National Tax Journal, Vol. 42, no. 2,
(June, 1989), pp. 191-203
MATCHING
GRANTS VERSUS BLOCK GRANTS WITH
IMPERFECT INFORMATIONT
NIRVIKAR
SINGH* AND RAVI THOMAS**
ABSTRACT
A matching grant for an input in public
good production is compared with a block
grant, where nonobservability
of public
good outputs and some inputs prevents the
use of the optimal grant system. The welfare comparison is shown to depend on the
technology of production. The second-best
grant is also compared with the optimal
(full-information)
grant.
1. Introduction
HE issue of design of intergovernTmental
grants in a federal system of
government is a complex one, encompassing political,
administrative
and economic considerations (see, for example,
Break (1980), Aronson and Hilley (1986)
and Petersen et al. (1986)). In this paper,
we examine a particular type of granta matching grant for an input in the production of a public good-and provide an
explicit economic rationale based on imperfections in information.'
These imperfections, as we explain below, rule out the
use of the first-best grant scheme, which
would be an output-matching
grant. The
grantor government's objective is to increase output of this good beyond the level
that would be chosen by the recipient
goverrunent acting totally on its own. This
could be for reasons of super-local spillover of benefits, or for merit good reasons
(Schultze (1974)). Subsidizing an input in
the good's production provides a secondbest way of achieving this goal. An alternative possibility is a more general lumpsum grant, which would also increase
production of the 'desirable' good, as well
as outputs of other goods. We compare the
input-matching grant to this alternative
'block' grant. Both are imperfect ways of
achieving the grantor government's goal
and, as we show, either may be better, de*University of California, Santa Cruz, CA 95064.
**Temple University, Philadelphia, PA 19122.
pending on preferences, technology and
prices.
Before we turn to the specifics of the
model, we briefly put the issue in context.
Recent debate, and grant-in-aid
reform
proposals have focused on the need to consolidate categorical grants, both formula
and project, within broad functional areas
(see Quigley and Rubinfeld (1986) for a
recent summary). The ability of the federal government to measure the fraction
of local output that spills over, and to implement a variable matching rate that a
Pigovian subsidy requires, has been questioned. This inability, along with the variety of options that recipients have to
convert categorical grants to fungible resources (McGuire (1979)) has led to the
following conclusion. Given that most
categorical grants restrict expenditure to
narrow program areas, the shift towards
block grants is seen to increase efficiency
through the elimination of the dead weight
loss resulting from selective inpUt2
subsidies, or attempts to convert contingent
funds to an income supplement.3
The debate, as well as the analysis of
the existing categorical grant system, for
the most part, ignores an important fact
about such grants. That is, very little attention seems to be have been paid to the
fact that a large proportion of categorical
Lrrants are capital grants. As such, these
-grants can be viewed as selective subsidies on intermediate outputs, or for specific inputs as we do in this paper. Project
grants, for example, are often inherently
grants for an intermediate output, such
as, sewage treatment plants and urban
mass transportation
projects. Of course
some grants are more obviously for inputs, rather than for an intermediate output, for example, grants for school buildings and audio-visual
equipment. Such
capital grants have accounted for between 21 to 26 percent of U.S. grants-inaid since 1975 (see Table 1). While the
policy winds have been shifting in favor
191
National Tax Journal, Vol. 42, no. 2,
(June, 1989), pp. 191-203
192
NATIONAL
TAX JOURNAL
[Vol. XLH
Table 1.
Composition of Grant-in-Aid
(billions of dollars)
Outlays
1975
1980
1985
1987
Payments to Individuals
Grants for Physical Capital Investment*
Highways
Mass Transit and Airports
Community and Regional Development+
Sewage Treatment Facilities
All Other
Other
16.4
10.9
4.6
1.0
2.5
1.9
0.8
22.4
31.9
22.5
9.0
2.6
5.8
4.3
0.8
37.0
48.1
24.9
12.7
3.2
5.0
2.9
1.1
33.0
56.4
23.8
12.5
3.5
4.0
3.0
0.9
28.2
TOTAL
49.7
91.4
105.9
108.4
33.0%
21.8%
45.1%
34.9%
24.5%
40.5%
45.4%
23.5%
31.1%
52.0%
22.0%
26.0%
Payments to Individuals
Capital Grants
Other
@ourc*:SpecialAnalyses,Budgetof the UnitedStatesGovernment,FiscalYearl9T7, 1982.1989.
Excludescapitalgrantsthat are includedas paymentsto individuals.
Dueto the,,froblems involvedin classifyingexpenditure,
all ofthe outlaysin this categoryhave beenclassifiedas capital
investment, a inputgroupin whichthe recipientsspendmostof the money.
of fewer restrictions
and broader
grants,
the past and continued use of input grants,
as indicated by the share of capital grants'
in total grant-in-aid,
deserves closer scrutiny. Previous discussions have focused on
administrative
and political
advantages
(see, for example, Tye (1973) and RaE;mussen (1976)). Here we provide a complementary
economic analysis.
