National Tax Journal
Vol. 46, no. 2, (June, 1993), pp. 121-34
SUBSIDIES FINANCED
WITH DISTORTING
TAXES
EDGAR K. BROWNING*
Abstract - Subsidizing the production
of
particular goods or services is frequently
justified on the basis of positive externalities, but the conventional analysis neglects the welfare cost of the taxes that
finance the subsidy. This paper shows
how the familiar partial equilibrium analysis of a subsidy can be extended to take
account of the marginal welfare cost of
raising the necessary tax revenue. The focus is on identifying
the marginal social
cost (MSC) of using the subsidy to expand
output which, in conjunction with the
marginal social benefit (MS8) of the output, determines the (second-best) optimal
subsidy. Three types of subsidies are analyzed and compared: excise subsidy, marginal excise subsidy, and in-kind transfer.
An important part of government spending
is devoted to the subsidization of the production (or consumption) of particular
goods and services. Externalities are often
cited as the justification for such subsidies,
and the familiar textbook analysis illustrates
how to determine the optimal Pigovian
subsidy, where the subsidy per unit should
equal the marginal external benefit (MEB)
at the efficient level of output. This analysis is, however, incomplete, because it implicitly assumes there is no welfare cost associated with raising the tax revenue
necessary to finance the subsidy. This is
well known, but exactly how to incorpo*Texas A&M Unwerslty, College Station, TX 77843
12 1
rate the welfare cost of taxation into the
traditional analysis has not been made
clear tn the literature. A major purpose of
this paper is to show how this extension
can be accomplished within a simple partial equilibrium model.
Closely related to this issue is the burgeoning recent literature on the marginal cost
of public funds.’ This literature emphasizes
how the welfare cost of taxation should be
measured for purposes of evaluating the
social cost of government spending. The
relevant measure of social cost is called the
marginal cost of funds (MCF) and equals
the direct resource cost of taxation (the
revenue raised) plus the marginal welfare
cost of collecting the revenue. With MCF
properly measured, the (second-best) optimal level of spending is where the marginal benefit of government spending
equals MU. In principle, that solves the
problem of determining efficient subsidies.
In practice, however, it is not always clear
how to measure the marginal benefit of
government spending, especially when the
expenditure takes the form of subsidization
of particular goods and services, with or
without externalities present. (In the MCF
literature, the expenditure is always assumed to be on a pure public good, so the
marginal benefit is simply taken to be the
summed marginal rates of substitution between the public and private good.) This
National Tax Journal
Vol. 46, no. 2, (June, 1993), pp. 121-34
paper basically extends the MCF literature
by assuming that we know the magnitude
of MCF and shows
how to apply it to the
analysis of different kinds of subsidies.
others, a conclusion that follows directly
from incorporating the welfare cost of taxation as part of the social cost of the polICY.
The analysis in this paper is based on a
partial (equilibrium model of the market
that is being s#ubsidized. The magnitude of
the marginal welfare cost of the taxation
used to substclize this, market is assumed to
be a constant and Independent of the type
or size of subsidy used. Note that this does
ignore the possible interdependencies between the subsidizecl market and the taxed
sector of the economy. While I think this is
often a reasonable assumption and certainly makes the analysis more accessible,
its limitations must be acknowledged. Ng
(1980) has investigated this issue within a
general equilibrium framework, and the
relatively limited conclusions derived serve
to emphasize the necessity of making
some sort of simplifying assumptions about
the relevant interdependencies.
Sandmo
(1975) also analyzes a similar problem
within an optlrnal commodity tax framework where it is assumed that all government expenditures and taxes are optimally
determined rather than assuming (as I do)
that there is ;ln unavoidable marginal welfare cost of taxation associated with financing a subsidy. These approaches lead
to an emphasis on the various possible
types of interdependencies between the
taxed and subsidized sectors, whereas the
approach adopted here allows us to focus
on the implications IlDr particular subsidies
in a given market.
