Public Finance Seminar Spring 2013, Professor Yinger Demand for Public Services: The Median Voter and Other Approaches.

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


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:


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.


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)


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


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


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


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


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


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.


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


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!


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.


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


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).


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.


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.


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


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


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


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!


The Cost Function

A multiplicative form:

implies that

{ }C S S W N P

1{ }C SMC S W N P

S


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


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


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


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!


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


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)*


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


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


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 Results from California


California, 2


California, 3

These variables are instruments in the cost equation.


Subject

Entry 1

Public Finance Seminar Spring 2013, Professor Yinger Demand for Public Services: The Median Voter and Other Approaches.

Documents

demand model

household demand slide

households demand

low demand

structure of demand

high demand

demand distribution

household demand function