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The Pricing of Gasoline Grades and the Third Law of Demand
R. Morris Coats, Nicholls State University;
Gary M. Pecquet, Shenandoah University;
and Leon Taylor, Tulane Universitya
Abstract:
Alchian and Allen’s “third law of demand” states that as a fixed cost increases by the same
amount for low- and high-quality goods, the ratio of the prices of high- to low-quality goods will
fall and the quantity demanded of high quality goods relative to low quality goods will increase.
We examine the more general hypothesis by estimating the ratio of the quantities of sales of
premium to regular grade gasoline using the ratio of premium to regular prices, controlling for
supply and demand factors. We find moderate evidence for the more general hypothesis.
JEL Codes: D120 - Consumer Economics: Empirical Analysis
Keywords: Third Law of Demand, Price Ratios, Gasoline Grades
a Correspondence for this article should be addressed to R. Morris Coats, Department of Finance and Economics,
Nicholls State University, Thibodaux, LA 70310, or to his email address: morris.coats@nicholls.edu. We thank
participants in an industrial organization session of the Southwestern Economics Association meetings for 2005,
especially Seung Mo Choi.
1. Introduction
Alchian and Allen’s (1967, pp. 62-64) famous “third law of demand” suggested that a
cost change that adds the same amount to the price of both low- and high-quality goods will
lower the price of the higher quality (and higher price) good relative to the lower quality good.
A decrease in the price ratio increases the quantity demanded of the higher quality good relative
to that of the lower quality good. Of course, perverse income effects could wipe out any
substitution effects, leaving the total effect ambiguous; but we will present evidence that the
income effect in the relative demand for premium is likely weak.
Earlier studies of Alchian and Allen’s “third law” of demand have depended upon adding
a fixed per unit shipping cost to both high-quality and low-quality substitute products. The
results have produced some confusion. Gould and Segall (1969) show that Alchian and Allen’s
theorem only holds in a two-good world, and may not hold in a world of more than two goods.
Borcherding and Silberberg (1978) show that if two goods are close substitutes, the Alchian and
Allen result still holds. Some ambiguities arose over the interpretation of “fixed charge.” Cowen
and Tabarrok (1995) pointed out that the “fixed charge” must be applied on a per unit basis.
Consequently, it mattered whether the goods were shipped to the consumers or the consumers
shipped to the goods.
Other articles have used the “third law” to examine the effects of quantity-based excise
taxes upon the quality consumed (James and Alston, 2002). Barzel (1976) applied the “third law”
to the per unit excise taxation of cigarettes, gasoline and alcoholic beverages. In the case of
cigarettes, he found that such taxes could be more than fully passed forward to consumers
because a tax increase lowered the ratio of prices of high-quality to low-quality cigarettes, so that
2
the higher quality and more expensive cigarettes would be substituted for the cheaper smokes.
With regard to gasoline and alcoholic beverages, Barzel found nothing of significance.
Razzolini, Shughart and Tollison (2003) have argued that the “third law” could not be
properly tested without considering the supply side. They argued that adding a fixed per unit cost
(i.e. transportation or taxes) to goods will affect the relative prices of high and low quality goods
depending upon whether or not the different qualities are produced in competitive or monopoly
markets and also upon the nature of long-run cost for each of the qualities (increasing, decreasing
or constant cost industries). While it is certainly true that these supply considerations affect the
relative prices of the two qualities, their interpretation of the “third law” recasts it from a demand
theorem to a hypothesis about markets.
We also believe that supply conditions should be taken into account, but only as controls.
In this paper, we examine the basic premise of the “third law” by analyzing the effect of relative
prices of premium to regular grades of gasoline upon the relative consumption of the grades after
controlling for other demand and supply influences. As the prices of both grades increase by the
same amount, the relative price of premium falls. To non-economists, focused only upon the
income effect, higher prices would induce people to switch towards the cheaper grades. On the
other hand, the “third law” (or substitution effect) suggests that people would switch towards the
higher quality premium grades. Our method of testing for the “third law” also avoids the pitfalls
of including transportation cost mentioned above. We put the “third law” to the test, not by
considering some common incremental change to prices (as in Nesbit, 2005), but by looking at
the more general case of changes in price ratios of high- and low-quality goods and the effect of
such changes on consumption ratios of high- to low-quality goods.
