# State Aid in the Budget Constraint Class Notes JY

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```							     State Aid in the Budget Constraint
Class Notes, JY 2/21/07

These notes discuss the impact of state aid on a
household’s budget constraint. This impact is the key to
studying the impact of state aid on the demand for local
public services. The notes focus on education, because
that is where state aid is most widely used and where the
principles are clearest, but the basic analysis applies to
other types of state aid, too.

Lump-Sum Aid

The household budget constraint:

Y = Z + PH + tV = Z + PH + t PH       (1)
r

The community budget constraint with aid per pupil, A

E (S ) = tV + A              (2)

Combining the two constraints gives
2

Y = Z + PH + [ E (S ) − A]V          (3)
V
This equation can easily be re-arranged to be

Y + AV = Z + PH + E (S ) V            (4)
V                   V

This leads to the first big insight about aid, which is that it
is just like income, except that it is weighted by tax share.
The reason for this is that the impact of \$1 of aid on the
median voter depends on her tax share—that is, on how
much it saves her in taxes. This was first pointed out by
Oates (see his book, Fiscal Federalism).

This is an equivalence result: \$1 time tax share of aid has
an impact on a voter’s budget constraint—and hence on
her choice of S, that is equivalent to \$1 of income.

But there are two important extensions.

1.   The impact of aid is difficult to estimate due to its
nonlinearity.

Remember the multiplicative demand function that is
so common in the literature. Replacing Y with the
left hand side of (4) gives us
3

θ                        θ                  μ
⎛     ⎞
S   = Κ ⎛Y
⎜    + A V ⎞ TP μ
⎟        = Κ ⎛Y
⎜      + AV ⎞
⎟    MC μ ⎜ V
⎜V    ⎟
⎟
(5)
⎝        V⎠             ⎝         V⎠          ⎝     ⎠

We can replace income with augmented income and
estimate this equation—but only by assuming that the
equivalence theorem holds. We cannot test this theorem
this way, that is, we cannot determine whether the
elasticity is the same for Y and tax-share-weighted A.

2.   Estimates of the impact of aid resoundingly reject
Oates equivalence theorem: V /V of aid has a much
bigger impact on the demand for local public
services than does \$1 of income.

This is known as the flypaper effect: money sticks
where it hits.

So how can we estimate the flypaper effect?

First, re-write the augmented income expression:

⎛             ⎞
Y + AV = Y ⎜1+ A V ⎟
⎜       ⎟
(6)
V       ⎝
YV   ⎠
4

Then plug this expression into the demand equation (5),
take logs, and use the approximation, ln(1+a) ≅ a:

⎛   ⎛           ⎞⎞
ln(S ) = ln(Κ ) + θ ln ⎜ Y ⎜1+ AV ⎟ ⎟ + μ ln(TP)
⎜ ⎜     Y V ⎟⎟
⎝ ⎝         ⎠⎠

⎛        ⎞
ln(S ) = ln(Κ ) + θ ln Y + θ ln ⎜1+ AV ⎟ + μ ln(TP)
⎜
(7)
⎝   YV⎟⎠

ln(S ) = ln(Κ ) + θ ln Y + θ AV + μ ln(TP)
YV

Now all we have to do is add a term for the flypaper
effect, say f, and estimate

ln(S ) = ln( K ) + θ ln Y + θ f AV + μ ln(TP)          (8)
YV

The flypaper effect is the ratio of the coefficient of the aid
variable to the coefficient of the Y variable.

Note: One also could add a flypaper effect to equation (5)
and estimate it with nonlinear techniques.

Why does the flypaper effect exist?                         Nobody really
knows!
5

For some reason, voters treat money that comes directly
into the public budget differently than money that comes
in through their own budget—even after correcting for the
tax-share effect.

Some kind of illusion is at work, but nobody, in my
opinion has been able to pin it down. The person who
does will be famous!

