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A qualification concerning the efficiency of tax expenditures
Journal
of Public Economics
33 (1987) 125-131.
A QUALIFICATION EFFICIENCY OF Patrick
North-Holland
CONCERNING THE TAX EXPENDITURES A. DRIESSEN*
US Treasury Department, Washington, DC, USA
Received January
This paper examines a problem in us to reconsider the normative highlights an implicit assumption model. The reformulation suggests the charity context.
1984, revised version
received November
1986
a welfare loss model of tax expenditures. This problem forces efftciency argument for tax expenditures. A reformulation of improper government allocation contained in the original that subsidization is an inappropriate marginal policy tool in
1. Introduction
The emergence of optimal tax-expenditure analyses can be understood as a logical balance to, and extension of, the wealth of optimal tax studies ushered in by Mirrlees (1971). One major difference between the optimal tax and optimal tax-expenditure analyses is the degree of specificity associated with the tax-expenditure literature: much of it has focused on subsidies for charitable contributions [e.g. Hochman and Rodgers (1977), Atkinson (1976), Warr (1982)]. In general, the conclusions reached in the tax-expenditure studies of charitable contributions have been equivocal, with only qualified judgments on the efficiency of subsidies. Feldstein’s (1980) paper is an exception: it presents a rather strong efficiency argument in favor of tax subsidization of charitable giving. Feldstein’s welfare loss model is an important contribution to taxexpenditure theory because it captures the somewhat elusive efficiency costs of tax expenditures. The model captures the choice between direct and tax expenditures, taking an increase in some favored good f (designed to proxy charity) as the policy goal. Feldstein’s results suggest that tax expenditures are more efficient than direct expenditures in achieving the policy goal for a wide range of parameter values. This paper discusses a problem in Feldstein’s formulation which brings these results into question. The comparison between direct and tax ex*The author wishes to thank Paul Courant, Edward Steuerle, and two referees for comments on earlier drafts.
W7-2727/87/$3.50
0
1987, Elsevier Science Publishers
Gramlich,
Russell
B.V. (North-Holland)
Krelove,
Eugene
126
P.A. Driessen, Efjciency of tax expenditures
penditure is faulty because the model’s government is generally unable to achieve a balanced-budget increase in f through direct provision. A reformulation is suggested which generally reverses the pro-subsidy results obtained previously.
2. The model Feldstein presents an economy with II homogeneous individuals, each of whom supplies L units of labor at a unit wage of w, and faces a constant tax rate t. These individuals have the following utility function:
u=uCc,(l -L),fl,
(1)
where c is a general consumption good with numeraire price, (1 -L.) is leisure, and f is a favored good. Individual i’s production function for f is
where x is a non-governmental input [with price (1 -s)p], g is the governmental input (with price p), the summation represents the ‘giving’ of x by i’s peers, and 5 is a publicness indicator, with 4 = 1 indicating ‘pure public’ and 5 = 0 indicating ‘pure private’. Individuals face a budget constraint of (1 -r)wL+a=c+(l
-s)px,
where a is non-labor income government. The governmental constraint: n( twL - spx) - epg = r,
and s is the subsidy rate chosen by the entity which provides g also faces a budget
(4)
where I refers to other government spending, and e is an efficiency indicator. If e > 1, the government is less efficient than non-governmental organizations, while e< 1 indicates the opposite case. The government appears to have two ways to increase f: either through s or g. Feldstein makes efficiency comparisons between these options by setting df= 1. In these comparisons the tax-expenditure option appears to be robustly superior to the direct expenditure option. However, these results are misleading because of a flaw in the model: the g alternative is non-existent. The government generally cannot increase f by increasing g in a balanced budget fashion, because the combination of contribution displacement and
P.A. Driessen, Efliciency of tax expenditures
127
increased taxation (both of which reduce x) preclude this option.’ In general, the model implies that any case in which g> 0 is non-optimal, an unrealistic implication for a model designed to compare s and g.2 The problem is clearly illustrated below. 3. The problem The general way to see the problem is to ask whether any balanced-budget increase in g can cause an increase J Explicitly building the government’s budget constraint into the problem, f can be represented as f = @C&L s), s, gl, d
(5)
(aflag)=(ah/ax)(ax/at)(at/ag)+(ah/ax)(dx/dg)+(ah/ag).
