- Email: [email protected]
A model of tax-cum-rebate oil (gasoline) policy
Resources
and Energy
A MODEL
3 (1981) 163-173.
North-Holland
OF TAX-CUM-REBATE
Publishing
OIL (GASOLINE)
Young Chin KIM and Nicholas Northern
Illinois University,
Company
POLICY
N. NOE*
DeKolb, IL 60115, USA
This paper develops a model to analyze the impact of alternative tax-cum-rebate policies currently being discussed as possible means of reducing gasoline consumption. The saving in gasoline from a 30-cent/gallon tax is simulated for four alternative rebate schemes and for a variety of elasticity estimates. The short-run impact is found to be small, while the long-run saving would probably be 3 to 6 Y, (and may even be as high as 9 Y,). The choice of rebate plans is shown to be an unimportant consideration in terms of the amount of gasoline consumption. It is also found that there would be no negative impact on the government budget under any of the four rebate schemes.
1. Introduction Prompted by precipitious increases in the price of oil in recent years, a great deal of interest has been expressed for some public policy to reduce the oil consumption. The purpose of this paper is develop a model to analyze and to simulate a particular type of policy, namely, to impose an excise tax on oil and then to rebate the tax revenue to the consumer. The analytical method is comparative static, partial equilibrium, competitive analysis; and the working of the model is illustrated for the case of U.S. gasoline market. Within the basic framework of tax-cum-rebate measures, four alternative schemes can be distinguished - depending on how the amount of rebate is to be determined. Let P, and Q, =price and quantity, respectively, of gasoline (x): P, and Q,=price and quantity, respectively, of ‘the other product’ (J); T=current per unit tax (in $‘s) on gasoline transactions; dT =additional per unit excise tax (in S’s) on gasoline transactions being considered; and M=amount of money income (in $‘s). Then, the first of the four alternative schemes (Plan A) is to impose a specific tax equal to dT with no rebate to the consumer. The second option (PlanB) is to levy dT and at the same time to rebate just enough money income to rnable the consumer to purchase the pre-Plan B quantities of x and y at the post-Plan B prices. The third scheme (Plan C) is to impose dT and then set the amount of rebate equal to the extra expenditure on gasoline if the consumer were to purchase some pre-determined, smaller-than-initial amount of gasoline at the *The authors would like to thank version helped to improve the paper.
016550572/81/000Cr0000/$02.75
an anonymous
referee,
whose
11:; 1981 North-Holland
comments
on an earlier
Y.C. Kim
I64
post-Plan utlditionul
and N.N. Nor,
C price of X. Finally, tax revenue generated
Tax-cum-rebutr
policies
Plan D involves giving a rebate equal through the introduction of dT.
to the
2. The model Let us now develop a model which can be used to analyze the expected impact of each of the four alternative plans in terms of both the amount of gasoline consumption and the governmental budgetary position. Consider the market demand and supply functions in the following general forms:
Q! =fV’,. P,, ML
(1)
Q:=gU’-7-1,
(2)
where the variables are defined as above. When eqs. (1) and differentiated and then expressed in their elasticity forms.
@Y/Q: = -4dP,lP,)+
(2) are totally
bW,IP,)+ c(dMIM 1,
dQS,/Q”,=e(d(P,-T)/(P,-T)),
(3) (4)
where u = price elasticity of demand for gasoline, b =cross-price elasticity of demand, c =income elasticity of demand for gasoline, and e=price elasticity of supply of gasoline. Alternative rebate schemes distinguish the four alternative plans as follows:
dM =dP,kQ,,,
k =O
under
Plan A,
k= 1
under
Plan B,
0< k< 1
under
Plan C.
under
Plan D.
where
dM=dTQ,,+dQ,T+dTdQ,,
The effect of a rebate on the demand for gasoline can be found substituting either eq. (5) or (6) for dM in eq. (3). Assuming that dP,=O, substitution of eq. (5) in eq. (3) yields dQ!/Q,” = ( - a + cs k ) dP,/P,,
(5)
(6) by the
(3a)
EC. Kim and N.N. Noe, Tax-cum-rebate policies
where s =PXOQXD/MO,i.e., the budget
share of gasoline.
165
Solving
eqs. (3a) and
(4) for dP,IP,,
dP,IP,= -(dQf/Q?)l(a-csk),
(7)
dP,/P, = (dQ;/Q:)(WN 1- TIP,) +dT/P,.
(8)
By combining eqs. (7) and (8) and solving for dQ,/Q,, we find the following expression for the percentage change in the equilibrium quantity of gasoline resulting from a given increase in the gasoline tax (expressed as a percentage of the current price) under Plans A, B, and C:
dQx Q,=
-(a-csk)
In addition, by solving shown that’
dP,_ PX -
l-
e
dT
eqs. (3a) and (4) simultaneously
e e+(a-csk)(l-T/P,)
(9)
P,
e+(a-csk)(l-T/P,)
for dP,/P,,
it can be
l-
dT
(10)
P;
It is to be noted that the first term in parentheses on the right-hand side of eq. (9) is the price elasticity of the ‘compensated’ demand curve, where the amount of compensation (i.e., rebate) is directly related to the value selected for k. The second term in brackets in eq. (9) turns out, by substitution of eq. (10) into eq. (9), to denote simply the consumer’s share of dT, i.e., the proportion of the new tax that will be reflected in an increase in the market price.2 In order to analyze the impact of Plan D, substitute eq. (6) into eq. (3). Solving for dP,/P,, dP,/P,= Next, by combining we obtain
dQ,_
-(dQ,“/Q,D)(l/a)(l-cs(dT+T)/P,)+(cs/a)(dT/P,). eqs. (7a) and (8) and then solving
QX - -(a-CS) e(1 -cs(dT+
for dQ,/Q,
e T)/P,)+a(l
(7a)
l-
as before,
dT
-T/P,)
P, ’
(11)
share of a ‘E q. (10) is a modified version of the more familiar result that the consumer’s specific tax is equal to e/(u +e). See Musgrave (1959, pp. 29C299). *By substituting eq. (10) into eq. (9) and then multiplying throughout by Q,, dQ, can be shown as the sum of substitution and income effects.