EXCISE SUBSIDY
I begin with an analysis of an excise subsidy, the familiar Pigovian remedy for a
market with external benefits. Then the
analysis is extended to two other types of
subsidies, a marginal excise subsidy and an
in-kind transfer. In each case, I show how
to idenltify the (second-best) optimal subsidy when the expenditure is financed with
taxes that result in welfare costs. The analysis also shows that some types of subsidies are Intrinsically imore efficient than
In what follows, I assume that the product
to be subsidized is produced by a constant-cost competitive industry. Income effects on the demand for the product,
either from changing the form of subsidy
or from collecting the tax revenue, are assumed to be negligible. Much of the analysis does not depend on whether or not
there are (jny external benefits associated
with the subsidized market, but when externalities are assumed to be present, they
are taken to be a function of total industry
output. Obviously, there are important
cases where this is not true-when
external benefits flow from consumption and
differ among individual consumers--but
extending the analysis to these cases is
straightforward.2
Tax revenue to finance any subsiclies is acquired through the use of taxes that produce excess burdens, or welfare costs. The
marginal welfare cost, that is, the additional welfare cost associated with each
dollar of revenue, is assumed to be constant and equal to W The assumption that
W is constant and independent of the size
of the subsidy will gelnerally be appropriate, becaL[se any given subsidy absorbs a
small portion of total tax revenue and
therefore does not significantly aifect the
magnitude of marginal welfare cost
(Browning, 1976).
Marginal welfare cost is closely related to
the concept of the m(arginal cost of funds;
Indeed, the MCF is simply one plus W, or
the direct cost of the revenue plus the indirect efficiency cost produced by increasing tax rates. However, much of the literature on MICF has emphasized that the
proper measure of MCF (or W) depends on
how the funds are spent (Wildasin, 1984).
I22
National Tax Journal
Vol. 46, no. 2, (June, 1993), pp. 121-34
SUBSIDIESFINANCED WITH DISTORTINGTAXES
This issue can be ignored here, since the
focus is on subsidizing a particular product,
and we can just assume that W is the appropriate measure for subsidization of this
product.
When there is a marginal welfare cost of
financing an excise subsidy, the social cost
of expanding output beyond the competitive level will exceed the private marginal
cost of producing the extra output. The
marginal social cost (MSC) of increasing
output with a subsidy is the sum of the
private marginal cost of producing the extra output and the welfare cost of the
taxes that finance the subsidy required to
stimulate the extra output. If W is zero, as
implicitly assumed in the usual textbook
analysis, MSC is simply the private marginal
cost, as given by the supply curve. When
W is positive, there is an additional social
cost and MSC will be greater than private
marginal cost. As we will see, MSC depends not just on W, but also on the type
of subsidy and on the price elasticity of demand, which affects how much government spending is required to expand output. Intuitively, this is obvious: MSC for an
excise subsidy will be smaller the more
elastic the demand for the subsidized
good, because this means that less tax revenue (and hence less welfare cost) is required to expand output by any given
amount. The crucial step in the analysis is
the determination of MSC for each subsidy. Once that is identified, the determination of the (second-best) efficient output
simply involves equating MSC and the marginal social benefit (MSB) of the good,
where the latter measure includes any external benefits that are related to output.
Note that this approach differs from that
in the literature on MCF. There, MCF is
equated to the marginal benefit of government spending to determine efficient
spending. Here, we equate the MSC of expanding output using a particular subsidy
with the MSB of that output, with the efficient level of spending determined implic-
itly by the efficient level of output. Carefully handled, these two approaches are
equivalent, but when applied to subsidies
in particular markets, I believe the approach adopted here is preferable. In part,
this is because it just extends the conventional partial equilibrium analysis of a subsidy In a particular market and so focuses
explicitly on the implications of incorporating the welfare cost of taxation.
Now let us turn to the derivation of MSC
for a per unit excise subsidy. In Figure 1,
the competitive demand and supply curves
are shown as D and 5, respectively, and
competitive output is Q-,. Let the subsidy
per unit be given by T and P be the price
to consumers, equal to MC - T, where
MC is marginal cost. The total budgetary
cost of the subsidy, C, is TQ. The marginal
budgetary cost per unit of output-how
much total spending on the subsidy must
rise to stimulate an extra unit of outputis given by
q
dC
dT
-=Qz+T
dQ
For a constant-cost industry, dT is equal to
dP, so equation 1 can be rewritten as
dC
P
--+T
z-7
where r) is the price elasticity of demand,
expressed as a positive number.
Equation 2 gives the marginal budgetary
cost associated with a change in output
brought about by changing the subsidy per
unit; it is how much taxes must be increased to finance the incremental subsidy
that increases output by a unit. This is not
itself a social cost; as with any subsidy, it
is only a transfer. However, if there is a
positive marginal welfare cost associated
National Tax Journal
Vol. 46, no. 2, (June, 1993), pp. 121-34
FIGURE 1. Marginal Soci.4 Cost Excise Subsidy
MC
with ralislng the tax revenue to finance this
budgetary cost, therie will be a social cost
Ihat results from the marginal budgetary
outlay: it will be given by (P/q t T)W.