(6)
Then
Now we can ask whether (aj/ag) >O, with the following results obtained from (2), (3) and (4) [all are similar to Feldstein’s results with the exception of (&lag), which he did not directly derive]?
(W%) = - CJ(1 - %Jll~,
(7)
(W&) = WfJIC~w’l, (dt/dg) = - [ntw(dL/rYg) - nsp(ax/dg)- ep]/[ntw(dL/&) -
w(wwl,
+ nwL
(9)
‘For those familiar with Feldstein’s model, the way to see this is to recognize that zl,, the denominator of dg, is generally negative, implying that dg is generally negative. For example, in the ‘pure private’ case cited often in Feldstein’s paper, with 5 = 0, e = 1, and s = 0, A * = - wLm,&C,,
< 0,
with m, (the marginal propensity to spend on x) positive, H=t/(l -t) positive, and E,, (the compensated labor supply elasticity) positive. Therefore dgO influences most of the results in his paper. Note also that Feldstein’s eq. (33) should read (df/dg)=[l +@n- l)][(l/n)+(dx/8g)], while the e(l-u) term of his eq. (41) should be e(l +a) in both of its occurrences. ‘Feldstein acknowledges that dg may be negative in certain cases (footnote 19, p. 113). However, dg is negative in almost every case, and this implies that the model is improperly formulated. Because of this sign problem, every tax/direct expenditure comparison in Feldstein’s paper is askew. In general, ds is positive, while dg is negative, so that the government in Feldstein’s model is providing too much g and not enough s. 3Eq. (7) is derived by noting that individuals will adjust x to equate demand for, and supply of, f; as Feldstein notes. See his eq. (37) and the paragraph which precedes it. Eq. (8) is the same as Feldstein’s eq. (40), capturing the labor supply adjustment to direct provision. Eq. (9) is derived by applying the Implicit Function Theorem to the government budget constraint, eq. (4). Note also that (ax/&)= -[8x/8( 1 -t)] = [-xr&( 1 -t)], while (dL/&) = -~3L/d(1 -t) = [-L&(1 -t)]. Eq. (10) results from partial differentiation of eq. (2) and a substitution from eq. (7). Eq. (12) is a strai&tforward partial differentiation of (2).
128
P.A. Driessen,
Efficiency
of
tax expenditures
(ww(w~~)
= Cl + an- 111c- x?,wl(1- 41,
(10)
(Wax)
= - C(l- %JJlnlCl + 5(n- l)l,
(11)
(dh/dg) = [ 1 + 5(fi - 1H/%
(12)
where m, and mL are the respective marginal propensities to purchase x and L (the latter is negative), rlij signifies an uncompensated elasticity, J= [l + [(n-l)]/[l+t(n-1)(1-m,)], w’=(l-_)w, andp’=(l-s)p. In general, with a=~/(1 -s), 8= t/(1 - t), n=(p’x)/(w’L), q,,=m,/n, and I/= (1 - &jLw + am,), we get: (Zffli3g)=[l
+<(n-l)][l
-(l
-m,)J-(m,/I/)(e+ea+am,J--aJ
- Bm,J)]/n.
(13)
Eq. (13) will be negative for a wide range of parameter values.4 For example, consider the ‘pure private’ case cited in Feldstein (1980), with 5 =O, J=l. and e=l:’
(V&d = m,(- W,,l V/n.
(14)
Provided the compensated labor supply elasticity (EL,,) is non-negative, eq. (14) will be non-positive for any number of realistic scenarios. This result means that a balanced-budget increase in g with 5 =0 and e = 1 cannot increase f:
4. Reformulation The problem with the model is that nothing constrains the government to maintain an appropriate mix of direct provision and subsidization. Because all the parameters, including e, are exogenously specified, the answer to the question about which policy tool is marginally appropriate hinges on which tool is being relatively overused. With Feldstein’s approach, the government
4Reasonable values for the parameters in (13) specific to the charity context Values for e which are close to unity 0.01, B=OS, a=0.5, qLW= mL= -0.2. negative, provided 0 5 5 s 1. 5The variables m,>O, The reduced
in (14) have the following n>O,
U>O,
with E,,
form in (14) is achieved
VLV= (ELW+ %.).
might be: m,= will make (13)
signs: and CO.
by using the definition
of uncompensated
labor
supply
as
P.A. Driessen, Efficiency of tax expenditures
is
129
initially
accumulation
aside.7
Meeting
this
constraint
requires:
where now the
budget
constraint
is
r’=ntwL-O[nspx+epg].
