An analysis of California gasoline sales was made, using Box and Jenkins' linear time series methods, in an attempt to detect the change due to a 17% change in price. There was no detectable change within the 2% noise limit of the method. Thus, either the elasticity is less than 2 parts in 17, and/or the tools must be further refined to detect it

Schneider, Alan M.

English

Caltech Authors - Main

$ 1.00 
eSC 
Visiting Associate in Environmental Engineering Science 
Fall 1973 
EQL 
Memorandum 
No. 10 
© California Institute of Technology 1973 
ENVIRONMENTAL QUALITY LABORATORY 
Supported in part by the National Science Foundation Research Applied to National Needs, (RANN), under Grant No. GI-29726 
EQL Memorandum No. 10 
ELASTICITY OF DEMAND FOR GASOLINE 
IN THE SOUTH COAST AIR BASIN 
by 
A M. Schneide r 
Visiting Ass ociate in Environmental Engineering Science 
Fall 1973 
Supported in part by the National Science Foundation, Research Applied to 
National Needs (RANN). under Grant No. GI-29726 
ABSTRACT 
An analysis of California gasoline sales was made, using Box and 
Jenkins I linear time series methods, in an attempt to detect the change 
due to' a 17% change in price. There was no detectable change within the 
2% noise limit of the method. Thus, either the elasticity is less than 2 
parts in 17, and/or the tools must be further refined to detect it. 
INTRODUCTION 
The elasticity of dem.and for gasoline is the (negative) ratio of the 
change in consumption per unit change in price. One would like to know 
this ratio so that one could predict the change in gasoline consumption 
by the motoring public due to price or tax increases. In particular, such 
measures have been proposed as part of a total strategy to reduce air 
pollution in the South Coast Air Basin. 
In July and August 1972, a step-function price increase to the 
consu.mer occurred in California; a result of (a) an increase in the price 
charged by the oil producers, and (b) imposition of the 5% state sales tax 
on gasoline. The price for regular grades of gasoline in the Los Angeles 
area, which, although subject to a great deal of weekly fluctuation, + had 
;::( 
been averaging 32,4 cents per gallon, jumped to 38. 0 cents per gallon, 
a 17.3% increase, and remained pretty nearly constant at that level until 
the end of March 1973, 
It was thought that this step increase, if accompanied by a c.orres-
ponding decrease in sales, would provide data for determining the elasticity, 
at least for price changes of the indicated amount. 
The objective of the present study then is to determine the elasticity 
of demand for gas oline in the South Coast A ir Basin by performing statisti-
cal analyses of the data on gasoline sales before and after the price change 
occurred in an attempt to determine what change in c onsu.mption took place. 
+ The Los Angeles market is considered to be an llunstable" one pricewise, 
A verage over preceding three months at maj or stations. 
- 2 -
It was known from previous studies that the elasticity was small, probably 
less than 0.1; hence, sophisticated means would have to be employed to 
extract, hopefully, a small signal buried in heavy noise. 
DATA 
The data obtained to carry out the study consisted of total taxable 
gasoline gallonage (less aviation gasoline) sold monthly in the State of 
California from January 1960 to March 1973 inclusive. The data were 
obtained from the State Board of Equalization, Sacramento. The agency 
compiled these figures as part of its task of collecting the tax on gasoline. 
These data appear as Appendix A. 
Price data were obtained as the weekly average for regular grade 
gasoline of approximately 4,000 major-brand retail stations in the Los 
Angeles area over the period January 3, 1971-January 28, 1973. These 
data are compiled regularly as a service to the petroleum industry by the 
Lundberg Co, The price data were used to determine the magnitude of the 
increase, which was simply treated as a step function. 
It would have been preferable to have had the sales and price data 
from the same geographic region. As it is, the sales of San Diego, San 
Francisc 0, and all other parts of the state are included in the total sales 
figures. Since the price use at the wholesale level was probably uniform 
throughout the state, and the 50/0 sales tax was applied everywhere in the 
state at the same time, it was felt that the. price rise was a step-function 
in these other areas as well. However, due to the fact that San Diego 
and San Francisco were historically more listable" pricewise (less sub-