This paper seeks to explain the importance given to this form of input subsidy.
In doing so our analysis differs from the
standard intergovernmental
grant model.
Treatments
of intergovernmental
grants
and their effects in terms of standard price
theory work in terms of the welfare function of the receiving government
and the
effects on consumption
of the public and
private goods, but do not model the objective fimction of the grants-giver
or the
production
of public goods (Rasmussen
(1976) is an exception for the latter). We
take account of both these aspects. In doing
so, we treat governments
or communities
as aggregate individuals,
with well defined objective functions and side-step issues associated with aggregation,
or how
government
level and individual
choices
may differ; this is, of course, standard in
this literature.
We may also place our analysis in the
context of the 'principal-agent'
literature,
which distinguishes
two types of asym-
metric information.
First, a higher level
government
(the 'principal')
may not observe preferences,
technology
or other
characteristics
at the local ('agent's') level.5
Second, and what we focus on, the higher
level government
may not observe some
local inputs and outputs, i.e., actions or
results of actions;6 this is in fact the novel
aspect of our approach .7 For example, education may not be perfectly measurable,
though there are achievement tests which
provide some information.
Some inputs,
such as 'general administration'
and education's share of it, are very hard to
monitor. On the other hand, some capital
inputs, such as school buildings,
and audio-visual
equipment,
are very easily
monitored.
Hence, the federal government, seeking to encourage the local production of 'education,' may not be able to
do so directly and optiMally.8
Instead, it
might have a choice between a general
monetary grant (revenue sharing or a
block grant-the
classification
depends on
the degree of aggregation
of the public
goods) which provides an income effect
stimulus,
but also encourages production
of less desirable public goods, and a subsidy of observable inputs, which directly
encourages output of the desirable good
but distorts input choices.
In Section 2, we set out the model and
assumptions.
In Section 3, we compare the
National Tax Journal, Vol. 42, no. 2,
(June, 1989), pp. 191-203
No. 2]
MATCHING
GRANTS VERSUS
alternate grant schemes by analytic means
or via numerical
simulations.
Section 4
summarizes our results.
2. Model
We assume that there are two local
public goods and a single composite private good. The quantities of these goods
are denoted by yi, y2 and x, respectively.
The lower level, local government has a
well defined objective fimetion, uL(yly2,x),
by assumption. Furthermore, we assume
that the higher level government also has
a welfare function, UH(YlY2,X)
over the locality's consumption. Hence the overall
social welfare is assumed to be separable
in the welfare of the locality in question
and that of the rest of society. A further
simplifying assumption is
Al.
UH(Yla2,X) = U"(PY1@Y2,X),P >
Al incorporates the idea that the higher
level goverrunent values consumption of
the first public good more than does the
local government. Implicitly, there is some
positive externality,
associated with this
good, that the local government neglects.
The relative valuation of the externality
is captured by the magnitude of p. Assuming p is constant permits tractablility
and does not qualitatively
affect our results. Different valuations for the other
goods could be similarly incorporated, but
we wish to focus on a single systematic
divergence of objectives. It should be noted
that Al also implies ulH > ul, where the
subscript denotes the derivative with respect to the first argument.
Since we shall be interested in the input choices in the production of public
goods, we make some assumptions about
this technology of production. Spelcifically,
A2. Public goods are produced under
constant returns to scale (CRS).
We shall assume that the inputs are
capital and labor, with quantities k and
1, respectively for public good i. We assume that inputs can be purchased competitively, at prices r and w respectively,
and there are no distortions in these input prices. By virtue of this and A2, the
BLOCK GRANTS
193
cost function for public good i is yic,(r,w).
Hence ci(r,w) is the marginal cost given
the input prices.
The private good, we assume, can be
purchased at a given price q, and the locality has a total income of I. Hence we
assume that the locality's income is independent of its consumption decisions.
We do not need to model the mechanism
whereby funds are raised by taxation for
public good expenditure.
By A2 (CRS), ci is the price of one unit
of the public good i, and the locality's decision problem is
Max
UL(YlY2,X)
Y*la2,-
s.t.
cl(r,w)yl
+ C2(r,W)Y2
+ qx
With our formulation, we have reduced
the locality's problem to a consumer's
utility maximization problem, and therefore the result of solving (1) is an indirect
utility function U"(Cl,C2,q,I),
with all the
standard properties. We next consider the
higher level government's viewpoint of
these local decisions, and its ability to observe and control the outcome. What follows is therefore the heart of our treatment of intergovernmental
grants.
First, in program (1), if we define yt
yllp, then (1) becomes
max
yi-,x
s.t.
U(PYI*,Y2,X)
cl(r,w)py*
+
C2(r,W)Y2
+ qx
But the objective function in (1') is
U'I(YtY2,X), by Al, and the higher level
government's welfare is thus V(PCI,C2,qJ).