Matching Aid

We can also add matching aid, which supplements local
revenue, to the community budget constraint. There are
two ways to set this up. One way is to express the
matching rate as the state share of total spending:
m=S/(S+L), where S stands for “state” and L stands for
“local.” In this case, the denominator equals total
spending less block grant aid, A; that is, states do not
supplement their own block grant aid with a match! In a
community budget constraint, we write

⎛
⎜   E(S ) − A⎞ ⎛1− m ⎞ = tV
⎟⎜      ⎟            (9)
⎝             ⎠⎝       ⎠

which, when combined with the individual budget
6

Y = Z + PH + [ E (S ) − A]V (1− m )        (10)
V
So now the tax-price term has a third component:

TP = MC V ⎛1− m ⎞
⎜     ⎟                   (11)
V⎝      ⎠

The other approach is to express the matching rate as the
ratio of state aid to local revenue: m* = S/L. With this
approach, the rate really is a match—how much state aid
is provided for each local dollar.

With this approach, we write

E (S ) = tV (1+ m*) + A              (12)

and the individual budget constraint becomes

⎛           ⎞
Y = Z + PH + [ E (S ) − A]V ⎜ 1         ⎟   (13)
⎝ 1+ m * ⎠
V⎜           ⎟

So you can see that the two matching rates are related:

⎛       ⎞
⎛
⎜
⎝
1− m ⎞ = ⎜ 1+1 * ⎟
⎟ ⎜
⎠       m ⎟
(14)
⎝       ⎠
7

However the matching rate is expressed, raising the
matching rate lowers the tax price—and stimulates
spending.

This leads to a widely known theorem:

A matching rate is thought to have a bigger
impact on the demand for public services than an
equal cost block grant.

A matching rate has a substitution effect as well as an
income effect—so it stimulates a larger shift toward the
subsidized activity. A corollary is that a matching grant
does not raise the utility of the median voter as much as
does a block grant—because it does not strictly follow her
preferences. The diagram supporting this comes from
intermediate microeconomics, and in particular from
comparing the consumption impact of a price subsidy
with that of an equal-cost income subsidy.

Recall that the demand for public services tends to be
inelastic. This applies to matching grants as well as to tax
shares or marginal cost.

But few studies estimate price elasticities for matching
grants. The reason is that matching grants mess up state
8

budgets: Their cost depends on the decisions of recipient
governments and therefore is hard to predict:
Legislatures can handle predicting revenues, but they
hate to have to predict spending! A strange puzzle.
Because of this uncertainty, legislatures usually make
matching grants closed-ended. They match only up to a
certain spending level. And then they make the cap so
low that most recipients hit it. In effect, this step
transforms the matching grant into a block grant.

But a few studies have estimated price elasticities for
matching grants, and found them to be in the same ball
park as price elasticities based on tax shares or marginal
costs. In other words, response to a matching grant is
inelastic.

One well known implication of this is that one should not
expect the demand response to be much greater with a
matching grant than with a block grant. With a small
substitution effect, the two grants have a similar impact
on demand.

But with the flypaper effect, this may not go far enough.
The theorem ignores the flypaper effect, which only
affects block grants. So it might be true that a block grant
9

is more stimulative than an equal-cost matching grant.
This topic is ripe for further research.

Interaction between Block and Matching Aid

For some reason, the literature has missed a key
implication of this set up if some block grant aid already
exists and the question concerns the impact of additional
aid:

Because matching grants alter the value of block
grant aid to the median voter, it is not at all clear
the matching grants have a greater stimulative
effect, even if the price elasticity is large!

Re-arrange the budget constraint we wrote down earlier

Y + fAV (1− m ) = Z + PH + E (S ) V (1− m )   (15)
V                          V

So the augmented income term has m in it, and so, of
course does the price term. Moreover, the impact of m on
the income term has the opposite sign from its impact on
the price term:
10

The higher the matching rate, the lower the value
of existing block grant aid for the median voter,
and the lower her demand for public services.

Grants and Efficiency
One final twist on this story arises because grants affect
efficiency.

A higher block grant leads to less efficiency through
the augmented income term.

A higher matching grant leads to less efficiency
through the tax-price term.

Thus, both types of grants not only have direct impacts on
demand, but also have offsetting indirect impacts on
demand through efficiency.

