Bulletin 592 March 1958
UNIVERSITY OF FLORIDA
AGRICULTURAL EXPERIMENT STATIONS
J. R. BECKENBACH, Director
GAINESVILLE, FLORIDA
Experimental Pricing As an Approach to
Demand Analysis
A Technical Study of the Retail Demand for Frozen
Orange Concentrate
By
L. A. POWELL, SR., WILLIAM G. O'REGAN
and
MARSHALL R. GODWIN
AG/?J
TECHNICAL BULLE
Single copies free to Florida residents upon request to
AGRICULTURAL EXPERIMENT STATION
GAINESVILLE, FLORIDA
CONTENTS
Page
SUMMARY .................. ........................................ ... 3
INTRODUCTION ............... ....... .... ..... ................. 5
THE ECONOMIC MODEL ............... ..........................  5
THE EXPERIMENTAL APPROACH ................. ..... ............... 9
THE STATISTICAL MODEL ..................... .....12
ANALYTICAL PROCEDURE ...............  ........ 16
Organization of the Data ........................ .. ... 16
Alternative Forms of the General Model .................. ..................... 18
Examination of Hypotheses for Selection of Specific Model ............... 19
ECONOMIC IMPLICATIONS OF THE ESTIMATING MODEL ................. .. ........ 28
The General Demand Function ..............  ........... 29
Demand Shifting Parameters .............. ..... ..... .. 30
APPENDIX I. Basic Data .................................... 33
APPENDIX II. Effect of Experimental Prices on Store Traffic .............. 34
APPENDIX III. Matrix Representation of the Analytical Model and
Data for Covariance Analysis ....................................... 36
APPENDIX IV. Normal Equations Used in Estimating the Regression,
Store, Week and Age Constants for the Final Model.. 44
ACKNOWLEDGMENTS
In the course of developing this study, the authors relied heavily upon
the advice, assistance and cooperation of many persons. While aid from
all sources was deeply appreciated, particular recognition is accorded the
following individuals who were most instrumental to the completion of
the work:
Dr. R. L. Anderson and Dr. H. L. Lucas, Department of Experimental
Statistics, North Carolina State College, developed the experimental design
and rendered valuable advisory and consultative assistance in connection
with analytical procedures.
Dr. H. G. Hamilton, head, Department of Agricultural Economics, Uni
versity of Florida, bore the brunt of the administrative burden created by
the study and gave unfailing support to the effort.
Dr. Vasant L. Mote, Department of Agricultural Economics, University
of Florida, critically reviewed preliminary drafts of the manuscript and
made several suggestions which contributed materially to an improvement
of the analytical presentation.
Mrs. Bonnye T. Leary and Mrs. Billie S. Lloyd performed the statistical
computations. Mrs. Jane M. Maddox and Mrs. Willie J. DuBose per
formed the difficult task of typing the manuscript.
SUMMARY
Unlike most approaches to demand analysis, the data for
this study were generated under semicontrolled conditions to
satisfy a previously specified economic model.
Generation of the underlying data involved measuring cus
tomer responses to a set of deliberately introduced retail prices.
Five price levels were tested: the prevailing market price, prices
representing discounts below the market of 3, 6 and 8 cents
per 6ounce can and one price representing a premium of 4
cents above the current market level. Test prices were intro
duced in 10 retail stores in conformance with an experimental
design especially derived to permit statistical isolation of the
price effect, while at the same time coping with the problems of
carryover effect and multiplicity of product brands.
Computational efficiency and prior economic theorization about
the problem led to the choice of a logarithmic "fixed unknown
constants" statistical model for analyzing the data. That is,
weekly purchases of orange concentrate per hundred customers
were assumed to be a function of the sum of a set of "class
constants" consisting of a price "age" effect, store effect, week
effect and effect of store x price "age" interaction, and of a re
gression on price, with all components of the model expressed
in terms of logarithms.
Given the general model, the choice of a specific model neces
sitated testing certain hypotheses about the model parameters.
In particular, the appropriate degree of the regression remained
to be determined and a choice made between fitting a single re
gression or individual regressions for each price age. The sig
nificance of the variation contributed by the specified "class
constants" also required examination.
Upon performing the necessary tests, the "best" estimating
model was found to be a quadratic single regression function
including all of the specified "class constants" with the excep
tion of store x age interaction, which turned out to be non
significant.
Application of the estimating model to the data yielded a
demand function for which price elasticity of demand varied
inversely with price. Derived estimates showed that demand
changed from an elastic to an inelastic relationship at the pivotal
price of about 12 cents per 6ounce can of concentrate. The
revenue function generated by this demand relationship was
convex to the origin with minimum revenue occurring at the
point of unitary elasticity on the demand curve.
Experimental Pricing As an Approach to
Demand Analysis
A Technical Study of the Retail Demand for Frozen
Orange Concentrate
INTRODUCTION
Measurement of the demand for agricultural products is one
of the major methodological problems in the field of agricultural
economics research. Since much of the difficulty encountered
in the study of demand is traceable to a lack of satisfactory data,
more attention to this aspect of demand research is urgently
needed.