Hence the locality acts as if it were the
higher level government faced with the
higher price pcl for public good 1. As a
result yt is lower than yl, the optimal
quantity for the higher level government.
The result is lower welfare, as viewed by
the higher level government than if it
could directly choose the levels of the
public goods, since indirect utility decreases in price, i.e.,
V,(C,,C2@q,I)> U"(PCI,C2,q,I).
National Tax Journal, Vol. 42, no. 2,
(June, 1989), pp. 191-203
194
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TAX JOURNAL
What can the higher level government
do to increase social welfare? We shall assume that it cannot directly choose the
levels of the local public good. This is institutionally realistic in the case of the
federal government because the U.S. constitution restricts the federal government
from intervening in areas considered
strictly in the state or local government's
domain.' The state government is not
constrained in the same manner, but in
our model the inability of the higher level
government to observe public good outputs and certain inputs effectively prevents it from choosing their levels. The
feasible alternative is to affect the incentives of the local government so as to increase consumption of public good 1, by
means of a grant. If the output of good 1
were observable, this would be straightforward: it could use a matching grant for
good 1 up to the point where the higher
level government's marginal benefit was
equal to the marginal opportunity cost of
the grant funds. However, it is in the nature of public goods, such as education,
that their output may not be easily observable or measurable." In that case the
above matching grant would not be feasible. Next, we explore the implications of
nonobservability of y, for grant design.
For simplicity we assume the extreme
of nonobservability, that the local government is able to divert matching funds
for one public good to another, without
beinf detected, i.e., the grant is fungible.' Specifically, this is possible due to
the nonobservability of a particular input
expenditure, combined with the nonobservability Of OUtpUtS.12,13 With this asswuption, a matching grant for a single
public good is equivalent in effect to a
block grant. This is therefore one avenue
open to the higher level government: it can
only encourage production of good 1 by
subsidizing both public goods. Hence the
first-best will not be attainable in this
case
In contrast to the output, say, of education, inputs in its production are more
likely to be measurable and observable.
With a known technology, if all inputs are
observable, output can be deduced from
the inputs. However, it is realistic that a
[Vol. XLII
higher level government might not be
completely informed about the technology; for example, it may not know the
elasticity of substitution between the inputs. In this paper we assume the technology is fully known. Instead, we assume
that only one input, capital, is observble.
This gives the higher level government a
way of increasing consumption of public
good 1, by a matching grant for capital
used in its production. This grant is also
not fully optimal, since it distorts input
choices. Hence we have two different
grants, a block grant and an inputmatching grant, with different defects. We
next compare the welfare effects of these
grants.
Our analysis proceeds by comparing the
two types of grants under the assumption
that the amounts transferred are equal for
the two grants, without computing optimal levels. This avoids having to make any
specific assumptions about the opportunity cost of grant funds.
In order to proceed we need further notation for the two types of grants. Let G
be the amount of a block grant, and let s
be the matching rate for capital used in
the production of good 1, i.e., 0 :s s < 1.
With the matching grant, the marginal
cost of good 1 is cl((l - s)r,w). We will use
cl(s) as an abbreviation. Clearly cl is decreasing in s. Similarly, let kl(s) be the
demand for capital in good l's production,
given s, i.e.,
L
kl(s) = kl((l - s)r, W yl(cl(s),c2,q,I)).
Hence the condition for the two types of
grants to involve the same dollar amount
is
A3. G = srkl(s).
We now turn to the analysis and results.
3. Analysis and Results
We proceed to compare welfare as perceived by the higher level government for
a block grant and an input matching grant,
under the assumption that the two types
of grants involve the same dollar expen-
National Tax Journal, Vol. 42, no. 2,
(June, 1989), pp. 191-203
No. 21
MATCHING
GRANTS VERSUS BLOCK GRANTS
diture by the higher level government.
That is, in terms of the notation introduced previously we compare
V'(PCI(O),C2,q,I
V'(PC1(S),C2,q,I)
+ G) and
given A3.
This is not possible for general welfare
functions because without specific assumptions it is impossible to compare the
two indirect utility functions. However,
we are able to provide a general result for
small grants. We are also able to compare
marginal benefits with a mild restriction
on the welfare functions. This is done in
Section 3.1. In Section 3.2 we examine the
case of Cobb-Douglas welfare fimetions and
CES technology, and make comparisons
via numerical simulations. In Section 3.3
we are able to get some analytical results
assuming Cobb-Douglas technology.
3.1 General Comparisons
The first result is for small grants.
PROPOSITION
1. Starting from the
situation of no grant, the marginal benefit of an input matching grant of the form
described above is greater than the marginal benefit of a block grant that involves the same transfer of funds.
PROOF: See Appendix.
The intuition for the result is as follows. Starting at a point of no grant, the
marginal distortionary effect on input
choice of the matching grant is zero. The
input matching grant is better than the
block grant because it directly corrects the
initial output distortion as perceived by
the higher level government. Note that
while in the proof we have assumed p
constant for simplicity, the result is actually more general. We only need p to be
a differentiable fimetion of the outputs and
have a value greater than one in the nogrant situation.