166
YC. Kim and N.N. Noe, Tax-cum-rebate policies
This expression gives the percentage change in the equilibrium quantity for a given increase in the tax (expressed as a percentage of the current price) under Plan D. In sum, the preceding analysis shows that eqs. (9) and (11) can be used to evaluate numerically the approximate impact of each alternative tax-cumrebate scheme on gasoline transactions, if some reasonable estimates can be agreed upon for u, c, e, and s.
3. Numerical
illustration
We will now illustrate how the model may be used to generate specific numerical results in the particular case of the U.S. gasoline market. In this endeavor, it is frustrating to find in the literature enormous ranges in the econometric estimates of elasticity coefficients. Given the situation, we do not presume to be able to settle the matter by conducting another econometric study. Instead, we have chosen the approach of reviewing the existing estimates and, then, of selecting several ranges of estimates for simulation. This approach gives the result of a sensitivity test. As can be found in surveys of literature by Kraft and Rodekohr (1978, of p. 53) and Verleger (1975, p. 278), available estimates of the elasticities gasoline-demand vary enormously in different studies. The estimates of the price elasticity of demand range between 0 and -0.9 for the short-run and between 0 and - 1.5 for the long-run. The range for income elasticity is from 0.01 to 0.85 in the short-run and from 0.02 to 1.5 in the long-run. As compiled by Moore (1971. p. 142) the estimates of the long-run supply elasticity of oil range between 0.3 and 0.89 for ‘new discoveries’. and 0.27 for ‘desired reserves’ in the U.S.3 Grouping the available estimates of demand elasticities into three median categories (‘low’, ‘middle’, and ‘high’), we may let the approximate estimate in each category represent its respective category. Then we find that 0.05, 0.1, and 0.5 are the low, middle, and high estimates for the price elasticity (i.e., a’s) in the short-run; and 0.1, 0.5, and 0.8 in the long-run. For the income elasticity (i.e., c’s), the corresponding values are 0.2, 0.4, and 0.6 in the short-run; and 0.6, 1.0, and 1.5 in the long-run. For the supply elasticity (i.e., e’s), a low estimate of 0.3 and a high estimate of 1.0 are of adopted. For the budget share (i.e., s’s), we find that the proportion consumers’ expenditures on gasoline and lubricating oil to disposable personal income is about 5 percent in the first three quarters of 1980 in the U.S.” Listed below are a variety of elasticity estimates (which are found
‘There is one study. however. which finds that the supply elasticity ‘far greater than unity at most prices’. See Mancke (1970, p. 750). %Zomputed from U.S. Department of Commerce (1980, p. 30).
of crude
oil in the U.S. is
EC. Kim und N.N. Noe, Tax-cum-rebate policies
167
sufficient to cover the entire spectrum, from ‘low’ to ‘high’, of final results) and numerical values of other parameters adopted for simulation: a=0.05, 0.1, 0.5, c=O.2, 0.6, 1.5, e=0.3, 1.0, s = 0.05, P, = $1.5/gallon T = SO.O4/gallon dT = $0.3/gallon
0.8,
(assumed current price of gasoline), (current federal gasoline tax), (assumed additional tax rate under the plans).
Table 1 shows expected reductions (in percent) in the equilibrium consumption of gasoline for a 30-cent/gallon tax under the four alternative rebate schemes. As can be verified from eq. (9), the figures in the table can easily be revised in response to any other assumed values of dT for the first three plans, If we wish to consider the impact of a new tax equal to (SO.3)1_, we only need to multiply the numbers in the table by the same factor of 2. For Plan D, however, using some other value of dT would change the numbers in the table slightly more than by the factor of 2 [see eq. (ll)]. The results in the table confirm what one would expect: the greater the supply elasticity, the larger the percentage reduction; the higher the price elasticity of demand, the larger the percentage decrease; and the higher the income elasticity, the smaller the percentage reduction for any given rebate plan. Using the ‘middle’ (M) estimates of demand elasticities as the likely or reasonable case, the short-run (SR) impact is no more than 1.5 on as shown in the rows marked SR/M. Moreover, since even this result is based on a ‘low’ but long-run supply elasticity (i.e., e=0.3). the short-run impact of 1.5 OC, is probably close to an upper bound. Hence, unless a gasoline tax substantially more than $0.3/gallon is contemplated, the short-term impact must be judged negligible. The long-run (LR) effects are substantial. Based on the ‘middle’ demand elasticity estimates, the long-run impact ranges between 3-plus ‘,,, when e =0.3 and 6-plus ‘lo when e= 1.0 (see rows marked LR/M). If we take ‘high’ (H) estimates of u and e, there emerges an optimistic scenario of 8 to 9 :, reduction (see row marked LR/H). It is interesting to compare the results of the different rebate plans. Ignoring the insignificant cases where the expected reduction amounts to less than, say, 2 Y;, it is found that the differences in the percentage reduction between Plan A (i.e., no rebate) and any of the rebate plans are very small never exceeding one percentage point. The reason the alternative schemes do not matter ‘much’ is that the differences in reduction among the alternative plans are due to different income (i.e., rebate) effects, but the income effects are ‘small’ because the budget share (i.e., s) of gasoline is small. Therefore, an implication is that the choice among alternative plans could be evaluated primarily in terms of political considerations as to which
168
YC. Kim
Noe, Tus-cum-rebute
und N.N.