(16)
Eq. (15) says that the government is indifferent to the choice between s and g on the margin. To ensure that (15) and (16) hold true, e and Q now become endogenous in the model: e and Q take on values which ‘justify’ the prevailing mix of direct provision and subsidization. After solving (15) and (16) for e and Sz, one can then proceed with Feldstein’s experiment of comparing the welfare loss of subsidization (Cs) with the loss attributable to direct provision (Cd) (as C” or Cd gets larger, the welfare loss becomes larger). If we again assume the ‘pure private’ case, with t=O, then Cd and C” can be compared using? Z=Cd-CS=
@,,Cnx + g - kmx/hx,))l - spm,E,,(nx
+g) +pmZ(g+ snx)( 1 +&I,)
-mz(ntwL-
r’) . (17)
6This built-in inefficiency in Feldstein’s model concerns how government spends its tax dollars, with revenue taken as given. This inefficiency is distinct from the Leviathan argument which asserts that government is too big. These two concerns may be exclusive: the government may be too big and yet be conditionally efficient in its spending choices, or the government may be taking in an appropriate amount of revenue but misallocating monies in its spending choices. Feldstein’s model implicitly suggests the latter, with subsidization being relatively underfunded. In light of the wave of tax reform proposals in the United States, which call for a reduction in tax expenditures [e.g. Courant and Gramlich (1982) Hall and Rabushka (1983), Galper (1983)], it seems questionable to build a model in which subsidies are underprovided relative to direct provision. ‘Conditional efftciency as defined by (15) and (16) can be achieved by either restructuring the f function or changing the government’s budget constraint. Following Feldstein, the latter option is undertaken here, although a form of duality makes these two options equivalent. ‘With 5 = 0, e = C(1-4/U&,)1 and
C%L,
- WI + s
P.A. Driessen, Efficiency
130
of tax expenditures
If Z>O, subsidization is more efficient than direct provision and vice versa. Note that the numerator of (17) is negative, so the sign of Z depends upon the sign of the denominator. The denominator will be positive for a number of reasonable parameter assumptions, implying that direct provision is the efficient choice.’ 5. Conclusion The government’s inability to increase the favored good in. Feldstein’s original model of the charity context is largely due to an implicit assumption that the government is misallocating public funds. This inability in turn guarantees that the marginal choice between subsidization and direct provision will generally favor the discriminated-against option (subsidization in this case). If the model is reformulated so that the government is properly allocating its funds between direct provision and subsidization, then the previously-obtained pro-subsidy results are reversed.
_
__
q,,[ntwL-r’-gm,0p’]-gm,m,.Bp’
using the definition
rtx,vfnx +d -gm,p’ of uncompensated
’
price elasticity,
nxp, as
nXp= E,, - m,, where E,, is the compensated price elasticity. Note also that the tax changes required to fund both policy increments will be identical, due to binding constraints (15) and (16) (that is, the taxcost to the government of either approach is the same). Finding the welfare losses, C” and Cd, requires solving for dg and ds (the policy changes) through a Cramer’s Rule manipulation, and some manipulations of the utility function and individual budget constraint [see Feldstein (1980)]. ‘For example, the following values are reasonable estimates for the charity context: m,=O.Ol, tJsp= - 1.01,
s=O.l5,
m,=-0.2,
nx=O.l5g,
t=0.3,
E,,=-1,
p=l,
g+snx=ntwL-r’=(O.l)ntwL.
With these parameter assumptions, Z
References Atkinson, A.B., 1976, The income tax treatment of charitable contributions, in: R.E. Grieson, ed., Public and urban economics (Heath, Lexington, MA) 13-27. Courant, P.N. and E.M. Gramlich, 1982, Tax reform: There must be a better way, National Papers 5 (National Policy Exchange, Washington, DC). Feldstein, MS., 1980, A contribution to the theory of tax expenditures: The case of charitable of taxation (Brookings, giving, in: H.J. Aaron and M.J. Boskin, eds., the economics Washington, DC) 99-122. Galper, H., 1983, Tax policy, in: J.A. Pechman, ed., Setting national priorities: The 1984 budget (Brookings, Washington, DC) 173-200.