This expression gives the added welfare
cost produced by financing the marglnal
budgetary cost with distortionary taxation.
rhat is part of the MSC associated wtth
changing output. The other part is, of
course, the MC of tne resources used to
produce more of thcl product. Thus, the
IMSC of changing output with a per unit
excise subsidy is given by
MSC == MC +- p t T IN.
i 77
)
T’o Interpret equation 3, let us initially assume that the demand curve IS of constant
unit elasticity
In that case, equation
3 simplifies to MC(‘I + PI/), since P -t- T = MC.
For example, if W equals 0.25, then MSC
IS simply 1.25 times the private marginal
cost o-f producing the output, and this
magnitude is independent of the level of
output (In excess of the competitive output). The MSC curve for a unit elastic demand curve (not for the linear demand
curve shown) is shown as MSC, in Figure
-____---
1. Note that It originates at the competltive output, since we are concerned with
the cost of expanding output beyond that
level.
The intuitlon behind MSC, is easy to explain. Whl?n demand is unit elastlic, direct
consumer outlays are constant relgardless
of the size of the subsidy and they are
equal to the resource cost of the first Qa
units. Therefore, the marginal budgetary
cost necer,sary to expand output is exactly
equal to the resource cost of producing
the increment in output. In other words, to
expand output by one unit, the budgetary
cost must increase by exactly MC when
demand io unit elastic. Financing that budgetary cost entails a welfare cost of
(MC)W, so the marginal social cclst is
higher than marginal private cost by the
proportion (1 + W).
When dernand is of constant elasticity not
equal to one, MSC is not a constant but
depends on the size of the subsidy per
unit, as shown In equation 3. For example,
if the price elasticity is 0.5, the additional
welfare cost component of marginal social
cost--the (P/T + T)W term in equation
3--for the first unit of output in excess of
Q. is exactly twice as large as for the unit
elastic case (since, in equation 3, T is equal
124
I
National Tax Journal
Vol. 46, no. 2, (June, 1993), pp. 121-34
SUBSIDIESFINANCED WITH DISTORTINGTAXES
to zero and P is equal to MC for the first
unit). The MSC curve when elasticity is 0.5
is shown as MSC 5 in Figure 1. It is negatively sloped and approaches MSC, as price
approaches zero. Its negative slope arises
because the additional budgetary cost of
successive units falls when demand is inelastic. When elasticity is greater than
unity, the MSC curve begins with a value
lower than MSC, and rises with output,
approaching MSC, from below as price approaches zero.3 MSCz is the MSC curve
when demand is of constant elasticity
equal to two.
The important implication of this analysis,
of course, is that the MSC of expanding
output with a per unit excise subsidy depends strongly on the price elasticity of demand for the product. (When demand is
not unit elastic, it also depends on the size
of the subsidy.) It is (socially) more costly
to increase output of a good in inelastic
demand, because the budgetary cost will
be higher and therefore the additionat welfare cost due to the taxes that finance the
outlay will be larger.
Now let us turn to the determination of
the (second-best) optimal subsidy and output when the competitive output is not efficient. Assume that there are external benefits associated with the output of good X.
The MSB of good X is then P + MEB,
where P is the marginal private benefit
given by the demand curve and MEB is the
marginal external benefit. Efficiency then
requires that output be expanded to the
point where MSB equals MSC, so the condition for efficient output is
II
P+MEB=MC+
This is illustrated in Figure 2. Marginal social cost is shown by MSC, drawn to refleet the assumption of a unit elastic demand curve (although the demand curve is
125
drawn as linear for simplicity). Marginal social benefit is shown by the MSB curve,
the vertical sum of the private demand
curve D and the MEB curve (not shown
separately). The efficient output when a
per unit excise subsidy is used to expand
output is given by the intersection of the
MSB and MSC curves at output Q,, and
the subsidy per unit is equal to GH. The
welfare gain is shown by area ALE. Area
BEFG is the additional welfare cost produced by the tax that finances the excise
subsidy.
By contrast, if the tax used to finance the
excise subsidy produces no marginal welfare cost, the efficient output would be Qz
and the subsidy per unit would be CJ.
(Note that when W is zero, equation 4
simplifies to MEB = MC - P = T: the
subsidy per unit is equal to the MEB, the
familiar conclusion from externality theory.)
The welfare gain is shown by area ABC.