Table Percentage SUPPlY elasticity e
policies
1
reductions
in the equilibrium
Price elasticity u
Income elasticity ‘
A (k=O)
0.05
0.2 0.6 1.5
0.86 0.86 0.86
0.2 0.6
1.51 1.51
1.39 1.14
1.45 1.33
1.35 I .06
1.5
1.51
0.46
I .04
0.38
0.5
0.2 0.6 1.5
3.81 3.81 3.81
3.78 3.12 3.51
3.80 3.77 3.70
3.74 3.59 3.26
0.8
0.2 0.6 1.5
4.45 4.45 4.45
4.44 4.40 4.33
0.05
0.2 0.6 1.5
0.95 0.95 0.95
0.77 0.3’) -0.51h
0.86 0.68 0.25
0.2 0.6 1.5
1.82
0.1
I .x3 1.x2
1.66 1.31 0.49
1.74 1.57 1.18
0.2
6.13
6.64
6.68
6.60
0.6 I.5
6.73 6.13 9.00 9.00 9.00
6.45 6.0 I
6.59 6.38 8.97 8.90 x.75
6.35 5.78 8.90 8.6Y x.23
0.1
quantity Rebate
of gasoline
for a tax increase
of $0.30/gallon.
plan B (k=l) 0.7 1 0.38 - 0.54b
c (k=0.5) 0.19 0.63 0.24
D Remarks” 0.69 0.35 - 0.44h
SR/L
i
SR/M
0.3
__.
LRiM
0.76 0.38 - 0.48b 1.64
1.28 0.46
1.0 0.5
0.8
0.’ 0.6 1.5
8.93 8.80 8.50
“SR =short-run; LR =long-run; L =‘low’; M =‘middle’; bThe negative sign indicates a percentage increase.
and H
1 LR,M (
LR/H
=‘high’
particular plan is least objectionable, rather than in terms of economic considerations as to which particular plan would be most effective after being put into effect. Let us now analyze the sensitivity of the results. If we ignore again the insignificant cases, the impact is shown to be remarkably insensitive to a wide range of income elasticity coefficients. In fact, the difference in the estimated impact are found to be no more than 0.8 percentage points even when the estimates of c vary by as much as a factor of 7.5 (i.e., between 0.2 and 1.5). This too is an expected outcome because of a small size of S. The price elasticity of demand turns out to be an important parameter partly
YC. Kim
because of the partly because results are also large values of countries which supply of oil.
und N.N.
Nor,
Tus-cxm-rebate
I 69
poiicirs
sensitivity in the impact depending on the value of u and of the wide variation in econometric estimates of a. Our quite sensitive to alternative estimates of supply elasticity at a. This finding takes on a special importance for those have to rely on the world market for all or most of their
4. Budgetary impact Another consideration of practical importance in the implementation of any rebate scheme is that of budgetary impact. Especially. if a rebate scheme requires a subsidization from the general government revenue, the scheme is less likely to be adopted. Our basic model allows us to analyze the expected budgetary impact under all alternative schemes. The initial (before dT and rebate) gasoline tax revenue, R. is R=
The u&lirionuI
TQ,,
government
(12) revenue
after tax-cum-rebate
scheme, dR, is
dR=(T+dT)(Q,,+dQ,)-R-dM,
(13)
where dM stands for the amount of rebate under the first three plans and is equal to dP,kQ,, as in eq. (5). Plan D was defined in such a way that dR=O. Therefore, under Plan D, dM =dTQ,, +dQ,T +dTdQ, as shown in eq. (6). To analyze the budgetary effect under Plans A, B, and C, find
WR
=dTIT+dQ,/Q,,~
+ (dQ,/Q,,,(dTP-I-
W’JP,)(P,IT)k. (14)
Substituting eqs. (9) and (10) for (dQ,/Q,,) equation and then solving for (dR/R)/(dTIT ),
and
(dP,/P,)
in the
above
If there is to be no deficit in the government budget resulting from any taxcum-rebate plan (i.e., if the additional revenue is to be non-negative), it is required that dR 2O(=>(dR/R)2O*(dR/R)/(dT/T)ZO). Therefore, the condition for a no-deficit outcome is that the sign of the right-hand side term , RE: c
in eq. (15) be non-negative.
Under
Plan A, k=O;
Then. it can be shown
therefore,
the no-deficit
that’
condition
is reduced
to
(17) Since T/P, is only 0.027 in our example, (l,!(l) + (1 - T,IP, )//c r (e + (I )jw. Then the condition can be restated as (t) f tr)j~ 2 (dT + T )jP, =0.X7. For the assumed values of elasticity coefficients in table 1. (e +tr)/trr> 2.2 in all cases. Consequently, Plan A, not surprisingly. is found not to pose a problem of budgetary nature. For Plan B, k= 1: hence, the no-deficit requirement, ignoring T/P, for being small, is that (dT+T)/P,i l/e. Since (dT+ T)/P,=0.227, the condition can be reduced to ~54.3. This leads to the conclusion, which is not intuitively apparent. that Plan B will mean a surplus in government revenue, unless the elasticity of supply exceeds 4.3. To analyze the budgetary impact of other schemes where O< /c < 1 (e.g.. 1;=0.5 for Plan B in the table), partially differentiate the right-hand-side (RHS) expression in eq. (16) with respect to k. It can then be shown that the sign of c?(RHS)/ik is determined by the sign of - (LI-C’S). Since ((l-cs)>O in our example (with the exceptions mentioned in footnote 5), C(RHS)/iktO. This implies that the critical value on the RHS increases as the value of k decreases. Because it has been shown above that Plan B with li = 1 would lead to a budget surplus, reducing li to a value less than one as in Plan C implies a (greater) surplus.
5. Qualifications The above analysis and empirical simulation are based on a set of constancy assumptions. First, both the demand and the supply functions are assumed to take constant elasticity forms. Second, the budget share of gasoline is taken to be constant. Third, the income elasticity is assumed to remain constant between the pre- and post-Plan periods, as well as among %rt the expression on the right-hand aide of eq. 115) to he non-negative and then solve for (dT + T),!Pr. In arriving at eq. (16). it is assumed that the numerical values of the coefklents are such that r+ (u-csk)(l -~ T/P)>0 and that r(tr-cks)>O. In light of the coefficients used in table I. the first assumption is satisfied in all cases, and the second assumption is met in all but two (out of 72) cases where the lowest value of tl (i.e.. 0.05) and the highest value of L’ (i.e.. 1.5) are assumed under Plan B.