P.A. Driessen, Efficiency of tax expenditures
131
Hall, R.E. and A. Rabushka, 1983, Low tax, simple tax, flat tax (McGraw-Hill, New York). Hochman, H.M. and J.D. Rodgers, 1977, The optimal tax treatment of charitable contributions, National Tax Journal 30, l-18. Mirrlees, J.A., 1971, An exploration in the theory of optimal income taxation, Review of Economic Studies 38, 175-208. Warr, P.G., 1982, Pareto optimal redistribution and private charity, Journal of Public Economics 19, 131-138.
of Public Economics
33 (1987) 125-131.
A QUALIFICATION EFFICIENCY OF Patrick
North-Holland
CONCERNING THE TAX EXPENDITURES A. DRIESSEN*
US Treasury Department, Washington, DC, USA
Received January
This paper examines a problem in us to reconsider the normative highlights an implicit assumption model. The reformulation suggests the charity context.
1984, revised version
received November
1986
a welfare loss model of tax expenditures. This problem forces efftciency argument for tax expenditures. A reformulation of improper government allocation contained in the original that subsidization is an inappropriate marginal policy tool in
1. Introduction
The emergence of optimal tax-expenditure analyses can be understood as a logical balance to, and extension of, the wealth of optimal tax studies ushered in by Mirrlees (1971). One major difference between the optimal tax and optimal tax-expenditure analyses is the degree of specificity associated with the tax-expenditure literature: much of it has focused on subsidies for charitable contributions [e.g. Hochman and Rodgers (1977), Atkinson (1976), Warr (1982)]. In general, the conclusions reached in the tax-expenditure studies of charitable contributions have been equivocal, with only qualified judgments on the efficiency of subsidies. Feldstein’s (1980) paper is an exception: it presents a rather strong efficiency argument in favor of tax subsidization of charitable giving. Feldstein’s welfare loss model is an important contribution to taxexpenditure theory because it captures the somewhat elusive efficiency costs of tax expenditures. The model captures the choice between direct and tax expenditures, taking an increase in some favored good f (designed to proxy charity) as the policy goal. Feldstein’s results suggest that tax expenditures are more efficient than direct expenditures in achieving the policy goal for a wide range of parameter values. This paper discusses a problem in Feldstein’s formulation which brings these results into question. The comparison between direct and tax ex*The author wishes to thank Paul Courant, Edward Steuerle, and two referees for comments on earlier drafts.
W7-2727/87/$3.50
0
1987, Elsevier Science Publishers
Gramlich,
Russell
B.V. (North-Holland)
Krelove,
Eugene
126
P.A. Driessen, Efjciency of tax expenditures
penditure is faulty because the model’s government is generally unable to achieve a balanced-budget increase in f through direct provision. A reformulation is suggested which generally reverses the pro-subsidy results obtained previously.
2. The model Feldstein presents an economy with II homogeneous individuals, each of whom supplies L units of labor at a unit wage of w, and faces a constant tax rate t. These individuals have the following utility function:
u=uCc,(l -L),fl,
(1)
where c is a general consumption good with numeraire price, (1 -L.) is leisure, and f is a favored good. Individual i’s production function for f is
where x is a non-governmental input [with price (1 -s)p], g is the governmental input (with price p), the summation represents the ‘giving’ of x by i’s peers, and 5 is a publicness indicator, with 4 = 1 indicating ‘pure public’ and 5 = 0 indicating ‘pure private’. Individuals face a budget constraint of (1 -r)wL+a=c+(l
-s)px,
where a is non-labor income government. The governmental constraint: n( twL - spx) - epg = r,
and s is the subsidy rate chosen by the entity which provides g also faces a budget
(4)
where I refers to other government spending, and e is an efficiency indicator. If e > 1, the government is less efficient than non-governmental organizations, while e< 1 indicates the opposite case. The government appears to have two ways to increase f: either through s or g. Feldstein makes efficiency comparisons between these options by setting df= 1. In these comparisons the tax-expenditure option appears to be robustly superior to the direct expenditure option. However, these results are misleading because of a flaw in the model: the g alternative is non-existent. The government generally cannot increase f by increasing g in a balanced budget fashion, because the combination of contribution displacement and
P.A. Driessen, Efliciency of tax expenditures
127
increased taxation (both of which reduce x) preclude this option.’ In general, the model implies that any case in which g> 0 is non-optimal, an unrealistic implication for a model designed to compare s and g.2 The problem is clearly illustrated below. 3. The problem The general way to see the problem is to ask whether any balanced-budget increase in g can cause an increase J Explicitly building the government’s budget constraint into the problem, f can be represented as f = @C&L s), s, gl, d
(5)
(aflag)=(ah/ax)(ax/at)(at/ag)+(ah/ax)(dx/dg)+(ah/ag).