The use of a distorting tax to finance the
excise subsidy therefore reduces the
achievable welfare gain from area ABC to
area AFE, a reduction of area EFCB. It
should also be noted that the efficient
level of government spending is always
smaller with a greater marginal welfare
cost of taxation (implying that MSC is
larger). This is because the efficient output
is always smaller when MSC is larger, implying T is smaller; therefore expenditures,
the product of T and output, will be
smaller.4
It is obvious from Figure 2 that, if MSC is
sufficiently large, it will be inefficient to
use any excise subsidy despite the presence
of positive MEBs. The necessary condition
for an excise subsidy to produce a welfare
gain is that the MSB evaluated at Q0 must
exceed MSC evaluated at QO. Recalling that
T = 0 and MC = P at Q,, the necessary
condition can be expressed as
II
MEB
W
\ -,
77
National Tax Journal
Vol. 46, no. 2, (June, 1993), pp. 121-34
FIGURE 2. Optrr~~~l Exctse Subsidy
\Nhen ldemanld is unit elastic at I&, this
simply says that MEB must be a larger proportion of the market price than W (which
is already expressed (3s a fraction). If MEB
is 25 percent of market price, marginal
welfare cost i’s 30 percent, and demand is
unit elastic, then efficiency calls for not
subsidizing this proditict with an excise subsidy. When demand is inelastic, MEB must
be larger to justify a subsidy, and they
must be smaller if demand is elastic.
put some quantitative perspective on
these conclustons and also to provide a ba!;IS for comparison with the different types
of subsidies considered subsequently, it will
be helpful to develop a simple numerical
example. Assume that demand is unit elastic and select units so that the competitive
price and quantity are 1 and 100, respeci:ively: Q = lOOF’. Assume further that
IMEBs are always half as large as marginal
Iprivate benefits (so the MEB curve is also
[unit elastic): IQ = 5O(MEB)-‘. Then marginal social benefits are given by
-r0
l\/lSB = 150Q-‘.
The welfare gain, G, from expanding
iput with a subsidy is, then given by
out-
Q
G=
I
(MSB - MSC‘IdQ.
QO
When financed with a tax that has a zero
marginal welfare cost, the MSC IS one and
the efficient output is 150 (from setting
equation 6 equal to one and solving). The
welfare g;-lin from the optimal subsidy IS
•91
Q
G =
I
(I ;OQ-’
= ITO In Q -
--
l)dQ
Q 1::: := 10.82.
The welfare gain is therefore just under 11
percent of total outlays on the product.
Since government outlays are $50, the
welfare g,-tin is equal to just under 22 percent of total spending.
Now suppose that W is equal to 0.25
(which is, I believe, a fairly conservative figure based on the estimates available in the
literature) so that MSC is 1.25. The efficient output when financed with this tax is
120, so the welfare gain is given by
I
National Tax Journal
Vol. 46, no. 2, (June, 1993), pp. 121-34
SUBSIDIESFINANCED WITH DISTORTINGTAXES
ary cost of the subsidy is equal to T(Q Q”), so the marginal budgetary cost of
changing output is given by
120
G=
I
(ISOQ-'
-
1.25)dQ
100
= 150 In Q - I .25Q 1;:: = 2.35.
The welfare gain in this case is only 2.35
percent of total outlays on the product,
less than one-fourth as large as achievable
when a nondistorting tax is used to finance the subsidy. It is apparent from this
example (as it is from inspection of Figure
2) that the welfare gains possible from using an excise subsidy are substantially reduced when a distorting tax is used to finance the subsidy. We will return to this
example later below to show how the situation differs for different types of subsidies.
MARGINAL
EXCISE SUBSIDY
A marginal excise subsidy is a subsidy that
lowers the price to consumers only for
units in excess of some exempted level of
consumption. For example, consumers
might have to pay the full market price
(MC) for the first 10 units purchased, and
then additional units would be available at
the subsidized price, MC - T. Such a subsidy is not easy to administer since it must
be provided directly to consumers and resale of the product must be prevented, but
when feasible, this type of subsidy offers
significant efficiency advantages over an
excise subsidy that applies to all units consumed.
Figure 3 illustrates a marginal excise subsidy. The term Q* is the level of exempted
consumption; this is the sum of the exempted levels of all the consumers (the
level can vary from consumer to consumer). A subsidy of T per unit applies to
consumption in excess of Q*, so the price
of additional units is P and total purchases
are Q.