3
2. Basic Theory
Gasoline is one item out of many that people buy. Of course, in the short run, people are
stuck living a certain distance from work, schools, shopping areas and friends, and they can do
little to change those distances. They are also stuck with their current means of transportation
and will have difficulty switching between alternative fuels. To some degree, a household may
switch between alternative grades of gasoline, and it can make these substitutions more easily
with certain vehicles than with others and more easily if it owns more than one vehicle.
The choice between gasoline (in general) and all other goods involves only a little
substitutability, as we observe long-run estimates of price elasticity of demand around -0.86 and
short-run estimates around -0.26 (Dahl and Sterner 1991, pp. 210).
While gasoline and all other goods are not close substitutes, one grade of gasoline should
be a relatively close substitute for another grade, if people are at least able to substitute gasoline
grades across the vehicles they drive. Substitution across grades is more likely to occur when
there is a change in price ratios across grades, as is the case when the price spread between two
grades of gasoline stays the same, yet the prices of both rise, cutting the ratio of high- to low-
priced grades. With such price ratio changes, the ratio of purchases of high- to low-priced grades
should increase.
3. Empirical tests
3.1 Data
The dependent variable, prgas, is the ratio of the number of gallons of premium gasoline
sold to the number of gallons of regular gasoline sold in a given month and in a given state in the
United States. The first difference of this variable is prgasdiff.
4
The current price variable is prprice, the ratio of the price of a gallon of premium
gasoline to the price of a gallon of regular gasoline in a given month and state. The prices
include local, state and federal fuel taxes. An n-month lag of a variable is denoted with the
prefix Ln. For example, the one-month lag of the dependent variable is L1.prgas. The
differenced variable is prpdiff.
Data on sales and prices excluding taxes come from the Web page of the U.S. Energy
Information Administration (http://tonto.eia.doe.gov/STEO_Query/app/papage.htm). Data on
state fuel taxes were collected from the Highway Statistics
(http://www.fhwa.dot.gov/policy/ohpi/hss/hsspubs.htm) and Monthly Motor Fuel Reported by
States (http://www.fhwa.dot.gov/ohim/mmfr/mmfrpage.htm) series of the Federal Highway
Administration; corrections were provided by e-mail or by phone by the transportation or finance
departments of the state governments. The state governments also provided data on local fuel
taxes, which we weighted by population when adding them to fuel prices. The state taxes
include sales taxes, environmental taxes (such as fees for a Leaking Underground Storage Tank
fund), and inspection fees based on gallons used. The local and federal taxes are excise taxes
that are quantity-based or (for many local taxes) sales-based.
A control for cost factors is Crude, which measures the real acquisition cost of crude oil
per barrel to the U.S. refineries each month, based on a composite of foreign and imported oil,
and deflated by the producer price index for oil refineries (in 1982 dollars). The cost data were
collected from various issues of the Monthly energy review of the U.S. Energy Information
Administration (http://tonto.eia.doe.gov/FTPROOT/monthlyhistory.htm). The deflator is from
the U.S. Bureau of Labor Statistics (http://data.bls.gov/cgi-bin/surveymost). The differenced
variable is crudediff.
5
Income represents disposable personal income per capita in a given month and state,
expressed in the number of gallons of regular gasoline that the income could purchase. It was
constructed from quarterly income and tax data as well as from annual population data, and it
was deflated by the monthly price of regular gasoline from the U.S. Energy Information
Administration.a Population data were from the Census Bureau’s Statistical abstract of the
United States (http://www.census.gov/prod/www/statistical-abstract-04.html). Income data were
from State quarterly personal income of the Bureau of Economic Analysis
(http://www.bea.gov/bea/regional/sqpi/). Tax data were from the Census Bureau’s Quarterly
summary of state and local tax revenues (http://www.census.gov/govs/www/qtax.html) and from
its Federal tax collections by state, based on the Internal Revenue Service data book for various
years (http://www.irs.gov/taxstats/article/0,,id=102174,00.html). Differenced income is
incomediff.
Time series of 75 months – from January 1998 through March 2004 – were collected for
each of the 50 states.