It may be argued that proposition 1 has
restricted application because in practice
matching rates are typically not small,
ranging from 50 to 94 percent (Aronson
and Hilley (1986)). Hence, our next result
is not restricted to small grants. To do this
we need to assume that the government
195
welfare function is quasi-linear, i.e., it has
the form W(YIY2)
+ x." The next proposition also compares marginal benefits
of the two types of grants. Hence, it does
not provide complete information about
the comparison of total benefits. We discuss this further after stating the proposition. We also need some additional notation. Let E be the price elasticity of
demand for capital used in production of
good 1.
PROPOSITION 2. If the welfare function has the form 0(yly2)
+ x then the
difference in marginal benefits for an input matching grant and a block grant is
positive if
s
p > 1 - --E.
1 - s
PROOF: See Appendix
The comparison between marginal and
total benefits is best understood through
illustration. Figure 1 plots the difference
in total benefits, um - uB, against the
matching rate s for particular CobbDouglas utility functions and CES production functions." In figure 1-A there is
only one turning point and proposition 2
identifies a subset of the region over which
the input matching grant is better. In figure 1B, for a different elasticity of substitution between inputs, proposition 2 is
less informative. Our numerical simulations with these fiinctional forms suggest
that figures 1A and 1B encompass the only
two possibilities. Note that proposition 1
H
tells us that vm
- VHB is always po8 itive
near s = 0.
With its limitations in mind, the condition in proposition 2 allows some conclusions about the relative desirability of
an input matching and a block grant. For
a given p and s, the inequality is more
stringent the greater the magnitude of E,
i.e., the more responsive is the demand for
capital to its price. Also, the condition will
not hold, for a given p and f, if s is sufficiently close to 1-unless of course E =
0, in which case it always holds since p is
greater than 1 by assumption. Hence, if
the subsidized input is very price sensitive but a high matching rate or a large
National Tax Journal, Vol. 42, no. 2,
(June, 1989), pp. 191-203
196
NATIONAL
FigurelA:
[Vol. XLII
Difference in total federal benefits as a function of the subsidy rate
( (7=0.5,
VI
TAX JOURNAL
r/w=1.1,p=1.5)
- VB
0
Input-Matching Grants Better :> 0.6
Figure 1B: Difference in total federal benefits as a function of the subsidy rate
( a = 40, r/w = 1.1, p = 1.5)
VM-
VB
0
Block Grants Better
0.1\\\\/
0.3
block grant is still optimal, depending on
which type of grant is chosen, it is more
likely that a block grant will be preferred
by the higher level government.
Finally, it is instructive to decompose
the input price elasticity F. It is easy to
show that
E @ Ek, + Eky * Tly@ * Ok,
(2)
where the right hand side terms are, respectively, the conditional input price
elasticity, the output elasticity of the input, the price elasticity of demand for the
aided good, and the share of the input in
total cost. Hence, for example, if the subsidized input is relatively more important, i.e., 0,, is higher, then, other things
equal, the condition in proposition 2 is less
likely to hold.
3.2 CES Technology
In this section we work with a CES production function. We are particularly in-
s
terested in how the elasticity of substitution between inputs affects the
comparison between an input matching
grant and a block grant. Note that the
limiting case of a zero elasticity of substitution is straightforward. In this case
the input matching grant causes no distortion in input use, and so the input
matching grant better achieves the higher
level goverwment's goals. It might seem
that as the elasticity of substitution increases, at some point the block grant
would become preferred and stay so for
further increases, but this turns out not
to be the case.
In this and subsequent sections we assume that the utilitk fimction has the
Cobb-Douglas form, u = a, In y, + OL2 In
Y2 + a3 In x, with (xl + (X2 + OL3 = 1. The
corresponding indirect utility is vH = In I
- CL1In pci - OL2In C2 - ot3In q.
For ease of exposition and computation
we assume the simplest form of CES pro-
National Tax Journal, Vol. 42, no. 2,
(June, 1989), pp. 191-203
MATCHING
No. 2]
GRANTS VERSUS BLOCK GRANTS
duction function, with capital and labor
entering symmetrically. Hence,
y, = (kPi+ ll)'IP, with -- < p < 1.
Therefore u = 1/(l - p) is the elasticity
of substitution. For this production function,
Cl =
(rl-a
+
w
1-a)lll-a
, and ki = (cllr)'yl.
With some further substitution,
simplification, we have
UHM _
VB
'q
0"
- or
(1
In 1 +
I
+
(wlr(l
(
1n
1
1 + (wlr(l
and
S
1
- s))'
Otl
s p
(3)
We would like to examine the behavior
of the last expression as a function of the
matching rate, input prices, and utility and
production function parameters. This is
not feasible by analytical techniques and
we proceed by using numerical simulations.