Y.C. Kim
and N.N. NW,
Ta-cum-rebate
policies
171
the alternative rebate schemes. All three (sets of) assumptions, although implicit and commonplace in the literature, need to be borne in mind in interpreting the results. If we are to scrutinize the first assumption, it is of course plausible that the elasticities would be non-constant. In particular, it may very well be that ic/?P, t0 and &x~(?M > 0.Then, due to ?u/GM >O, a growth in the economy in the long-run would have the effect of raising the value of LI. Referring to table 1. this would tend to move the estimates of (I upward. That ic/iP, ~0. however, would matter little. Given the insensitivity of our results for alternative estimates of C, the effect of a higher price (by an excise tax) on gasoline consumption (through a lower value of c) would be negligible. Secondly, the budget share of gasoline (s) may be postulated to change over time in relation to price elasticity coefficients. If c = 1, the elasticity effect could be ignored. However, since most studies report that U-C 1 even in the long-run, the size of s would rise as the price of gasoline is raised through dT. The importance of changes in s, nevertheless, should not be overstated: the size of s has increased by only one percentage point since when the price of gasoline was $0.5/gallon. Thirdly. any murkrt elasticity of demand (u or c) is some weighted arerugr or their subgroups’ elasticities. Hence, elasticity of individual consumers’ whenever the weights are altered, the (average) market elasticity is affected. This problem takes us to the quagmire of distributive impact at disaggregate levels of arl~’ policy based on the coefficients at an aggregate level. In the context of this paper, such distributive impact looms in two respects: (i) distributive impact of the excise tax (i.e., the incidence of taxation), and (ii) (re-)distributive impact of a particular form of rebate. Moreover, one can perceive a variety of different sorts of subgroup-distributive impact, depending on how the subgrouping is classified (e.g., by income, age, geographical location, and so on). Given the enormous range of elasticity coefficients even when the market is treated as a single group, there is little to be gained by expanding this exercise to deal with the elasticities for a variety of subgroups. Hence, let it suffice by pointing out just two issues of some import. First, contrary to popular, public claim, the little evidence there is appears to indicate that the incidence of gasoline tax will not be regressive [Data Resources (1973, pp. IV.34-37), Hella and Jayson (undated), and Ramsey, Rasche and Allen (1975)]. Second, it may be argued that a rebate plan implies a change in income distribution between the pre-Plan and the post-Plan situations; and the changing income distribution will alter the aforementioned weights and thus aggregate (average) income elasticity. It has been hypothesized [Ramsey, Rasche and Allen (1975, p. 503)], as some support can be found in implied income elasticities computable from a BLS’s consumer expenditure study [BLS (1966, table 29A)], that the income elasticity decreases as income
177
EC. Kim
und N.N. NW.
Tuz-cunt-rchute
policies
increases. Yet, that Cc/CM ~0 is not apparent either in time-series estimates for the U.S. [Data Resources (1973)] or in inter-country estimates for O.E.C.D. [Lebanon (1977)]. Above all, as discussed earlier, our findings are very insensitive to wide variations in the income elasticity. Thus, we can conclude that any likely change in the estimates for c, due to incomeredistributional impact, will have a negligible consequence in terms of the market as a whole.
6. Summary
and closing remarks
We have developed a partial equilibrium model to analyze several alternative tax-cum-rebate schemes. Then the model is applied to simulate the gasoline market in the U.S. The alternative rebate plans are shown to matter very little in terms of the simulated reduction in gasoline consumption. Moreover, it is shown that the four tax-cum-rebate schemes would be at least self-financing in all likelihood. For a hypothetical tax rate of $0,3/gallon, the short-run reduction is unlikely to exceed 1.5 I’“. The projected long-run effects are substantial, but vary considerably depending on the assumed values of supply elasticity (c) and price elasticity of demand (u). For ‘middle’ estimates of ~1,the reduction ranges from 3-plus “o to 6-plus Oo, depending on whether 4 is as low as 0.3 or as high as 1.0, respectively. However, even an optimistic scenario (i.e., for ‘high’ estimates of LI and e) does not project a reduction beyond 9 (‘o. (The low bounds for the long-run approximately coincide with the ‘middle’ estimates for the short-run.) Although the impetus for this study is the current ‘oil problem’, the basic model is applicable to other situations where a reduction in consumption is desired (e.g.. marijuana). Moreover, due to the symmetrical nature of the effect of any specific tax and also of rebate, each with respect to the sign (i.e., direction) of change in tax and rebate, the basic model is also applicable in situations where an increuse in consumption of a given item (e.g.. housing) may be desired through a reduction in a specific tax (e.g.. property tax) combined with a negative rebate (i.e.. an increase in income tax). References Bureau of Labor Statistics, 1966, Survey of consumer expenditures. 196G61. Suppl. 3-Part A to BLS Report no. 237793 (Washington, DC). Data Resources. 1973. A study of the quarterly demand for gasoline and impacts of alternative gasoline taxes, Prepared for the Environmental Protection Agency and Council on Environment Quality (Lexington, MA). Jlella. Karl N. and June Jayson. undated, Gasoline tax regrrssivity: An empirical finding. Mimeo. (Washington University, St. Louis. MO). Kraft, John and Mark Rodekohr, 1978, Regional demand for gasoline: A temporal cross-section specification, Journal of Regional Science 18, April, 45-55. Lebanon, Alexander, 1977, The household demand for energy and fuels in OECD countries. European Economic Review 9, April, 61 81.
Yc’. Kim
und N.N. NW. Tuu-cum-rehute
policies
173
Mancke, Richard B., 1970, The longrun supply curve of crude oil produced in the U.S.. Antitrust Bulletin 15. Winter, 727-756. Moore, T.G., 1971. The petroleum industry. in: W. Adams. ed., The structure of American industry. 4th ed. (Macmillan, New York). Musgrave, Richard A., 1959. The theory of public finance (McGraw-Hill, New Yorh). Ramsey, J., R. Rasche and B. Allen. 1975, An analysis of the private and commercial demand for gasoline, Review of Economics and Statistics 58. Nov.. 502 507. U.S. Department of Commerce, 1980. Survey of current business 60, Dec. Verleger, Philip K.. 1975, The demand for gasoline: Some further analyses. ASA 1974 proceedings of the business and economic statistics section. 273 281.
and Energy
A MODEL
3 (1981) 163-173.