ject to price wars and price fluctuations), the average priee in these 
- 3 -
areas before July 1972 was probably sOITlewhat higher than in Los Angeles. 
The price in San Diego, San Francisco, and Los Angeles seeITl to have been 
ITlore nearly equal and constant after July 1972. In sUITlITlary, the step-
function in price occurred statewide, although the average magnitude for 
the state as a whole may be less than the 17% experienced in the Los Angeles 
area. 
PROCEDURE 
The .methods of tiITle series analysis as described by Box and Jen-
kins (Ref. I) were used to analyze the sales data. The following assuITlp-
tions were ITlade: 
(1) The tiITle series Zt representing the sales data were 
the result of a stochastic process in which white 
noise at is the only input (Figure 1). 
(2) The stochastic process at and linear filter were 
stationary over the period January 1960 - July 1972. 
In particular, assuITlption (I) implies that the customary weekly 1 to 4 cent 
price fluctuations prior to the big step of 1972 were inconsequential, and 
that no other variables (lileading indicators"), such as unemployment rate, 
were of iITlportance in deterITlining sales. The second assuITlption iITlplies 
that the saITle (unknown) forces were at work to determine sales throughout 
the entire period. These forces could include growth in car population, 
growth in per car ITlonthly gasoline consumption, etc. The assuITlption 
iITlplies that the increased per ITlile consu.mption of new cars with pollution 
control devices was not a recognizably distinct change in the input process. 
- 4 -
The procedure consisted of the following steps: 
(1) finding an appropriate form of the model; 
(2) estimating the parameters of the model; 
(3) diagnostic checking to verify the adequacy of the 
proposed model; 
(4) forecasting the monthly sales beginning with July 1972 
with the model, on the assumption that no price 
change occurred (i. e., the same model applied); 
(5) obtaining the standard deviation associat'ed with 
the forecasts; and 
(6) comparing actual sales with forecast sales to see 
if a significant change in sale s, actual relative to 
forecast, had occurr~d. 
These steps are discussed in more detail in the paragraphs which follow. 
Step 1 Finding the form of the model 
The time series input data consisted of the 150 monthly sales figures 
from January 1960 through June 1972 inclusive. These are plotted in Figure 
2. As can be seen, there is a strong growth component over the twelve-year 
period and also a marked seasonality over a one-year period. 
The natural logarithm of the data was taken to remove a constant 
exponential growth factor. The result of this is, by definition, the time 
series Zt of Figure 1. Then, the first (\7) and twelfth (\712) backward differ-
':< 
ences were taken, yielding a time series w to 
(1) 
Notation follows that of Reference 1. 
- 5 -
The mean of the w
t 
series, w, was subtracted from we and the resulting 
ti.me series was fit by a moving average seasonal/non-seasonal model of 
the form: 
(2) 
where 
12 
W t - W = (1 - 6 1 B ) ( 1 - 6 sIB ) at 
B = backward shift operator. i. e., Bat = a t _1 
6 1 , asl = unknown model parameters, to be 
determined by fitting the model 
This particular form of the model was arrived at by first following 
the preliminary identification procedures for non-seasonal models as 
sugge sted in Chapter 6, Ref. 1. 
Step 2 . E s Hma tingmodel parameters 
The parameters of the model were determined in each case by the 
Marquardt algorithm for nonlinear least-square estimation, p. 504, 
Ref. 1. In essence, that set of parameters is selected which permits 
the sum of squares of the residuals, at in model 2, to be the smallest. 
The program used to perform the needed calculations was patterned after 
the estimation procedure of pp. 500-505, Ref. 1. 
Step 3 Diagnostic checking 
To determine whether the assumed model form, together with its 
estimated parameters, constituted a reasonable one, having satisfactory 
stochastic properties, the residuals at' and their first 32 autocorrelations 
Ref. 1 (p. 291) describes the chi-square test as a general, or llport-
manteau, If test of the hypothesis of model adequacy. 
- 6 -
r (k)(k = I, 32), were calculated according to the procedure of p. 503, 
aa 
Ref. 1. The chi-square Fltatistic 
(3) X 2 = 
32 
n E 
k=l 
r ... " (k) 
aa 
was computed where n is the number of data points in the w
t 
series 
(n = 150-1-12 = 137), and was compared with a chi-square distribution with 
v = 32 - 2 = 30 
degrees of freedom, 2 being the number of parameters estimated in the 