Both types of grants have less impact on service
quality (if not on spending) than one would think
because they undermine efficiency—and thereby
boost the effective tax price for a voter.
11

Moreover, if the indirect impact of matching grants is
greater than the indirect impact of block grants, the
standard theorem could be reversed!

One final implication of the link to efficiency, which is
not proved formally here, is that when efficiency is
omitted from the demand equation, the estimated
“income” and “price” elasticities will pick up both the
direct and indirect effects through efficiency, and
therefore cannot be interpreted as standard income and
price elasticities. In fact, the true income and price
elasticities will generally be larger than the estimated ones.

Types of Aid to Education

There are 2 main types of aid to education (3 if you count
ad hoc!): foundation grants and power equalizing
grants.    Power equalizing grants are also called
guaranteed tax base grants (GTB).

Foundation Grants

Most states use a foundation grant for some of their
education spending. It is also the form most widely
12

discussed for recent reforms in most states, including
New York.

With a foundation grant, aid per pupil to district j equals a
foundation spending level per pupil, E*, minus what the
district could raise at what is presumed to be a fair tax rate,
t*. This revenue equals t* multiplied by the district’s
property tax base per pupil, Vj.

Thus, the standard formula for a foundation grant is

Aj = E* − t*V j            (16)

This formula does not recognize cost variation, but this is

Aj = S *C j − t*V j
(17)
NW *
Aj   = S*    j
−t V j
Nj

Note that it is not appropriate to multiply the entire aid
expression the first line of (19) by weighted pupils. This
would link pupil weights to property values, which makes
no sense. A low-value district should not get less aid
because it has more weighted pupils!!
13

One important decision is whether to make t* required. If
it is not required, as in New York and many other states,
then a troubled district is unlikely to reach the foundation
spending level because some of the aid it receives will be
devoted to a tax cut. The City of Syracuse is a good
example.

Power Equalizing Grants (GTB)

A power equalizing grant (GTB) is based on the
philosophy that the amount of money a district spends on
education should depend only on the rate at which it is
willing to tax itself, not on its tax base. It is used in three
or four states.

In symbols:

E j = t jV *                      (18)

Now since spending equals aid, A, plus the local
contribution tjVj, we can also write

E j = Aj + t jV j = t jV * or Aj = t j (V * −V j )   (19)
14

Combining this with equation (21) gives the final form for
aid, namely,

⎛   V ⎞
Aj = E j ⎜1 − j ⎟            (20)
⎜   V* ⎟
⎝      ⎠

Comparing this with equation (19) reveals that GTB aid is
similar to foundation aid except that it is based on actual
parameter.

Equation (23) also reveals that GTB aid is a form of
matching aid, in which the state share of the total, the
expression in parentheses (also m), increases as a
district’s property value declines.

Power-equalizing grants were introduced at about the
same time as the Serrano decision in California, which
was the first major case to throw out a state’s education
finance system. At the time, these grants were thought to
satisfy the criterion of wealth-neutrality, which is said to
exist when there is no correlation between wealth and
spending (or, better, performance, and which was part of
the Serrano decision. Another way to put it is that with
wealth neutrality, a regression of spending (performance)
15

on wealth will have a zero slope. A GTB grant was
thought to do this because it seems to say that spending
does depend on district wealth.

In a well-known article, Feldstein showed that this is not
the case. The impact of a power-equalizing grant on
performance depends on the behavioral response to the
matching rate, which is much higher for poorer districts.
It would be an amazing coincidence if this behavioral
response exactly produced wealth neutrality. (Feldstein
also found that a GTB grant went past wealth-neutrality to
a negative slope; Bill and I find that it moves toward
wealth-neutrality, but does not go past it.)

If the behavioral response is known, one can build it into
the grant to get wealth-neutrality. Alternatively, one can
put in a parameter and adjust it over time until wealth
neutrality is achieved.

Finally, Bill and I show that it is possible to add costs to a
GTB grant. In this case, the cost index simply goes into
equation (23):

⎛         Vj ⎞
Aj = E j ⎜ C j −      ⎟        (21)
⎜          *⎟
V ⎠
⎝

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