North-Holland
OF TAX-CUM-REBATE
Publishing
OIL (GASOLINE)
Young Chin KIM and Nicholas Northern
Illinois University,
Company
POLICY
N. NOE*
DeKolb, IL 60115, USA
This paper develops a model to analyze the impact of alternative tax-cum-rebate policies currently being discussed as possible means of reducing gasoline consumption. The saving in gasoline from a 30-cent/gallon tax is simulated for four alternative rebate schemes and for a variety of elasticity estimates. The short-run impact is found to be small, while the long-run saving would probably be 3 to 6 Y, (and may even be as high as 9 Y,). The choice of rebate plans is shown to be an unimportant consideration in terms of the amount of gasoline consumption. It is also found that there would be no negative impact on the government budget under any of the four rebate schemes.
1. Introduction Prompted by precipitious increases in the price of oil in recent years, a great deal of interest has been expressed for some public policy to reduce the oil consumption. The purpose of this paper is develop a model to analyze and to simulate a particular type of policy, namely, to impose an excise tax on oil and then to rebate the tax revenue to the consumer. The analytical method is comparative static, partial equilibrium, competitive analysis; and the working of the model is illustrated for the case of U.S. gasoline market. Within the basic framework of tax-cum-rebate measures, four alternative schemes can be distinguished - depending on how the amount of rebate is to be determined. Let P, and Q, =price and quantity, respectively, of gasoline (x): P, and Q,=price and quantity, respectively, of ‘the other product’ (J); T=current per unit tax (in $‘s) on gasoline transactions; dT =additional per unit excise tax (in S’s) on gasoline transactions being considered; and M=amount of money income (in $‘s). Then, the first of the four alternative schemes (Plan A) is to impose a specific tax equal to dT with no rebate to the consumer. The second option (PlanB) is to levy dT and at the same time to rebate just enough money income to rnable the consumer to purchase the pre-Plan B quantities of x and y at the post-Plan B prices. The third scheme (Plan C) is to impose dT and then set the amount of rebate equal to the extra expenditure on gasoline if the consumer were to purchase some pre-determined, smaller-than-initial amount of gasoline at the *The authors would like to thank version helped to improve the paper.
016550572/81/000Cr0000/$02.75
an anonymous
referee,
whose
11:; 1981 North-Holland
comments
on an earlier
Y.C. Kim
I64
post-Plan utlditionul
and N.N. Nor,
C price of X. Finally, tax revenue generated
Tax-cum-rebutr
policies
Plan D involves giving a rebate equal through the introduction of dT.
to the
2. The model Let us now develop a model which can be used to analyze the expected impact of each of the four alternative plans in terms of both the amount of gasoline consumption and the governmental budgetary position. Consider the market demand and supply functions in the following general forms:
Q! =fV’,. P,, ML
(1)
Q:=gU’-7-1,
(2)
where the variables are defined as above. When eqs. (1) and differentiated and then expressed in their elasticity forms.
@Y/Q: = -4dP,lP,)+
(2) are totally
bW,IP,)+ c(dMIM 1,
dQS,/Q”,=e(d(P,-T)/(P,-T)),
(3) (4)
where u = price elasticity of demand for gasoline, b =cross-price elasticity of demand, c =income elasticity of demand for gasoline, and e=price elasticity of supply of gasoline. Alternative rebate schemes distinguish the four alternative plans as follows:
dM =dP,kQ,,,
k =O
under
Plan A,
k= 1
under
Plan B,
0< k< 1
under
Plan C.
under
Plan D.
where
dM=dTQ,,+dQ,T+dTdQ,,
The effect of a rebate on the demand for gasoline can be found substituting either eq. (5) or (6) for dM in eq. (3). Assuming that dP,=O, substitution of eq. (5) in eq. (3) yields dQ!/Q,” = ( - a + cs k ) dP,/P,,
(5)
(6) by the
(3a)
EC. Kim and N.N. Noe, Tax-cum-rebate policies
where s =PXOQXD/MO,i.e., the budget
share of gasoline.
165
Solving
eqs. (3a) and
(4) for dP,IP,,
dP,IP,= -(dQf/Q?)l(a-csk),
(7)
dP,/P, = (dQ;/Q:)(WN 1- TIP,) +dT/P,.
(8)
By combining eqs. (7) and (8) and solving for dQ,/Q,, we find the following expression for the percentage change in the equilibrium quantity of gasoline resulting from a given increase in the gasoline tax (expressed as a percentage of the current price) under Plans A, B, and C:
dQx Q,=
-(a-csk)
In addition, by solving shown that’
dP,_ PX -
l-
e
dT
eqs. (3a) and (4) simultaneously
e e+(a-csk)(l-T/P,)
(9)
P,
e+(a-csk)(l-T/P,)
for dP,/P,,
it can be
l-
dT
(10)
P;
It is to be noted that the first term in parentheses on the right-hand side of eq. (9) is the price elasticity of the ‘compensated’ demand curve, where the amount of compensation (i.e., rebate) is directly related to the value selected for k. The second term in brackets in eq. (9) turns out, by substitution of eq. (10) into eq. (9), to denote simply the consumer’s share of dT, i.e., the proportion of the new tax that will be reflected in an increase in the market price.2 In order to analyze the impact of Plan D, substitute eq. (6) into eq. (3). Solving for dP,/P,, dP,/P,= Next, by combining we obtain
dQ,_
-(dQ,“/Q,D)(l/a)(l-cs(dT+T)/P,)+(cs/a)(dT/P,). eqs. (7a) and (8) and then solving
QX - -(a-CS) e(1 -cs(dT+
for dQ,/Q,
e T)/P,)+a(l
(7a)
l-
as before,
dT
-T/P,)
P, ’
(11)
share of a ‘E q. (10) is a modified version of the more familiar result that the consumer’s specific tax is equal to e/(u +e). See Musgrave (1959, pp. 29C299). *By substituting eq. (10) into eq. (9) and then multiplying throughout by Q,, dQ, can be shown as the sum of substitution and income effects.