Now consider the MSC of expanding output with this type of subsidy. Total budgetIL,
dC
dT
-=Qz+T-Q*%
dT
dQ
which simplifies to
Equation I I gives the marginal budgetary
cost of increasing output by a unit, and
multiplying this by marginal welfare cost
gives the part of MSC associated with the
financing of the subsidy. Adding this to
the direct resource cost of additional output gives MSC:
u
MSC=MC+[;(I-$)+T]W
Equation 12 shows that, for given values
of 77 and W, the MSC of a marginal excise
subsidy is lower than that of an excise subsidy, and it will be lower the higher the exempted level of consumption as a fraction
of total consumption (Q*/Q) is. This is not
surpnsing since a marginal excise subsidy
involves a smaller total (and marginal) budgetary outlay to stimulate additional consumption and therefore a smaller added
welfare cost from the financing of the subsidy.
Figure 3 shows the MSC curves for three
different levels of exempted consumption
(with demand assumed to be unit elastic in
all cases). MSC is the relationship when
the exempted level is zero; this is simply
the curve for the per unit excise subsidy
from the previous section. MSC* is the relationship when the exempted consump-
National Tax Journal
Vol. 46, no. 2, (June, 1993), pp. 121-34
FlGURE 3. Marginal Social Cost: Marglnal Excise Subsidy
!ion is half the competittve level of Qo. It
begins at exactly half the level of MSC and
rises with output. Finally, MSCo IS drawn
for a marginal excise subsidy when the exempted level is exactly Q,,; note that MSC
begins at MC in this case.
,4s in the case of an excise subsidy, we can
give the necessary condition for there to
exist a margrnal excise subsidy that will rmprove welfare by evaluating the MET3 and
IMSC at the competitive output level.
The result is:
IMEB W
.->. P
11
I\Aarginal external benefits do not have to
be as large to justify a marginal excise sublsidy as to justify an excise subsidy (comIpare equations 13 and 5). Of course, this is
(obvious from the relative positions of the
MSC curves
This analysis makes it clear that there are
Iefficiency advantages to the use of a mar(ginal excise subsidy rather than an excise
<subsidy, and .these advantages are greater
the larger the exempted level of consump-
tion is. To suggest the yuantitatrve magnitudes that may be Involved, let us extend
the numerical example from the iiast section. The same assumptions regarding demand, supply, MEB, rnarginal welfare cost,
and demand elasticity will be used; the
only difference is in tlhe type of subsidy
used. Let us assume t:hat the exempted
level of consumption of the marginal excise
subsidy is 50, exactly half the competitive
level of output, 100. Inserting this and the
other assumed values into equati,on 12, we
find that MSC is equal to 1.25-1 250Q2.
(Evaluating this at the competitive output,
we see that MSC of the first unit is 1 .I 25
compared to 1.25 for an excise subsidy applying to all units.) Equating MSC and MSB
(equation 6), we find that the optimal output for the marginal (excise subsidy (conditional on Q* equal to 50) is 127.8. Thus,
the welfare gain from the use of this subsidy is
1273
G=
r
J100
(150Q-'
- 1.25 + 125OQ-*)dQ
= 150 In Q - 1.2512
-- 1250 Q-' I;;;"
IL8
= 4.76.
I
National Tax Journal
Vol. 46, no. 2, (June, 1993), pp. 121-34
SUBSIDIESFINANCED WITH DISTORTINGTAXES
This welfare gain is more than twice as
large as that possible with an excise subsidy (2.35) but still is less than half as large
as that achievable if nondistorting taxes
can be used to finance the subsidy (10.82).
Applying the same procedure for a marginal excise subsidy that exempts the competitive output (Q* = 100)5, a larger exemption than is likely to be practical, we
can calculate that the optimal output is
134.8 and the welfare gain from the optimal subsidy is 7.74.
These examples suggest that there are substantial welfare advantages to the use of
marginal excise subsidies when distorting
taxes are used to finance the outlays.
IN-KIND TRANSFER
There are a variety of in-kind transfers that
can be used to subsidize consumption of a
good, and the type I will examine in this
section is the unpriced voucher. With this
subsidy, consumers receive a voucher
which can be used only to purchase a
specified product. The voucher is denominated in dollars so that, if the face value is
Y dollars, Y/MC units of consumption can
be purchased with the voucher. (The food
stamp program is an example of this type
of in-kind transfer.) Resale of the product
is not permitted.
One characteristic of this type of subsidy is
that it will not increase consumption of the
good beyond the competitive level until
the face value of the voucher exceeds total
outlays of consumers at the competitive
equilibrium. For example, if a consumer is
given a voucher worth $100 and his or her
unsubsidized outlay on the product is
$150, private purchases will fall to $50
and total consumption will be unaffected.
At least this is true if we maintain the assumption of zero income effects on demand. That assumption is a reasonable approximation if the consumers are also the
taxpayers. However, in many instances
where there is substantial redistribution to
129
all or a subset of consumers (as in the case
of the food stamp program), ignoring income effects would be unwarranted. I will
consrder first the case of zero income effects and then examine how the analysis
differs when income effects are taken into
account.