(6)
Then
Now we can ask whether (aj/ag) >O, with the following results obtained from (2), (3) and (4) [all are similar to Feldstein’s results with the exception of (&lag), which he did not directly derive]?
(W%) = - CJ(1 - %Jll~,
(7)
(W&) = WfJIC~w’l, (dt/dg) = - [ntw(dL/rYg) - nsp(ax/dg)- ep]/[ntw(dL/&) -
w(wwl,
+ nwL
(9)
‘For those familiar with Feldstein’s model, the way to see this is to recognize that zl,, the denominator of dg, is generally negative, implying that dg is generally negative. For example, in the ‘pure private’ case cited often in Feldstein’s paper, with 5 = 0, e = 1, and s = 0, A * = - wLm,&C,,
< 0,
with m, (the marginal propensity to spend on x) positive, H=t/(l -t) positive, and E,, (the compensated labor supply elasticity) positive. Therefore dg
128
P.A. Driessen,
Efficiency
of
tax expenditures
(ww(w~~)
= Cl + an- 111c- x?,wl(1- 41,
(10)
(Wax)
= - C(l- %JJlnlCl + 5(n- l)l,
(11)
(dh/dg) = [ 1 + 5(fi - 1H/%
(12)
where m, and mL are the respective marginal propensities to purchase x and L (the latter is negative), rlij signifies an uncompensated elasticity, J= [l + [(n-l)]/[l+t(n-1)(1-m,)], w’=(l-_)w, andp’=(l-s)p. In general, with a=~/(1 -s), 8= t/(1 - t), n=(p’x)/(w’L), q,,=m,/n, and I/= (1 - &jLw + am,), we get: (Zffli3g)=[l
+<(n-l)][l
-(l
-m,)J-(m,/I/)(e+ea+am,J--aJ
- Bm,J)]/n.
(13)
Eq. (13) will be negative for a wide range of parameter values.4 For example, consider the ‘pure private’ case cited in Feldstein (1980), with 5 =O, J=l. and e=l:’
(V&d = m,(- W,,l V/n.
(14)
Provided the compensated labor supply elasticity (EL,,) is non-negative, eq. (14) will be non-positive for any number of realistic scenarios. This result means that a balanced-budget increase in g with 5 =0 and e = 1 cannot increase f:
4. Reformulation The problem with the model is that nothing constrains the government to maintain an appropriate mix of direct provision and subsidization. Because all the parameters, including e, are exogenously specified, the answer to the question about which policy tool is marginally appropriate hinges on which tool is being relatively overused. With Feldstein’s approach, the government
4Reasonable values for the parameters in (13) specific to the charity context Values for e which are close to unity 0.01, B=OS, a=0.5, qLW= mL= -0.2. negative, provided 0 5 5 s 1. 5The variables m,>O, The reduced
in (14) have the following n>O,
U>O,
with E,,
form in (14) is achieved
VLV= (ELW+ %.).
might be: m,= will make (13)
signs: and CO.
by using the definition
of uncompensated
labor
supply
as
P.A. Driessen, Efficiency of tax expenditures
is
129
initially
accumulation
aside.7
Meeting
this
constraint
requires:
where now the
budget
constraint
is
r’=ntwL-O[nspx+epg].