166
YC. Kim and N.N. Noe, Tax-cum-rebate policies
This expression gives the percentage change in the equilibrium quantity for a given increase in the tax (expressed as a percentage of the current price) under Plan D. In sum, the preceding analysis shows that eqs. (9) and (11) can be used to evaluate numerically the approximate impact of each alternative tax-cumrebate scheme on gasoline transactions, if some reasonable estimates can be agreed upon for u, c, e, and s.
3. Numerical
illustration
We will now illustrate how the model may be used to generate specific numerical results in the particular case of the U.S. gasoline market. In this endeavor, it is frustrating to find in the literature enormous ranges in the econometric estimates of elasticity coefficients. Given the situation, we do not presume to be able to settle the matter by conducting another econometric study. Instead, we have chosen the approach of reviewing the existing estimates and, then, of selecting several ranges of estimates for simulation. This approach gives the result of a sensitivity test. As can be found in surveys of literature by Kraft and Rodekohr (1978, of p. 53) and Verleger (1975, p. 278), available estimates of the elasticities gasoline-demand vary enormously in different studies. The estimates of the price elasticity of demand range between 0 and -0.9 for the short-run and between 0 and - 1.5 for the long-run. The range for income elasticity is from 0.01 to 0.85 in the short-run and from 0.02 to 1.5 in the long-run. As compiled by Moore (1971. p. 142) the estimates of the long-run supply elasticity of oil range between 0.3 and 0.89 for ‘new discoveries’. and 0.27 for ‘desired reserves’ in the U.S.3 Grouping the available estimates of demand elasticities into three median categories (‘low’, ‘middle’, and ‘high’), we may let the approximate estimate in each category represent its respective category. Then we find that 0.05, 0.1, and 0.5 are the low, middle, and high estimates for the price elasticity (i.e., a’s) in the short-run; and 0.1, 0.5, and 0.8 in the long-run. For the income elasticity (i.e., c’s), the corresponding values are 0.2, 0.4, and 0.6 in the short-run; and 0.6, 1.0, and 1.5 in the long-run. For the supply elasticity (i.e., e’s), a low estimate of 0.3 and a high estimate of 1.0 are of adopted. For the budget share (i.e., s’s), we find that the proportion consumers’ expenditures on gasoline and lubricating oil to disposable personal income is about 5 percent in the first three quarters of 1980 in the U.S.” Listed below are a variety of elasticity estimates (which are found
‘There is one study. however. which finds that the supply elasticity ‘far greater than unity at most prices’. See Mancke (1970, p. 750). %Zomputed from U.S. Department of Commerce (1980, p. 30).
of crude
oil in the U.S. is
EC. Kim und N.N. Noe, Tax-cum-rebate policies
167
sufficient to cover the entire spectrum, from ‘low’ to ‘high’, of final results) and numerical values of other parameters adopted for simulation: a=0.05, 0.1, 0.5, c=O.2, 0.6, 1.5, e=0.3, 1.0, s = 0.05, P, = $1.5/gallon T = SO.O4/gallon dT = $0.3/gallon
0.8,
(assumed current price of gasoline), (current federal gasoline tax), (assumed additional tax rate under the plans).
Table 1 shows expected reductions (in percent) in the equilibrium consumption of gasoline for a 30-cent/gallon tax under the four alternative rebate schemes. As can be verified from eq. (9), the figures in the table can easily be revised in response to any other assumed values of dT for the first three plans, If we wish to consider the impact of a new tax equal to (SO.3)1_, we only need to multiply the numbers in the table by the same factor of 2. For Plan D, however, using some other value of dT would change the numbers in the table slightly more than by the factor of 2 [see eq. (ll)]. The results in the table confirm what one would expect: the greater the supply elasticity, the larger the percentage reduction; the higher the price elasticity of demand, the larger the percentage decrease; and the higher the income elasticity, the smaller the percentage reduction for any given rebate plan. Using the ‘middle’ (M) estimates of demand elasticities as the likely or reasonable case, the short-run (SR) impact is no more than 1.5 on as shown in the rows marked SR/M. Moreover, since even this result is based on a ‘low’ but long-run supply elasticity (i.e., e=0.3). the short-run impact of 1.5 OC, is probably close to an upper bound. Hence, unless a gasoline tax substantially more than $0.3/gallon is contemplated, the short-term impact must be judged negligible. The long-run (LR) effects are substantial. Based on the ‘middle’ demand elasticity estimates, the long-run impact ranges between 3-plus ‘,,, when e =0.3 and 6-plus ‘lo when e= 1.0 (see rows marked LR/M). If we take ‘high’ (H) estimates of u and e, there emerges an optimistic scenario of 8 to 9 :, reduction (see row marked LR/H). It is interesting to compare the results of the different rebate plans. Ignoring the insignificant cases where the expected reduction amounts to less than, say, 2 Y;, it is found that the differences in the percentage reduction between Plan A (i.e., no rebate) and any of the rebate plans are very small never exceeding one percentage point. The reason the alternative schemes do not matter ‘much’ is that the differences in reduction among the alternative plans are due to different income (i.e., rebate) effects, but the income effects are ‘small’ because the budget share (i.e., s) of gasoline is small. Therefore, an implication is that the choice among alternative plans could be evaluated primarily in terms of political considerations as to which
168
YC. Kim
Noe, Tus-cum-rebute
und N.N.