Figure 2 shows the combinations of the
matching rate and the elasticity of substitution for which one or the other grant
scheme is preferred by the higher level
government. The graph is divided into two
parts. In figure 2A, u exceeds 1, while in
figure 2B it ranges from 0 to 1. This division allows us to clearly display the different behavior of em - uB' for low and
high elasticities of substitution. The dividing line is, of course, the Cobb-Douglas case. For low elasticities of substitution the results agree with an obvious
intuition: up to some level of the matching rate, the input distortion, which is
negligible for small s, continues to be outweighed by the fact that the input matching grant directly affects the desired output. Hence, for these values of s, the input
matching grant is preferred. For higher s,
the input distortion is too great and the
197
block grant is preferred. In figure 2B the
region to the left of each line represents
values of u and s for which the input
matching grant is preferable, for a particular factor price ratio. Each line is labeled with the associated factor price ratio.
We may note two other features of figure 2B. First, as the elasticity of substitution approaches zero the critical value
of s approaches 1. As noted earlier this is
what we would expect since in this case
the input distortion associated with the
input matching grant becomes zero. More
generally, the dividing value of s increases as the elasticity of substitution
decreases. This also means that for a given
matching rate there is a value of the elasticity of substitution such that the input
matching grant is preferable only below
that value.
For high levels of the elasticity of substitution the situation is considerably more
complex, as may be seen from figure 2A.
There, we again display the regions over
which one or the other grant scheme is
better for the same three values of r/w as
in figure 2B. For r/w = 1.1 the shaded
side indicates the direction in which the
input matching grant is preferable. For
other values of r/w, the area to the left
of the line represents values for which the
input matching grant is better. When r/
w = 0.5, then for each value of the elasticity of substitution there is a critical
value of s below which the input matching grant is preferable and above which
the block grant is better. This is not true
for r/w = 1.1. For high enough (Tthe input matching grant if; preferred for values of s very close to zero, and then again
for intermediate values of s. The intuition
for this phenomenon is as follows. Increases in s beyond zero very quickly lead
to a substantial input distortion, which in
turn makes the block grant preferable.
Further increases in s, however, stimulate output of the desired good and the input matching grant becomes preferred
again. For still higher s the increased
output is valued less and the input distortion matters more. For r/w = 2 we
again have a single dividing line between
the two grant schemes. Now the initial
factor price ratio is so unequal that the
National Tax Journal, Vol. 42, no. 2,
(June, 1989), pp. 191-203
NATIONAL
198
TAX JOURNAL
Figure 2ADividing lines between grant types
80
rlw
r/w
1.1
2.0
[Vol. XLII
p = 1.5)
r/w
r/w
0.5
70
60
50
40
30
20
10
subsidy rate
0
0.4
0.2
Dividing
0.6
1.0
0.8
Figure 2B:
line between grant types
(p
1.5)
Block Grants
Better
0.6
0.5
Input -Matching
Better
o.4
0.3
0.2
rlw = 0.5
0.1
0
r/w = 1.1
r/w = 2.0
02
0.4
phenomenon described above disappears.
If we see how the dividing line changes
as the elasticity of substitution changes
for a given matching rate, there are similarities for all three values of the factor
price ratio. In each case, for s high enough,
there is a value of cr below which the input matching grant is preferred, and above
which the block grant is preferred. For
0.6
0.8
1.0 subsidy rate
lower S, however, as cr increases further,
the input matching grant may again become preferred. The intuition is again in
terms of the relative importance of input
distortions and effects on the desired output.
From figure 2A, the effect of the factor
price ratio on the relative desirability of
the two grant schemes is also seen to be
National Tax Journal, Vol. 42, no. 2,
(June, 1989), pp. 191-203
No. 21
MATCHING
GRANTS VERSUS BLOCK GRANTS
complex. As r/w increases from 0.5 to 1.1
the region over which the input matching
grant is better shrinks. However, this does
not happen as r/w increases from 1.1 to
2.0 if (r is above a certain level. Finally,
the effect of changes in p is more
straightforward. In fact, by differentiating the expression in equation (3), it can
be shown that as p increases, the region
over which the input matching grant is
better increases.
199
From proposition 1 we know this is positive for s close to zero. For this special
case we can deduce more by differentiating the last expression. It is easy to show
that
a(vhmas
vBH)
> 0 if andonlyif
(p - 1)
s<(p - aotl)
(5)
Statement(5)allows us to assert that
vB' > 0 for an interval of s of the form
In this section we employ the Cobb- (0,9). 9 is not analytically obtainable, but
Douglas technology to examine several it is implicitly defined by setting the difadditional issues. In addition to a further ference between the matching and block
analysis of how the utility function and grants equal to zero. Hence we may exproduction function parameters affect the plore how it varies with the parameters
relative desirability of equally costly in- p, a and a,.