Thus, with zero income effects, thts in-kind
transfer increases consumption only when
the government finances the entire consumption of the product6 Total cost of the
in-kind transfer is (MC)Q, where Q is the
subsidized quantity and MC is constant
and equal to price since I continue to assume constant-cost conditions. Marginal
budgetary cost, dC/dQ, is undefined up to
the point where government is financing
exactly Q. units; thereafter, it is equal to
MC, and the social cost associated with financing the marginal unit is equal to
(MC)W. Thus, MSC is given by
MSC = MC(l
+ W).
Note that MSC is constant; it does not depend on output or the demand elasticity.
Figure 4 illustrates the analysis for an inkind transfer. Assume the subsidy leads to
consumption of Q units. Total budgetary
cost IS then equal to (MC)Q, and the associated total welfare cost of the taxes that
finance this outlay is equal to W times
(MC)Q. This is shown in the graph as the
sum of the shaded rectangular area (the
welfare cost of the tax necessary to finance the Q. units that would be consumed in the absence of the subsidy) plus
the rectangular area ACFE (the welfare
cost of the tax necessary to finance the expansion in consumption from Q0 to Q). In
effect, there is a fixed cost with an in-kind
transfer, as shown by the shaded area, because the first Q, that are subsidized do
not result in any increase in total consumption. After that cost is incurred, the MSC
of expanding output beyond Q. is shown
by MSC, as given by equation 15.
National Tax Journal
Vol. 46, no. 2, (June, 1993), pp. 121-34
In-Kind Transfer
FIGURE 4. Marginal Social
i
./
MC/
I
1
I_-
The welfare effect of an in-kind transfer
that expands consumption from Q, to Q
(where Q is greater than QJ is now the
sum of two terms. The first is the welfare
cost of subsidizing the first Q. units of
consumption, given by W(MC)Q,. The second telrm is the welfare effect of expanding output from Q0 to Q, given by the difference between MSC and MSB summed
over thlese units. In terms of the graph, the
net gain (or loss) from the subsidy is
shown by area ABC minus the shaded
area. In general, the welfare gain (positive
or negative) is given by
-0
cc;= - W(MC:Q,
+
I
(MSB - MSC)dQ
~Qo
It is clear that MEB must be quite large for
an in-kind transfer to be capable of producing a welfare gain. For example, using
the azumptions of the earlier numerical
example (with unit elasticity and marginal
external benefit equal to half the consumer’s marginal valuation), if the in-kind
transfer is used to finance consumption at
the level where MSC and MSB are equal,
there vvill be canet loss of 22.65. (The tri-
_-----Qo
Q
angle ABC- equals 2 .35, and the shaded
rectangle equals 25 in this case.)
It IS straightforward
to determine how
large MEB must be for the in-kind transfer
to produce a welfare gain in the context
of the earlier numerical example Varying
MEB as a fraction of marginal private beneflts alter the MSB given by equation 6;
for example, if MEB is equal to marginal
pnvate benefit, MSB will equal 2OOQ-‘.
Equating MSB and MSC (equation 15) allows us to determine Q, and equation 16
can be used to solve for the welfare gain.
Following this procedure, I find that MEB
must be greater than 1.12 times as large
as marginal private benefit before there
can be a positive welfare gain from the
use of this type of in-kind transfer. Of
course, the required level of MEB is lower
if marginal welfare cost is lower than the
0.25 value assumed.
Now let us turn to the case where income
effects are important, as would be the
case when the consumers do not pay the
taxes that finance the in-kind transfer. Let
Q, be the level of consumption where the
constraint prohibiting resale becomes binding. Then the in-kind transfer is equivalent
to a lump sum transfer and will expand
130
I
National Tax Journal
Vol. 46, no. 2, (June, 1993), pp. 121-34
SUBSIDIESFINANCED WITH DISTORTINGTAXES
consumption through its income effect up
to the point where consumption equals Q,.
The marginal budgetary cost of expanding
output over this range is then MC/M,
where M is the marginal propensity to
spend on good X out of income. (For example, if MC equals $1 and M equals
0.25, the government would have to
spend $4 to increase consumption by one
unit.) The social cost associated with financing the marginal unit is thus equal to
(MC)W/M, and marginal social cost is
given by
Equation 17 gives MSC up to the point
where the subsidy totally displaces private
purchases, Le., up to consumption of Q,.