Summary statistics are presented in Table 1. The price of premium gasoline exceeds the
price of regular gasoline by only 13 percent, but Americans buy only a sixth as much of premium
as of regular. The relative price of premium varies from 1.3 (Georgia, February 1999) to 1
(North Dakota, October 2001). Relative purchases of premium vary from .51 (New Jersey,
February 1999) to .035 (North Dakota, March 2000). Annual per capita disposable income
a This method of estimating monthly income has the advantage of using all available information; but, because we
lack direct data for monthly income, we have likely introduced a measurement error. Let our measure of monthly
income be Income; and let actual monthly income be Income*. Then it is likely that Income = Income* + e, where e
is a measurement error. Because income tends to rise over time, our measure of income likely underestimates actual
income. This implies that the measurement error has a negative mean and is correlated negatively with actual
income. OLS estimators are thus biased and inconsistent. However, as we see it, the main problem with the
measurement error is that it biases the Income coefficient toward zero; this coefficient is not the main concern of our
analysis.
6
averages 16,283 gallons of regular gasoline, ranging from 9,928 gallons in Georgia (January
2001) to 29,600 gallons in Florida (February 1999).
[Table 1 here]
3.2 Empirical Results
We estimated the model by three-stage least squares in order to disentangle the effects of
supply and demand. The model is in first differences, rather than levels, in part because our
variables for the relative consumption of premium gasoline and for its relative price, prgas and
prprice, may follow random walks and thus induce a spurious correlationb; and in part because
we wished to remove the effects of unobservable variables that were fixed over time and that
may correlate with explanatory variables, since this could render the coefficient estimators
biased and inconsistent. We calculated robust standard errors (White, 1980; Wooldridge, 2003)
because tests indicated heteroskedasticity in the 3SLS equations. Serial correlation is also
present, but it is not significant in magnitude.c
We first discuss the demand model. Generally, the evidence for the third law of demand
is moderate. In differences, the relative sales of premium gasoline respond negatively and highly
significantly to an increase in the relative price of premium gasoline (prprice) with a two-month
lag; the negative coefficient on the three-month lag is significant at the 10% level for a one-tailed
test but not at the 5% level. (The current relative price, and its one-month lag, were dropped
from the model for statistical insignificance.) The price elasticity of demand for premium
gasoline, relative to regular gasoline, is -.097 with a two-month lag and -.05 with a three-month
lag.d
b A regression of each variable on its lag produces a coefficient over .93.
c A regression of the residual of each equation on its lag turned up a coefficient of about -.1.
d Elasticities were estimated from a 3SLS model of the first differences of natural logs.
7
The relative demand for premium may be falling over time. Large vehicles today may be
more likely to use regular gasoline than they were before 1998. Figures 1 and 2 suggest a fall in
the relative demand for premium gasoline: Relative sales of premium gasoline have fallen along
with its relative price.
The fall in the relative price of premium gasoline is not due to the reduction, through
inflation, in the effective tax rate on gasoline, which is indicated by Figure 3. The bulk of the
gasoline tax consists of per-gallon taxes that are the same for premium and regular gasoline (as
opposed to ad valorem taxes). Since 1998, the per-gallon taxes, relative to the untaxed price of
gasoline, have dropped fairly steadily (Figure 3): While few states have lowered their nominal
taxes, few have raised them as rapidly as the rise in gasoline prices. The relative reduction in the
effective tax rate has been greater for regular gasoline than for premium gasoline, since the price
of regular has risen relative to the price of premium. Instead, the fall in the relative price of
premium gasoline seems due either to the fall in relative demand for premium or to the rise in the
cost of crude oil. If the cost of crude is about the same to manufacture a gallon of regular
gasoline as for a gallon of premium gasoline, then a rise in oil costs will raise the relative price of
regular gas – that is, lower the relative price of premium gas.
[Figure 1 here]
[Figure 2 here]
[Figure 3 here]
8
The coefficient on per capita personal disposable income, Incomediff, expressed in
gallons of regular gasoline, is positive but not significant. The income elasticity of relative
demand for premium gasoline is .668.