put-matching and block grants, we look
The following result is stated without
at the welfare loss due to the use of an proof
input-matching grant rather than the fullPROPOSITION
3. For a Cobb-Douginformation optimal output-matching
las welfare function and Cobb-Douglas
grant. Also, we explore the issue of the technology in producing public good 1, the
optimal grant for a particular specifica- range over which an input matching grant
tion of the cost of grant fimds for the is better than a block grant increases with
higher level government.
the magnitude of the distortion, p. The efThe Cobb-Douglas technology, with fects on 9 of changes in the elasticity of
constant returns to scale, is y, = k'111-4. output of good 1 with respect to the subThe technology of the other good need not sidized input, a, and of changes in the
be specified, and can be very general. The elasticity of welfare with respect to conmarginal cost, conditional demand for sumption of good 1, ocl, are ambiguous.
capital and the output functions are reWhile the direction of variation of 9 with
spectively;
respect to 'a' is ambiguous, numerical
simulations suggest that in practice 'a@has
(1-a)
w
little effect on 9. The conclusion for oL,is
1-a
cl = Aew
, ki = aA
Yi
the same since it enters (4) in the same
manner as 'a'.
Next, we examine the welfare loss due
and y, = (xi
to the use of the input matching grant
pcl
rather than the output matching grant,
which would be optimal if the higher level
where A = a-'(1 - a)'-'. Hence, the equi- government had full information.
librium use of capital with no grant is k,
Let t be the matching rate for the production of public good 1, if its output is
a(ot'I
observable.Then an input matching grant
pr).
involver, the same transfer of funds
With some further derivation we have which
satisfies
V'M' - VB" = -cL,a ln(l
- s)
srkl(s) = t(rki(t) + wli(t)).
s
aoti).
-In 1+-.(4)
(1 - S) p
Substituting for the input demands and
3.3 Cobb-Douglas
Technology
t#m -
National Tax Journal, Vol. 42, no. 2,
(June, 1989), pp. 191-203
200
NATIONAL
TAX JOURNAL
simplifying, we obtain s = t/(a + (1 - a)t).
From this we can obtain the welfare loss
from the lack of observability of outputs,
and show that this is increasing in t. 16
Hence, as might be expected, the welfare
loss due to the lack of information is
greater for a larger grant.
We can also examine the effect of an increase in 'a' on the welfare loss. It can be
shown that an increase in the elasticity
of output with respect to capital reduces
the welfare loss from using the input
matching grant. This is what we would
expect.
Finally, we explore the issue of the optimal input-matching grant. To do this,
we must explicitly introduce the opportunity cost of grant funds. If the cost is
linear, the higher level government will
wish to increase the grant without limit
given the Cobb-Douglas welfare and production functions. Therefore we let the cost
be eG - 1, where G is the amount of grant
funds. Even this simple case is not analytically tractable, but we may extract
some further information from numerical
simulations. Figure 3 plots the optimal
input matching grant, characterized by s*,
for various values of p and I (a and ot,
constant). We observe that for sufficiently
high p and I the optimal input matching
[Vol. XLII
grant lies in the region where the input
matching grant is better than a block
grant, the region to the left of 9 in figure
3.
4. Conclusion
In this paper, we have provided an economic rationale for input matching grants
as a second-best alternative in the face of
certain kinds of imperfections in information. We have examined how the relative desirability of an input matching
grant over a block grant depends on various factors. For example, an inputmatching grant is always preferable for
small grants. Furthermore it is preferable over a wider range of technologies and
subsidy rates if the higher level government has a strong relative preference for
the good it wishes to aid. For low elasticities of substitution in production, this is
also true as the elasticity of substitution
decreases, or as the relative price of the
matched input decreases. However, for
high elasticities of substitution, such
comparative statics are ambiguous. This
is because of the differing importance of
substitution and output/utility
effects.
Further comparisons are made in terms
of various input and output elasticities.
Figure 3:
(a = 0.5, (XI= 0.5)
p
1.9
s
1.8
(I= 0.9) s*(l 0.7) .
s (I = 0.5)
1.7
I-s
1.6
1.5
1.4
400
1.2
' o
0
0.2
0.4
0.6
0.8
1.0
subsidy
rate
National Tax Journal, Vol. 42, no. 2,
(June, 1989), pp. 191-203
No. 21
MATCHING
GRANTS VERSUS
Hence empirical work which can estimate
such quantities, combined with our analysis, may have implications for designing
grants according to economic criteria. This
remains a subject for further research.
ENDNOTES
tAn earlier version of this paper was presented at
the Econometric Society Winter Meetings in New Orleans, December, 1986. We are grateful to Steven Craig
for his extremely helpful comments as a discussant
on that occasion. We also thank Robert Adams, George
Break, Dan Friedman, Ron Grieson, Peggy Musgrave, Richard Musgrave, John Quigley and Don
Wittman for useful comments and discussions on this
and related work. Finally, the comments of two anonymous referees were very valuable. Remaining errors
are ours. The first author acknowledges with thanks,
financial support from NSF Grant SES 860643a ana
various UCSC Faculty Research grants.