Table Percentage SUPPlY elasticity e
policies
1
reductions
in the equilibrium
Price elasticity u
Income elasticity ‘
A (k=O)
0.05
0.2 0.6 1.5
0.86 0.86 0.86
0.2 0.6
1.51 1.51
1.39 1.14
1.45 1.33
1.35 I .06
1.5
1.51
0.46
I .04
0.38
0.5
0.2 0.6 1.5
3.81 3.81 3.81
3.78 3.12 3.51
3.80 3.77 3.70
3.74 3.59 3.26
0.8
0.2 0.6 1.5
4.45 4.45 4.45
4.44 4.40 4.33
0.05
0.2 0.6 1.5
0.95 0.95 0.95
0.77 0.3’) -0.51h
0.86 0.68 0.25
0.2 0.6 1.5
1.82
0.1
I .x3 1.x2
1.66 1.31 0.49
1.74 1.57 1.18
0.2
6.13
6.64
6.68
6.60
0.6 I.5
6.73 6.13 9.00 9.00 9.00
6.45 6.0 I
6.59 6.38 8.97 8.90 x.75
6.35 5.78 8.90 8.6Y x.23
0.1
quantity Rebate
of gasoline
for a tax increase
of $0.30/gallon.
plan B (k=l) 0.7 1 0.38 - 0.54b
c (k=0.5) 0.19 0.63 0.24
D Remarks” 0.69 0.35 - 0.44h
SR/L
i
SR/M
0.3
__.
LRiM
0.76 0.38 - 0.48b 1.64
1.28 0.46
1.0 0.5
0.8
0.’ 0.6 1.5
8.93 8.80 8.50
“SR =short-run; LR =long-run; L =‘low’; M =‘middle’; bThe negative sign indicates a percentage increase.
and H
1 LR,M (
LR/H
=‘high’
particular plan is least objectionable, rather than in terms of economic considerations as to which particular plan would be most effective after being put into effect. Let us now analyze the sensitivity of the results. If we ignore again the insignificant cases, the impact is shown to be remarkably insensitive to a wide range of income elasticity coefficients. In fact, the difference in the estimated impact are found to be no more than 0.8 percentage points even when the estimates of c vary by as much as a factor of 7.5 (i.e., between 0.2 and 1.5). This too is an expected outcome because of a small size of S. The price elasticity of demand turns out to be an important parameter partly
YC. Kim
because of the partly because results are also large values of countries which supply of oil.
und N.N.
Nor,
Tus-cxm-rebate
I 69
poiicirs
sensitivity in the impact depending on the value of u and of the wide variation in econometric estimates of a. Our quite sensitive to alternative estimates of supply elasticity at a. This finding takes on a special importance for those have to rely on the world market for all or most of their
4. Budgetary impact Another consideration of practical importance in the implementation of any rebate scheme is that of budgetary impact. Especially. if a rebate scheme requires a subsidization from the general government revenue, the scheme is less likely to be adopted. Our basic model allows us to analyze the expected budgetary impact under all alternative schemes. The initial (before dT and rebate) gasoline tax revenue, R. is R=
The u&lirionuI
TQ,,
government
(12) revenue
after tax-cum-rebate
scheme, dR, is
dR=(T+dT)(Q,,+dQ,)-R-dM,
(13)
where dM stands for the amount of rebate under the first three plans and is equal to dP,kQ,, as in eq. (5). Plan D was defined in such a way that dR=O. Therefore, under Plan D, dM =dTQ,, +dQ,T +dTdQ, as shown in eq. (6). To analyze the budgetary effect under Plans A, B, and C, find
WR
=dTIT+dQ,/Q,,~
+ (dQ,/Q,,,(dTP-I-
W’JP,)(P,IT)k. (14)
Substituting eqs. (9) and (10) for (dQ,/Q,,) equation and then solving for (dR/R)/(dTIT ),
and
(dP,/P,)
in the
above
If there is to be no deficit in the government budget resulting from any taxcum-rebate plan (i.e., if the additional revenue is to be non-negative), it is required that dR 2O(=>(dR/R)2O*(dR/R)/(dT/T)ZO). Therefore, the condition for a no-deficit outcome is that the sign of the right-hand side term , RE: c
in eq. (15) be non-negative.
Under
Plan A, k=O;
Then. it can be shown
therefore,
the no-deficit
that’
condition
is reduced
to
(17) Since T/P, is only 0.027 in our example, (l,!(l) + (1 - T,IP, )//c r (e + (I )jw. Then the condition can be restated as (t) f tr)j~ 2 (dT + T )jP, =0.X7. For the assumed values of elasticity coefficients in table 1. (e +tr)/trr> 2.2 in all cases. Consequently, Plan A, not surprisingly. is found not to pose a problem of budgetary nature. For Plan B, k= 1: hence, the no-deficit requirement, ignoring T/P, for being small, is that (dT+T)/P,i l/e. Since (dT+ T)/P,=0.227, the condition can be reduced to ~54.3. This leads to the conclusion, which is not intuitively apparent. that Plan B will mean a surplus in government revenue, unless the elasticity of supply exceeds 4.3. To analyze the budgetary impact of other schemes where O< /c < 1 (e.g.. 1;=0.5 for Plan B in the table), partially differentiate the right-hand-side (RHS) expression in eq. (16) with respect to k. It can then be shown that the sign of c?(RHS)/ik is determined by the sign of - (LI-C’S). Since ((l-cs)>O in our example (with the exceptions mentioned in footnote 5), C(RHS)/iktO. This implies that the critical value on the RHS increases as the value of k decreases. Because it has been shown above that Plan B with li = 1 would lead to a budget surplus, reducing li to a value less than one as in Plan C implies a (greater) surplus.
5. Qualifications The above analysis and empirical simulation are based on a set of constancy assumptions. First, both the demand and the supply functions are assumed to take constant elasticity forms. Second, the budget share of gasoline is taken to be constant. Third, the income elasticity is assumed to remain constant between the pre- and post-Plan periods, as well as among %rt the expression on the right-hand aide of eq. 115) to he non-negative and then solve for (dT + T),!Pr. In arriving at eq. (16). it is assumed that the numerical values of the coefklents are such that r+ (u-csk)(l -~ T/P)>0 and that r(tr-cks)>O. In light of the coefficients used in table I. the first assumption is satisfied in all cases, and the second assumption is met in all but two (out of 72) cases where the lowest value of tl (i.e.. 0.05) and the highest value of L’ (i.e.. 1.5) are assumed under Plan B.
