A
V V
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:
where the expressions with asterisks can all be identified with the expenditure results.
Note that the simultaneity problem is solved with algebra, not econometrics.
* *
1* * *V VS K Y f A C e
V V
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 overridesNo independent school districts, so
actions may depend on costs of other services
Observe 296 districts over 6 yearsYear dummies, but no fixed effectsService 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 non-school costs ()
interacted with regional dummy
(RD)
Income and price variables (1) (2) (3)
Chapter 70 aid component of adjusted income1.576 1.871 1.917
(2.47)** (2.57)** (2.58)**
Log of median income 0.082 0.076 0.075
(2.09)** (1.96)* (2.01)**
Log of tax share -0.265 -0.288 -0.287
(-4.05)*** (-3.87)*** (-3.75)***
Log of cost index -0.472 -0.513 -0.504
(-6.38)*** (-6.33)*** (-6.08)***
Log of efficiency index 1.548 1.547 1.705
(4.01)*** (3.87)*** (3.67)***
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)**
RD -0.249
(-2.48)**
Percent of college graduates 0.003 0.004 0.003
(3.63)*** (3.60)*** (3.75)***
Percent of senior citizens 0.000 0.000 0.000
(0.02) (0.07) (0.18)Percent of low-income students in comparison districts -0.001 -0.001 -0.001
(-2.33)** (-2.30)** (-2.13)**Percent of special ed students in comparison districts 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
Demand for Public Services
Tax Price with Parcel Tax
The budget constraints
Solve for P and substitute
{ }C SE tV A NP
e
{ }tV A C SY Z tV
N eN
Spending 1dC MCTP
S dS eN eN
Y Z tV P
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.
Demand for Public Services
Subject
Entry 1
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