'Although
the appropriateness
of using inputmatching grants to correct for spillover benefits has
been questioned in the literature (see Schultze (1974),
McGuire (1971)), a satisfactory formal economic rationale has not been developed for its use. In fact, the
inefficiency generated by distorting input prices argues against its use.
2We may think of grants for inputs as being an extreme case of narrowness. If one were to di8tin gtiish
between final and intermediate outputs in the provision of a public good, then most categorical gra nts
are associated with inputs (Bradford, Malt and Oates
(1969)). Of course our analysis does not rely on this
view of public goo&.
sthrough the resale of aided goods at a disc ount,
or entitlements at a discount. For example, smal ler
communities in California sell Federal-Aid Urban
System fxmds to large cities and counties at a discount. The smaller jurisdictions
accept the discount
because they don't have to match funds and save on
processing ooets. In many instances the State has acted
as an agent and had standardized the discount rate
at 65 percent (OHP (1985)). Other examples are in
McGuire (1978), (1979).
'Although only grants for physical capital inv estment are listed in this table we do not intend to imply
that they are the only input grants possible. Neither
do we mean to imply all capital grants are matching,
open-ended, categorical grants.
'Such lack of information and its implications for
decentralization
in general are discussed in Holmstrom (1984) and Guesnerie and Laffont (1984). Specffic applications to the design of grant schemes are
in Bohn (1984) and Singh and Thomas (1984a). An
application to user charge requirements for grants is
in Singh and Thomas (1986).
'This leads to moral-hazard-type
problems, analyzed abstractly in Holmstrom (1979), Shavell (1979)
and numerous other papers (references in Holmstrom). In the context of grants this is the fungibility
problem (see McGuire (1978, 1979) and Singh and
Thomas (1984b)). A discussion of the "agency approach" and its empirical application to intergovernmental grants is in Craig and Kohlase (1985).
BLOCK
GRANTS
201
7Rasmussen mentions some reasons for funding re'trict"'
which he traces to administrative and political considerations. Some of these "noneconomic
reasons" are control of grant spillovers and avoidance
of indefinite commitment of funds. We suggest that
these noneconomic reasons are really problems of imperfection of information, and in some cases can he
e licitly modeled.
XP,Here,and throughout the paper 'optimal' means
according to the objective function of the federal government.
9However, no Court decisions have disputed the right
of the federal government to involve itself in grantin-aid activity (see ACIR, A-62 (1978), for a more detailed discussion of the constitutional issues).
"Dffficulties in isolating measures of output, and
the resulting problems it posesfor deternuning unit
costs are discussed in Brafford, Malt, and Oates (1969).
"Grant moneys are considered to be 'Tungibl&' when
the recipient is able to use the fimds for purposes other
than those specified in the grant authorization
(see
ACIR, Summary and Concluding Observations, Report A-62, June 1978). We note that fxmgibility may
anse even though the donor may observe the recipients' actions, i.e., through the inability
to control.
However, our concern is the conversion of categorical
funds which arises through trading or reclassification.
IzThe process by which a grant is made fungible is
illustrated here (the example is from Friedman (1984)).
A local government requiring medical equipment for
its hospital could transfer hospital security guards to
the police payroll. Then, it could apply for criminal
justice funds. The federal funds plus local resources,
would then be used to maintain police services plus
the guards. The hospital would rind its costs decreased freeing the required funds for the desired
ent.
e'@@-Wp
' emneed nonobservability
of output as well as certain inputs; otherwise, the federal government with
knowledge of the technology may be able to infer the
input use through the observation of output.
"'This is a common simplifying assumption in theoretical treatments. Why it is necessary here will be
clear from the proof of proposition 2 in the appendix.
We should note this form implies zero income elasticity for the public good.
"These are used in subsequent analysis and nunierical simulations.
'cdetails of this and other derivations are available
from the authors.
REFERENCES
Advisory Commission on Intergovernmental
Relations, I;ilmmary and Concluding Observations, A62 (U.S. Government Printing Office, 1978).
Aronson, J. R. and J. L. Hilley, Financing State and
Local Governments, The Brookings Institution,
Washington, D.C., 1986.
Bohn, H., "Taxation and Intergovernmental
Grants
in a Federal Governmental
System," Graduate
School of Business, Stanford University, April, 1984.
Bradford, D. F., R. A. Malt, and W. E. Oates, "The
Rising Cost of Local Public Services: Some Evi-
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(June, 1989), pp. 191-203
202
NATIONAL
TAX JOURNAL
denceand Reflections," National Tax Journal,
22:185-202,1969.
Break, G. F., Financing Government in a Federal System, Washington D.C.: The Brookings Institution,
1980.