We turn now to the supply model. The coefficients on price and its one-month lag are
positive and highly significant: Measured in differences, the relative supply of premium gasoline
rises with the relative price. Producers are quite sensitive to price: The price elasticity of relative
supply for premium gasoline is 7.85 in the current month and .615 with a one-month lag. The
coefficient on Crude is negative but insignificant. The elasticity of relative supply of premium
gasoline with respect to crude costs is small, as expected: -.03.
The results of the basic model are shown in Table 2.e The constants, Cons, indicate that
the relative quantity of premium gasoline may have fallen over time, holding prices, income and
crude costs constant.
[Table 2 here]
Since 3SLS estimates coefficients by using instrumental variables, it may introduce
noise. For comparison, we provide in Table 3 an OLS model of the relative demand for
premium gasoline.f Lags of the dependent variable are highly significant, and the price
coefficients are positive. We suspect that this model reflects supply factors.
[Table 3 here]
e One can show that the linear demand function used here derives from an indirect money metric utility function
that is quasi-linear and quasi-exponential, in which “income” denotes the budget allocation to purchases of premium
and regular gasoline. We use personal disposable income per capita as a proxy for the budget allocation.
9
Conclusions and discussion
Overall, we find moderate support for the third law of demand in the market for premium
gasoline, in a simultaneous-equations model of demand and supply. We thus avoid conflating
demand and supply factors.
We cannot claim to have isolated the substitution effect, since in theory the income effect
may also increase the relative quantity demanded of premium gasoline. For example, if premium
and regular gasoline are both normal goods, and the absolute price of premium gasoline drops,
then the consumer may spend relatively more of the resulting increase in real income on
premium than on regular gasoline. But we believe that the income effect in the relative demand
for premium gasoline is likely weak: The estimated income elasticity of relative demand is .66,
and the income coefficient in the demand function is statistically insignificant. Also, the usual
$0.20 per gallon price spread between regular and premium is unlikely to have much effect on
most Americans’ buying power. We believe that we are observing primarily the substitution
effect.
f A regression of the residual upon its lag did not indicate serial correlation in this model. The dependent lags have
probably controlled for much of the innate serial correlation.
10
References
Alchian, A.A. and W. R. Allen, 1967, University Economics, 2nd ed. (Wadsworth, Belmont,
Calif.).
Barzel, Y., 1976, An alternative approach to the analysis of taxation, Journal of Political
Economy 84, 1177-97.
Borcherding, T. E. and E. Silberberg, 1978, Shipping the good apples out: The Alchian and
Allen theorem reconsidered, Journal of Political Economy 86, 131–38.
Cowen, T. and A. Tabarrok, 1995, Good grapes and bad lobsters: applying the Alchian and Allen
theorem, Economic Inquiry 33: 253-256.
Dahl, C. and T. Sterner, 1991, Analyzing gasoline elasticities: a survey. Energy Economics 13,
203-210.
Gould, J. P. and J. Segall, 1969, The substitution effects of transportation costs, Journal of
Political Economy 77, 130–37.
James, J. S. and J.M. Alston, 2002, Taxes and quality: a market-level analysis, The Australian
Journal of Agricultural and Resource Economics 46, 417-445.
Nesbit, T.M., 2005, Taxation and product quality: The gasoline market, working paper (West
Virginia University, Morgantown, W.V.).
Razzoni, L., W.F. Shughart II and R.D. Tollison, 2003, On the third law of demand, Economic
Inquiry 41, 292-298.
U.S. Bureau of Economic Analysis, 2005, State quarterly personal income, available at URL:
http://www.bea.gov/bea/regional/sqpi/ (downloaded 5/1/2005).
U.S. Bureau of Labor Statistics, 2004, Consumer price index, available at URL:
http://www.bls.gov/cpi/home.htm (downloaded 10/10/2004).
U.S. Bureau of Labor Statistics, 2004, Producer price index for petroleum refineries, available at
URL: http://data.bls.gov/cgi-bin/surveymost (downloaded 7/1/2004).
U.S. Census Bureau, 2005, Quarterly summary of state and local tax revenues, available at URL:
http://www.census.gov/govs/www/qtax.html (downloaded 5/1/2005).
U.S. Census Bureau, 2005, Statistical abstract of the United States, available at URL:
http://www.census.gov/prod/www/statistical-abstract-04.html (downloaded 5/1/2005).