model. Only if the model passes this check do we regard it as in any way 
being representative of the process which generates the data. (Even if it 
passes this test, it still may not represent the data. ) 
Step 4 Forecasting 
The time series used the given data up to and including June 1972, 
and then the model was used to forecast data beginning with July 1972 under 
the assumption that the model was still valid. That is, the forecast figures 
are those sales which we would have expected had there been no sudden 
change in price. The forecasting procedures of Chapter 9, Ref. 1 were 
used. Forecasts of w t were converted back to ze and the anti-log of this 
-,-
',' becomes the sales forecast. 
Step 5 Standard deviation of the forecasts 
There is an uncertainty in the forecasts which grows as the lead interval 
(the length of time over which the forecast is made) increases. This 
The anti-log of a normally distributed random variable generates a log 
normally distributed random variable, whose mean is offset from the 
anti-log of the mean of the former by an amount which depends on the 
variance. In this case, the offset affects the predicted sales by one part 
in 5000, a negligible amount. 
- 7 -
uncertainty is expressible in terms of the variance V( l) of the forecast for 
lead time 1 (1 = 1,2,.0. for one-month, two-month, etc., forecasts). The 
standard deviation of the forecast at lead time 1 is the square root of the 
variance. The model itself provides the basis for estimating the variance of 
the forecasts. 
Step 6 Comparison of actual and forecast sales 
The foreca st sales are those which we expect if there had been no 
change in conditions, i. e., if the same model (2) with its estimated parameters 
applied. If the actual sales differ significantly (say more than I, 2 or 3 
standard deviations) from the forecast values, then we can say that the time 
series analysis has detected a change in consumption. The ratio of the 
change in consumption (expressed in percent) to the change in price (also 
expressed in percent) is the elasticity. 
RESULTS 
As scussed above. preliminary estimation procedures led to selec-
tion of model (2) as the working hypothesis. 
The diagnostic check for model (2) gave an observed chi-square of 
30. 0 for 32 residual autocorrelations and 30 degrees of freedo.m. The 50% 
and 25% points for model (2) with 30 degrees of freedom are 29.3 and 34.8 
respectively. Therefore, the check does not give any evidence of the inade ... 
quacy of the model. The estimated parameters of the model are: 
36 9 ± 01 
± • 
The data do not warrant the inclusion of all of the digits produced by the com-
puter; however, results are given as generated rather than introducing round-
- 8 -
off. The plus-minus figures are the standard deviation of the esti.mates. 
The model (2) selected gave better results than any of the following, 
which were also tested. 
(4 ) :: 
(5 ) :: 
(6) :: 
(7) 
The forecasts obtained with the model (2) are shown in Figure 3. The 
latter figure also shows the one-sigma tolerance on the forecasts and the 
actual sales. The one-sigma tolerance, expressed as a percent of sales, 
varied from 1. 99 for the I-month lead (the July forecast) to 2.01 for the 9-
month lead. This tells us. essentially, that the assumed model, and the data 
it was fitted to, together were capable of detecting changes in sales of about 2% 
but that changes in sales of less than this a.mount would be buried within the 
still-remaining model noise. 
Comparing the actual sales (with the price increase) to the sales fore-
cast on the basis of no change in price, it is seen that the differences still lie 
within the noise band of the model. Thus, it is not possible. from the analysis 
performed. to know whether a change in sales level occurred or not when the 
price rose 17%. 
- 9 -
CONCLUSIONS 
It is not clear that the increase in price in the Los Angeles area of 
17% produced any observable change in gas oline consuITlption in the state. 
1£ there was a net change in sales, it was less than 2%. the noise limit of the 
present analysis. 
It ITlust be recognized that a 170/0 increase in price in gasoline represents 
probably no ITlore than a 3% increase in the cost of operating a car, and just 
a few tenths of a percent of total average incom.e. Thus, small changes in 
gasoline price m.ay not show the saITle elasticity effects as large changes. 
Also, the present study looked for short-term response, whereas long-term 
adjustments may be more significant. 
RECOMMENDATIONS 
1£ one were interested in carrying the analysis still further, in an 