Thereafter, each dollar of additional spending on the in-kind transfer will increase
consumption by $1. Thus, MSC for output
in excess of Q, is given by equation 15.
Figure 5 illustrates this case. Marginal social cost is shown by the solid MSC curve,
which is discontinuous at an output of Q,.
(Ignore the downward sloping dashed
FIGURE 5. Comparison of Subsidies
MC
curve for the moment.) Note that, because
of the discontinuity, it is possible for there
to be two (local) optima in this case. It
should also be noted that as M goes to
zero, this case converges to the earlier case
of no income effects.
It should also be pointed out that this
analysis of an in-kind transfer can be extended to apply to public production and
free distribution, as with public schools. At
least if public production takes place at the
same resource cost as private production,
the financing cost of any given output in
excess of Q0 will be the same as for the inkind transfer. If public production is more
inefficient than private production, as a
large number of studies have suggested,7
then this can be incorporated into the
analysis as an upward shift in the MSC
curve(s) in Figures 4 and 5.
RANKING
OF SUBSIDIES
This analysis suggests that the three types
of subsidies can be ranked according to
their social costs of expanding output. It
suffices to compare the social cost of
achieving any given output in excess of the
competitive level for the various subsidies.
The comparison of an excise subsidy and a
National Tax Journal
Vol. 46, no. 2, (June, 1993), pp. 121-34
marginal excise subsidy is simple. We have
shown that tl’le MSC of a marginal excise
subsidy IS lower at every level of output
than that of <anexcise subsidy-compare
equations 3 and 12 Thus, on efficiency
grounds, a mlarginai excise subsrdy is preferable to an #excise subsidy.
The comparison of an in-kind transfer with
an excise subsidy is more difficult, but it
can be shown that the excise subsidy is
more efficient than the in-kind transfer.
First, consider the case when demand is
unit elastic. Then the marginal social cost
of an excise subsidy is MC(l -t VV); see
equatilon 3 and the discussion immediately
following. Note that this is the same as
MSC for the ill-kind transfer for output in
excess of Q, (or Q0 with zero income effects). However, since MSC of the in-kind
transfer exceeds this value for previous
units, the total social cost of expanding
output to any level is larger for the In-kind
transfer. Since the MSC of an excise subsidy when dernand is of greater than unit
elasticity is uniformly less than when dernand is unit elastic, this shows that the
excise subsidy is more efficient than the inkind transfer in this case also.
When demand IS inelastic, the comparison
is slightly more involved. In this c.ase, the
MSC of an excise subsidy is greater than
MC(l + W), and declines as output expands. Such a case IS illustrated in Figure 5
by the downward sloping dashed curve. To
see th’at the total social cost of achieving
any level of output is still always less for
the excise subsidy than for the in-kind
transfer, consider first the case when output falls between Q. and Q,. The MSC of
the first unit of output in excess of Q. for
the excise subsidy is MC t (MC)W/q,
from setting 7-equal to 0 in equation 3.
For the in-kind transfer, MSC is MC +
(MC)Vi/lM. Thus, if M is less than r~, the
MSC curve for the tilxcise subsidy begins at
a lower value than the MSC curve for the
in-kind transfer (as they are drawn in Figure 5) That this must always be the case
follows from the Slutsky equation in elasticity form, because M equals the budget
share times the income elasticity (recalling
that I am treating price elasticities as positrve numbers). Finally, since MSC for the
excise subsidy is downward sloping, the
MSC for ,311units between Q. and Q, is
lower for the excise subsidy. Therefore, the
total social cost of achieving any output
between Q. and Q, is lower for an excise
subsidy, regardless of how low the price
elasticity of demand IIS.
If the subsidized output is greater than Q,,
at first gl,jnce it might appear to be possible for the advantage to go to the in-kind
transfer because its MSC is lower than that
of the excise subsidy over this range, as illustrated In Figure 5. That this is not the
case is easily seen, however. Over this
range of output, government ou tlays on
the in -kind transfer are equal to the total
resource cost of output, since private purchases have been displaced. By contrast,
consumers are still paying part o-f this resource cost when an excise subsidy is used
(as long as the demand price is positive).