Craig, S. G. and J. Kohlase, "Why There is not an
Unified Welfare System: Fiscal Federalism from an
Agency Approach," in Perspectives on Local Public
Finance and Public Policy 11,J. Quigley ed., Grenwich, JAI Press, 1985.
Friedman, L. S., Microeconomic Policy Analysis,
McGraw Hill, N.Y., 1984.
Guesnerie, R. and J.-J. Laffont, "A Complete Solution
to a Class of Principal-Agent Problems with an Applicatiton to the Control of a Self-Managed Firm,"
Journal of Public Economics, 25:329-369, 1984.
Hohnstrom, B., 'Moral Hazard and Observability," Bell
Journal of Economics, 10:75-91, 1979.
Holmstrom, B., "The Theory of Delegation," in M.
Boyer and R. Kihlstrom, Bayesian Models of Decision-Making, North-Holland, Amsterdam, 1984.
McGuire, M. C., "Cost Versus Performance Subsidies
as Tools of Intergovenunental Finance," National
T= Journal, 24:13-18, 1971.
McGuire, M. C., "A Method for Estimating the Effect
of a Subsidy on the Receiver's Resource Constraint," Journal of PublicEconomics,16:25-44,
1978.
McGuire, M. C., "The Analysis of Federal Grants into
Price and IncomeComponenta,"in Fiscal Federalism and Grants-In-Aid, P. Mieszkowski et al.
Washington, D.C.: The Urban Institute, 1979.
Oates, W., Fiscal Federalism, New York: Harcourt
Brace Jovanovich, Inc., 1972.
Office of Highway Planning, Review of Federal-Aid
Urban System Program, U.S. Department of Transportation, Washington, D.C., January 1985.
Petersen, P. E., B. G. Rabe and K. K. Wong, When
Federalism Works, Washington, D.C.: The Brookinp Institution, 1986.
Quigley, J. M., and D. L. Rubinfeld, "Budget Reform
and the Theory of Fiscal Federalism," American
EconomicReview,76:132-136, 1986.
Rasmussen,J., "The Allocative Effectsof Grants-InAid," National Tax Journal, 29:211-219, 1976.
Schultze, C. L., "Sorting Out the Social Grant Programs: An Economist's Criteria," American EconomicReview,Vol. 64, No. 2, May 1974.
Shavell, S., "Risk Sharing and Incentives in the Principal and Agent Relationship," Bell Journal of Economics, 10, 55-74, 1979.
Silkman, R. and D. R. Young, Subsidizing Inefficiency:A Study of State Aid and Local Government
Productivity, Praeger, New York, 1985.
Singh, N. and R. Thomas, "Intergovernment Grants
and Information Asymmetries," unpublished, U.C.
Santa Cruz, 19&4a.
Singh, N. and R. Thomas, "Fungibility of Categorical
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Singh, N. and R. Thomas, "User Charges as a Delegation Mechanism," National TaxJourn4al, 39:log113,1986.
Tye, W. B., "The Capital Grant as a Subsidy Device:
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sion,TheEconomicsof FederalSubsidyPrograms,
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Wilde, J. A., "Grants-In-Aid: The Analytics of Design
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1971.
5. Appendix
5.1 Proof
of Proposition
I
The marginal benefit of a block grant is
aV(PCI(O),C2,q,I)IaI.
From A3, an equivalent input matching grant
satisfies
rki(s) + sr
ds
- = 1.
dG
as
1
ds
0, this becomes- = -.
dG rk,(O)
The marginal benefit from the input matching grant is
As s -
auff
ac, cls
*p -- -(evaluated
as dG
a(pc,)
at s = 0).
Substituting
in for ds/dG and using ac,las
auff ac, p
-rac,lar,
this becomes
Howa(pc,) ar k, *
ever, from Shephard's Lemma, y,ac,lar
k,.
aul . p
Substituting again, we get
- Fia(pcl) Y,
au"la(pc@)
nally, from Roy's Identit,, ,
au-1ai
This last substitution gives the following result for the marginal benefit of the input
matching grant, p -aulai, which is greater than
aumlai, since p > 1.
Q.E.D.
5.2 Proof of Propostion
The general
ds
dG
expression
rk, + sr(ak,las)
ak,
ak,
- = -(-r).
a(l
- s)r
as
Hence
2
for
National Tax Journal, Vol. 42, no. 2,
(June, 1989), pp. 191-203
MATCHING
No. 21
ak,
(1 - s)r
E = -
k,
GRANTS VERSUS BLOCK GRANTS
(1 - s) ak,
a(l- s)r
k,
as
Hence
1, is therefore,
auff
pl-aI 1
se/(l
s)'
whilethe marginal
ds
dG
Proposition
-
SE/(l
benefit of
theequally
costly
allH
1
rkl(l
203
blockgrant
-
8))'
The marginal benefit from the input matching grant, following the steps in the proof of
ear,-
auH
a(I + G)
is -.
Since
a(I + G)
aum
= -. Hence
aI
UH
is quasi-lin-
the result follows.
Q.E.D.