U.S. Energy Information Administration, 2004, Monthly Energy Review, available at URL:
http://tonto.eia.doe.gov/FTPROOT/monthlyhistory.htm (downloaded 7/1/2004).
11
U.S. Energy Information Administration, 2004, Monthly petroleum data, available at URL:
http://tonto.eia.doe.gov/STEO_Query/app/papage.htm (downloaded 10/10/2004).
U.S. Highway Administration, 1996, Highway statistics, available at URL:
http://www.fhwa.dot.gov/ohim/1996/index.html (downloaded 11/1/2004).
U.S. Highway Administration, 2004, Monthly motor fuel reported by states, available at URL:
(http://www.fhwa.dot.gov/ohim/mmfr/mmfrpage.htm (downloaded 7/1/2004).
U.S. Internal Revenue Service, 2005, Internal Revenue Service data book, available at URL:
http://www.irs.gov/taxstats/article/0,,id=102174,00.html (downloaded 5/1/2005).
White, H., 1980, A heteroskedasticity-consistent covariance matrix estimator and a direct test for
heteroskedasticity, Econometrica 48, 817-838.
Wooldridge, J., 2003, Introductory econometrics: A modern approach, 2nd edition (South-
Western, Mason, Ohio).
12
Table 1. Summary Statistics
Variable Obs Mean Std. Dev. Min Max
prprice 3727 1.129 .0353 1.002 1.299
prgas 3705 .176 .082 .036 .508
crude 3750 23.31 2.24 18.63 27.77
income 3750 16283.0 2896.0 9297.7 29600.2
prgdiff 3644 -.0012 .0129 -.0974 .294
prpdiff 3655 -.00029 .0218 -.0675 1.135
incomedif 3700 -34.8 972.8 -4065.4 4937.8
crudediff 3700 .044 1.26 -3.03 3.45
13
Relative sales of premium gas drop
0
0.05
0.1
0.15
0.2
0.25
0.3
Mar-97 Jul-98 Dec-99 Apr-01 Sep-02 Jan-04 May-05
Premium gas/Regular gas
Figure 1
14
Relative price of premium gas falls
1.08
1.1
1.12
1.14
1.16
1.18
1.2
Mar-97 Jul-98 Dec-99 Apr-01 Sep-02 Jan-04 May-05
Premium price/relative price
Figure 2
15
Gas tax rates fall
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Mar-97 Jul-98 Dec-99 Apr-01 Sep-02 Jan-04 May-05
Tax/Untaxed gas price
Tax/reg Tax/prem
Figure 3
16
Table 2. Estimates of the 3SLS model
Equation Obs Parms RMSE "R-sq" chi2 P
qDemand 3409 3 .009826 0.2824 1341.000 0.0000
qSupply 3409 3 .011652 -0.0091 1028.583 0.0000
Coef. Std. Err. Rob. Std. Err.
qDemand
prpdiff
L2 -.1049 .0111 .0101
L3 -.0151 .0053 .0107
incomedf 5.99e-06 1.67e-07 .0114
_cons -.0011 .0002
qSupply
prpdiff
-- .9641 .0317 .0250
L1 .0845 .0126 .0141
crudedff -.0003 .0001 .0109
_cons -.0005 .0002
17
Table 3. Estimates of an OLS model
Number of obs = 3325
F(9, 3315) = 167.61
Prob > F = 0.0000
R-squared = 0.3682
Root MSE = .00909
Robust
prgdiff Coef. Std. Err. t P>|t| [95% Conf. Interval]
prgdiff
L1 -.1161 .025 -4.65 0.000 -.165 -.0671
L2 -.1215 .021 -5.76 0.000 -.1629 -.0801
L3 -.0717 .0189 -3.80 0.000 -.1087 -.0348
L4 -.0482 .0186 -2.59 0.010 -.0847 -.0117
prpdiff
-- .2657 .0237 11.22 0.000 .2192 .3121
L1 .1661 .0235 7.08 0.000 .1201 .2121
L2 .0005 .0198 0.03 0.979 -.0383 .0394
L3 .0206 .0167 1.24 0.217 -.0121 .0534
Income 4.12e-06 2.34e-07 17.61 0.000 3.66e-06 4.58e-06
Constant -.0012308 .000165 -7.46 0.000 -.0016 -.0009