(16)
Eq. (15) says that the government is indifferent to the choice between s and g on the margin. To ensure that (15) and (16) hold true, e and Q now become endogenous in the model: e and Q take on values which ‘justify’ the prevailing mix of direct provision and subsidization. After solving (15) and (16) for e and Sz, one can then proceed with Feldstein’s experiment of comparing the welfare loss of subsidization (Cs) with the loss attributable to direct provision (Cd) (as C” or Cd gets larger, the welfare loss becomes larger). If we again assume the ‘pure private’ case, with t=O, then Cd and C” can be compared using? Z=Cd-CS=
@,,Cnx + g - kmx/hx,))l - spm,E,,(nx
+g) +pmZ(g+ snx)( 1 +&I,)
-mz(ntwL-
r’) . (17)
6This built-in inefficiency in Feldstein’s model concerns how government spends its tax dollars, with revenue taken as given. This inefficiency is distinct from the Leviathan argument which asserts that government is too big. These two concerns may be exclusive: the government may be too big and yet be conditionally efficient in its spending choices, or the government may be taking in an appropriate amount of revenue but misallocating monies in its spending choices. Feldstein’s model implicitly suggests the latter, with subsidization being relatively underfunded. In light of the wave of tax reform proposals in the United States, which call for a reduction in tax expenditures [e.g. Courant and Gramlich (1982) Hall and Rabushka (1983), Galper (1983)], it seems questionable to build a model in which subsidies are underprovided relative to direct provision. ‘Conditional efftciency as defined by (15) and (16) can be achieved by either restructuring the f function or changing the government’s budget constraint. Following Feldstein, the latter option is undertaken here, although a form of duality makes these two options equivalent. ‘With 5 = 0, e = C(1-4/U&,)1 and
C%L,
- WI + s
P.A. Driessen, Efficiency
130
of tax expenditures
If Z>O, subsidization is more efficient than direct provision and vice versa. Note that the numerator of (17) is negative, so the sign of Z depends upon the sign of the denominator. The denominator will be positive for a number of reasonable parameter assumptions, implying that direct provision is the efficient choice.’ 5. Conclusion The government’s inability to increase the favored good in. Feldstein’s original model of the charity context is largely due to an implicit assumption that the government is misallocating public funds. This inability in turn guarantees that the marginal choice between subsidization and direct provision will generally favor the discriminated-against option (subsidization in this case). If the model is reformulated so that the government is properly allocating its funds between direct provision and subsidization, then the previously-obtained pro-subsidy results are reversed.
_
__
q,,[ntwL-r’-gm,0p’]-gm,m,.Bp’
using the definition
rtx,vfnx +d -gm,p’ of uncompensated
’
price elasticity,
nxp, as
nXp= E,, - m,, where E,, is the compensated price elasticity. Note also that the tax changes required to fund both policy increments will be identical, due to binding constraints (15) and (16) (that is, the taxcost to the government of either approach is the same). Finding the welfare losses, C” and Cd, requires solving for dg and ds (the policy changes) through a Cramer’s Rule manipulation, and some manipulations of the utility function and individual budget constraint [see Feldstein (1980)]. ‘For example, the following values are reasonable estimates for the charity context: m,=O.Ol, tJsp= - 1.01,
s=O.l5,
m,=-0.2,
nx=O.l5g,
t=0.3,
E,,=-1,
p=l,
g+snx=ntwL-r’=(O.l)ntwL.
With these parameter assumptions, Z
References Atkinson, A.B., 1976, The income tax treatment of charitable contributions, in: R.E. Grieson, ed., Public and urban economics (Heath, Lexington, MA) 13-27. Courant, P.N. and E.M. Gramlich, 1982, Tax reform: There must be a better way, National Papers 5 (National Policy Exchange, Washington, DC). Feldstein, MS., 1980, A contribution to the theory of tax expenditures: The case of charitable of taxation (Brookings, giving, in: H.J. Aaron and M.J. Boskin, eds., the economics Washington, DC) 99-122. Galper, H., 1983, Tax policy, in: J.A. Pechman, ed., Setting national priorities: The 1984 budget (Brookings, Washington, DC) 173-200.
P.A. Driessen, Efficiency of tax expenditures
131
Hall, R.E. and A. Rabushka, 1983, Low tax, simple tax, flat tax (McGraw-Hill, New York). Hochman, H.M. and J.D. Rodgers, 1977, The optimal tax treatment of charitable contributions, National Tax Journal 30, l-18. Mirrlees, J.A., 1971, An exploration in the theory of optimal income taxation, Review of Economic Studies 38, 175-208. Warr, P.G., 1982, Pareto optimal redistribution and private charity, Journal of Public Economics 19, 131-138.