attempt, for example, to reduce the noise level on the forecasts (and to iITlprove 
the forecasts in the process), then the following should help. 
(1) Obtain data on both price and sales for the same ge agraphic 
region. 
(2) Include in the forITlulation a linear dynamic system which 
represents a deterministic dependence of the ITlodel on 
price. For even better results, include other factors 
which are believed to have a significant effect on gasoline 
consumption. One such factor would be the decreased 
miles per gallon realized with the newest ITlodel cars. 
This, of course, requires complete price data over the 
same time period as the sales data. Another factor to 
include might be s orne prosperity indicator, or negative 
prosperity indicator (unemployment rate in the area) 
which presumably would have some influence on gasoline 
- 10 -
sales. Then the model becomes the multiple-input model 
shown in Figure 4. Methods for analyzing such models are 
a direct extension of the methods applied here, and are dis-
cus sed in Chapter 11 of Ref. l. 
(3) Continue the analysis as time progresses. As this is being 
written the price is climbing at an accelerated pace. 
White 
noise 
Figure 1 
Linear 
Filter 
The Stochastic Process 
1/ 
TiITle series 
(ITlonthly sales) 
(/) 
~ 
..J 
..J 
« 
C) 
I.i.. 
o 
(/) 
z 
o 
..J 
..J 
2 
1000 
1960 
Fioure 2 
MONTHLY SALES OF GASOLINE IN 
CALIFORNIA 
1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 
-- ..... 
-- ..... 
..... _-
'" <I) 
" (j') 
"0 
<I) 
.... 
<..> 
'/loN "0 ~ b Il.. 
'>< '~:JO 
+ 
...J 
'" 
b 
<I) « 
" 
:::::> (j') 
'" 
.... 
"0 \ 
'" 
U 
II) 
\ 
« .... 
(.) 
'~das 
I:) 
(j') \ "0 
) 
"0 
II) 
<I) 
Ii) 
... 
..-
> 
Il.. 
./ 
0. ~ .... // ;' ~ 
0 
I.IJ 
.... 
.,,-
.... ( 
\ 
\ 
\ 
\ 
.(Inr 
N 
I"-
m 
0 0 
0 
0 
10 
I"-
0 
0 
I"-
0 
10 
m 
0 
0 
m 
it) 
0'1 0'1 
lllW 
at Linear 
..... Filter , 
White noise 
lXt 
Linear 
....... 1.\:>. 
-'" Dynalllic .;- /~Y· 
Gas oline price Systelll I 
zXt Linear 
"-
Dynalllic 
'" Systelll Miles per gallon 
of average of this 
year's production 
3Xt Linear 
Dynalllic 
-'-
/" Systelll 
Prosperity 
IndIcator 
Figure 4 
Multiple -Input Deterlllinistic /Stochastic -Model 
\It 
.~ 
'i' 
Noise 
Nt 
--------'1> 
Yt 
Sales 
REFERENCE 
1. Box, George P •• and Gwilym M. Jenkins, Time Series Analysis, 
Forecasting, and Control, Copyright 1970, Second Printing 1971, 
Holden-Day, San Francisco. 
419525966 
490443236 
448176905 
507760806 
479157595 
537756552 
512513491 
580935092 
543969522 
615948480 
544401794 
639228416 
593413574 
659462378 
593278660 
683354587 
650302605 
740879137 
661497199 
765914293 
704161429 
814846854 
717759909 
817775618 
771559455 
847761632 
809773370 
APPENDIX A 
Total Monthly Gasoline (less Aviation) Sales 
in California (Gallons) 
January 1960 - March 1973 Inclusive 
426648441 471355767 468579504 
509990974 476570597 473323177 
424021439 494401792 470221431 
525805876 492905287 493830971 
426691513 507608741 499951034 
560099067 505188157 527819708 
462678605 523365025 529412632 
592866525 538805405 552373267 
527299671 549272767 555980663 
608705525 588011227 582087772 
531359326 605921762 560788840 
629159189 608954524 608790856 
528548553 6192'03521 625405286 
689180380 631606786 628412744 
573680460 639718321 602921336 
708990207 654237227 668952664 
606857777 666110515 686603193 
747311910 689125677 707951626 
628947871 706965083 715861781 
764888787 750609087 747916545 
667839557 764958389 739530638 
798427743 758250071 766100930 
692579960 765440919 777442698 
822949774 790058781 788992405 
758301373 845233081 806046434 
899385235 831569009 836898326 
760666260 879012700 
Note: Read horiz ontally, then down one row at a tim.e. 
476871751 488728081 
457312473 476025319 
499261765 505011073 
480675775 488444300 
522575859 526863764 
508842006 518388590 
554244136 552311172 
527487962 551953739 
574400274 599626856 
548347603 586726967 
605382754 624020730 
580213242 615355004 
636973982 659434045 
604798619 641171006 
657360979 684601313 
632059576 666182877 
702221786 688027598 
658555091 696850990 
768411771 722660760 
705170400 733540728 
774130469 779636361 
721869531 771607070 
789538807 815086188 
779558984 801108894 
856752072 866595658 
822385628 852003692 


Elasticity of demand for gasoline in the South Coast Air Basin

Time Series Analysis, Forecasting, and Control, Copyright

Time Series Analysis, Forecasting, and Control, Copyright 1970, Second Printing

http://authors.library.caltech.edu/25759/1/EQLmemo10.pdf

Elasticity of demand for gasoline in the South Coast Air Basin

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