Therefore, the tax revenue required for an
excise subsidy to achieve any output in this
range is smaller than for an in-kind transfer, and, from this it follows that the total
social cost of achieving any output in this
range is smaller for ti’le excise subsidy
The conclusion to be drawn from this analysis is, therefore, that the social cost of
achieving any level elf output is lower for a
marginal excise subsidy than for an excise
subsidy and lower for an excise subsidy
than for on In-kind transfer. (It should be
recalled, however, that I have restricted attention tCJ a particular type of in-kind
transfer; for an in-kind transfer of the
priced voucher variety, the ranklng can be
National Tax Journal
Vol. 46, no. 2, (June, 1993), pp. 121-34
SUBSIDIESFINANCED WITH DISTORTINGTAXES
Conclusions
3 In the case of linear demand,
additional
The major goal of this paper has been to
show how the marginal welfare cost of
taxation can be incorporated into the analysis of some familiar types of subsidies. In
developing the analysis, I have ignored a
number of factors that could be relevant in
evaluating actual policies. For example, the
informational requirements necessary to
determine the efficient subsidy and the administrative and compliance cost of the
subsidy have not been taken into account.
Apart from these omissions, it should also
be emphasized that the attention has been
on efficiency considerations. Subsidies are
also often intended to redistribute income,
and if this goal is important, some of the
conclusions of this paper could be
changed. In general, we have found that
subsidies are more efficient when they involve lower levels of spending to achieve
desired output levels (thereby minimizing
the welfare cost from raising the tax revenue)*. But lower spending means smaller
direct benefits to the consumers of the
product, so the most efficient subsidies are
also often the ones that redistribute the
smallest amounts to the consumers.
demand
output
‘n equation
3 The MSC curve will then be up-
ward slop’ng, because the demand elast’c’ty at higher rates
of output
will be lower (see equat’on
4 Equat’on 4 can be rewritten
VT) The left-hand
ENDNOTES
3)
in the form that shows the
marg’nal benef’ts from government
the marginal cost of funds
spending
must equal
(1 + w) = (P + q(MEB)]/(P
+
s’de IS the marginal cost of funds, and
the right--hand s’de IS the marg’nal benef’t from government spending-when
excise subsidy
the spend’ng takes the form of an
In th’s formulat’on,
from government
spending
the marginal benef’ts
~111 differ w’th the type of sub-
s’dy used I prefer the formulat’on
‘n the text, because ‘t
srmply extends a familiar graph’cal analys’s of external benefits
5 Marg’nal social cost ‘n th’s case IS given by 1 252500Q-2
’ I am assumrng that the vouchers are distributed
among
consumers such that the marg’nal values (demand
prices,
or marginal rates of subst’tut’on)
rema’n
equal
Otherwise.
one consumer
It would be poss’ble to subs’d’ze only
thereby total consumpt’on)
sumers
of all consumers
and increase h’s or her consumpt’on
Alternatively,
without
subs’d’z’ng
(and
other con-
tt can be assumed that all consumers
have ‘dent’cal demands
and rece’ve vouchers of the same
value
7 Mueller (1989, pp 261 -66) surveys some 50 stud’es that
have compared
the prov’son of s’m’lar services by publ’c
and pnvate firms
He concludes
“The evidence that publ’c
prov’s’on of a servtce reduces the eff’c’ency of Its prov’s’on
seems overwhelm’ng
Obviously, further work needs to be done
to apply this approach to actual policies,
but this paper does suggest that the marginal welfare cost of taxation has important implications for the evaluation of subsidies. Not only is the appropriate size of a
subsidy sensitive to the marginal welfare
cost, but different subsidies differ as to
their capacity to produce welfare gains.
the MSC of the f’rst un’t of
is given by us’ng the potnt elast’c’ty of
” (p 266)
’ This also suggests that regulat’ons
vate spending should be cons’dered
requ’nng
mandated
as alternat’ves
pr’-
to sub-
sidles, since this avo’ds the use of tax revenue altogether
In this regard, see Flowers (1991)
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Edgar K. “The Marg’nal
Cost of Publ’c Funds ”
Journal of Pohtd
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Flowers,
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Marilyn
Spend’ng ” Working
283-98
Paper, Munc’e, IN Ball State Un’vers’ty,
1991
I would like to thank Timothy Gronberg,
Slemrod, and two anonymous
Ph’l’p Trostel, Joel
referees for helpful com-
Fullerton,
Don. “Reconc’l’ng
’ See Fullerton (1991) and the references g’ven there’n
81 (March, 1991). 302-08
’ Pogue and Sgontz (1989) analyze the symmetrical
Mueller,
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the s’tuat’on
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IS to apply d’fferent
Recent Est’mates of the Mar-
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case
They focus on
Dennis
Economic Review
C. Pub/~ Chorce II Cambridge
Cambndge
University Press, 1989
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the first-best theorettax rates to different
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Correct’ve
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133
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1980)
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Vol. 46, no. 2, (June, 1993), pp. 121-34
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