Y.C. Kim
and N.N. NW,
Ta-cum-rebate
policies
171
the alternative rebate schemes. All three (sets of) assumptions, although implicit and commonplace in the literature, need to be borne in mind in interpreting the results. If we are to scrutinize the first assumption, it is of course plausible that the elasticities would be non-constant. In particular, it may very well be that ic/?P, t0 and &x~(?M > 0.Then, due to ?u/GM >O, a growth in the economy in the long-run would have the effect of raising the value of LI. Referring to table 1. this would tend to move the estimates of (I upward. That ic/iP, ~0. however, would matter little. Given the insensitivity of our results for alternative estimates of C, the effect of a higher price (by an excise tax) on gasoline consumption (through a lower value of c) would be negligible. Secondly, the budget share of gasoline (s) may be postulated to change over time in relation to price elasticity coefficients. If c = 1, the elasticity effect could be ignored. However, since most studies report that U-C 1 even in the long-run, the size of s would rise as the price of gasoline is raised through dT. The importance of changes in s, nevertheless, should not be overstated: the size of s has increased by only one percentage point since when the price of gasoline was $0.5/gallon. Thirdly. any murkrt elasticity of demand (u or c) is some weighted arerugr or their subgroups’ elasticities. Hence, elasticity of individual consumers’ whenever the weights are altered, the (average) market elasticity is affected. This problem takes us to the quagmire of distributive impact at disaggregate levels of arl~’ policy based on the coefficients at an aggregate level. In the context of this paper, such distributive impact looms in two respects: (i) distributive impact of the excise tax (i.e., the incidence of taxation), and (ii) (re-)distributive impact of a particular form of rebate. Moreover, one can perceive a variety of different sorts of subgroup-distributive impact, depending on how the subgrouping is classified (e.g., by income, age, geographical location, and so on). Given the enormous range of elasticity coefficients even when the market is treated as a single group, there is little to be gained by expanding this exercise to deal with the elasticities for a variety of subgroups. Hence, let it suffice by pointing out just two issues of some import. First, contrary to popular, public claim, the little evidence there is appears to indicate that the incidence of gasoline tax will not be regressive [Data Resources (1973, pp. IV.34-37), Hella and Jayson (undated), and Ramsey, Rasche and Allen (1975)]. Second, it may be argued that a rebate plan implies a change in income distribution between the pre-Plan and the post-Plan situations; and the changing income distribution will alter the aforementioned weights and thus aggregate (average) income elasticity. It has been hypothesized [Ramsey, Rasche and Allen (1975, p. 503)], as some support can be found in implied income elasticities computable from a BLS’s consumer expenditure study [BLS (1966, table 29A)], that the income elasticity decreases as income
177
EC. Kim
und N.N. NW.
Tuz-cunt-rchute
policies
increases. Yet, that Cc/CM ~0 is not apparent either in time-series estimates for the U.S. [Data Resources (1973)] or in inter-country estimates for O.E.C.D. [Lebanon (1977)]. Above all, as discussed earlier, our findings are very insensitive to wide variations in the income elasticity. Thus, we can conclude that any likely change in the estimates for c, due to incomeredistributional impact, will have a negligible consequence in terms of the market as a whole.
6. Summary
and closing remarks
We have developed a partial equilibrium model to analyze several alternative tax-cum-rebate schemes. Then the model is applied to simulate the gasoline market in the U.S. The alternative rebate plans are shown to matter very little in terms of the simulated reduction in gasoline consumption. Moreover, it is shown that the four tax-cum-rebate schemes would be at least self-financing in all likelihood. For a hypothetical tax rate of $0,3/gallon, the short-run reduction is unlikely to exceed 1.5 I’“. The projected long-run effects are substantial, but vary considerably depending on the assumed values of supply elasticity (c) and price elasticity of demand (u). For ‘middle’ estimates of ~1,the reduction ranges from 3-plus “o to 6-plus Oo, depending on whether 4 is as low as 0.3 or as high as 1.0, respectively. However, even an optimistic scenario (i.e., for ‘high’ estimates of LI and e) does not project a reduction beyond 9 (‘o. (The low bounds for the long-run approximately coincide with the ‘middle’ estimates for the short-run.) Although the impetus for this study is the current ‘oil problem’, the basic model is applicable to other situations where a reduction in consumption is desired (e.g.. marijuana). Moreover, due to the symmetrical nature of the effect of any specific tax and also of rebate, each with respect to the sign (i.e., direction) of change in tax and rebate, the basic model is also applicable in situations where an increuse in consumption of a given item (e.g.. housing) may be desired through a reduction in a specific tax (e.g.. property tax) combined with a negative rebate (i.e.. an increase in income tax). References Bureau of Labor Statistics, 1966, Survey of consumer expenditures. 196G61. Suppl. 3-Part A to BLS Report no. 237793 (Washington, DC). Data Resources. 1973. A study of the quarterly demand for gasoline and impacts of alternative gasoline taxes, Prepared for the Environmental Protection Agency and Council on Environment Quality (Lexington, MA). Jlella. Karl N. and June Jayson. undated, Gasoline tax regrrssivity: An empirical finding. Mimeo. (Washington University, St. Louis. MO). Kraft, John and Mark Rodekohr, 1978, Regional demand for gasoline: A temporal cross-section specification, Journal of Regional Science 18, April, 45-55. Lebanon, Alexander, 1977, The household demand for energy and fuels in OECD countries. European Economic Review 9, April, 61 81.
Yc’. Kim
und N.N. NW. Tuu-cum-rehute
policies
173
Mancke, Richard B., 1970, The longrun supply curve of crude oil produced in the U.S.. Antitrust Bulletin 15. Winter, 727-756. Moore, T.G., 1971. The petroleum industry. in: W. Adams. ed., The structure of American industry. 4th ed. (Macmillan, New York). Musgrave, Richard A., 1959. The theory of public finance (McGraw-Hill, New Yorh). Ramsey, J., R. Rasche and B. Allen. 1975, An analysis of the private and commercial demand for gasoline, Review of Economics and Statistics 58. Nov.. 502 507. U.S. Department of Commerce, 1980. Survey of current business 60, Dec. Verleger, Philip K.. 1975, The demand for gasoline: Some further analyses. ASA 1974 proceedings of the business and economic statistics section. 273 281.