The research method employed in this study differs from
orthodox approaches to demand analysis in one important re
spectthe basic data were generated under semicontrolled con
ditions to satisfy a previously specified economic model. Instead
of depending upon the market mechanism as a source of data,
measurements were taken of customer response to a set of de
liberately introduced prices. Data obtained in this fashion have
two distinct advantages over market information customarily
available. First, such information permits the study of cus
tomer response to a range of price circumstances considerably
wider than that afforded by the workings of the market. Second,
the data can be collected within a relatively short time period,
thereby partially obviating the analytical and interpretive prob
lems associated with changes in consumer income, tastes, prefer
ences and similar troublesome factors.
The purpose of this bulletin is to describe the datagenerating
technique and the procedure followed in an analysis of the de
mand for frozen orange concentrate. Data serving as a basis
for the study were obtained from 10 retail food stores in the
vicinity of Trenton, New Jersey, during the period June 7 through
August 7, 1954. Descriptive aspects of the research undertaking
are dealt with comprehensively in Florida Agricultural Experi
ment Station Bulletin 589: "Consumer Reaction to Varying Prices
for Frozen Orange Concentrate."
THE ECONOMIC MODEL
Although neoclassical demand theory calls for certain re
quirements virtually unattainable via empirical analysis, it was
6 Florida Agricultural Experiment Stations
regarded as the instrumental ideal for explaining consumer de
mand for frozen concentrated orange juice. Accordingly, the
quantity of orange juice purchased by a group of consumers per
unit of time, as of a certain time, was presumed to be dependent
upon the price of orange concentrate, the prices of closely related
commodities, and the tastes, preferences and real income (com
mand over goods and services in general) of the consumer group.
Since the effect of price upon purchases was of primary interest,
the goal became that of approximating the typical Marshallian
demand curve, which summarizes the functional relationship be
tween quantity purchased and price with the other factors (real
income, tastes, etc.) remaining invariant.1
Customarily, empirical quests of this sort cannot completely
conform to theoretical concepts. Certain compromises must be
effected between desired objectives and reality. Some theoreti
cal requisites are assumed to be nonexistent or inoperative for
the particular problem at hand, while other essentials, though
not directly measurable, are assumed to be associated with cer
tain identifiable variables and thus are accounted for indirectly.
As a practical measure in formulating the analytical approach,
customers patronizing each store were assumed to constitute a
distinct consuming group characterized by a particular prefer
ence pattern and income status. Adjustment for differences in
the number of individuals comprising the patronage of each
store was to be accomplished by reducing concentrate purchases
for each group to some per customer unit basis through the use
of customer count data. The quantity of concentrate purchased
at a particular price by each representative group supposedly
would vary by a proportional amount. Given this supposition,
the adjusted or customer unit purchase observations would rep
resent points on separate members of a family of proportional
demand curvesone for each store or group. For a particular
period, the differences in these curves presumably would largely
reflect differences in the preference patterns and income circum
stances of the clientele among the several stores.
With one exception, it was assumed that all temporal disturb
ances would effect proportional shifts in the demand curves that
1Milton Friedman, "The Marshallian Demand Curve," The Journal of
Political Economy, LVII, No. 6 (December, 1949). As Friedman points out,
holding the prices of closely related products constant is only a provisional
measure designed to isolate the immediate, direct effect of a change in price
of the commodity in question from the indirect effects that eventually fol
low. However, because of the short duration and restricted nature of this
study, such indirect effects did not materialize and, hence, are of no concern.
Experimental Pricing As an Approach to Demand Analysis 7
could be related to the time element. Prices of closely related
products, which, incidentally, changed simultaneously and uni
formly in all stores, were included in this category. In addition,
this class contained other factors (temperature, holidays, pay
days, etc.) that varied with time and which, very likely, affected
the demand for concentrate. The exceptional caseassumed
to be a function of timerelated to the influence of price "age,"
i.e., the length of time a particular price for concentrate was
in effect, on the demand for concentrate. Presumably, by use
of this measure, a distinction could be drawn between ultra
shortrun and shortrun demand curves. It was assumed that
price "age" could affect demand by shifting the demand curve,
changing the slope of the curve or both.
No practical means was available for holding real income con
stant along the demand curve. However, since expenditures for
orange concentrate represent only a minute fraction of the total
Price D
\ P3
Sp1
Sp2
D'
Quantity
S = Supply at pth price (p = 0, ...4)
8 Florida Agricultural Experiment Stations
consumer budget, the income effect of a change in the price of
orange concentrate was considered tq be negligible.
Finally, the problem of a nonhomogeneous product required
attention because concentrate was sold under different brands
and in two types of containers. Arbitrary price differentials
assigned to the other brands and can size in relation to the
major brand and can size were assumed to accommodate this
difficulty.
Within the framework of these assumptions, the experiment
constituted an attempt to generate a demand curve for concen
trate by confronting consumers with a completely elastic supply
of concentrate at selected prices, as schematically depicted on
page 7.
Conceptually, the representative demand curves, one for
each store, would form a proportional cluster about the general
demand function for concentrate derived from purchase observa
tions for the entire group.
Price
Dsi Da
D = General demand curve
D = Demand curve for ith store (i = 1,...10)
Ds = Demand curve for ith store (i = I, .. .10)
Experimental Pricing As an Approach to Demand Analysis 9
THE EXPERIMENTAL APPROACH
Informational requirements of the envisioned economic an
alysis necessitated the development of a specialized, datagener
ating technique. In planning the experiment, several matters
of particular emphasis were necessarily involved. Although not
entirely independent, these considerations logically may be classi
fied as either primarily structural or operational.
Along with the usual statistical features that would permit
isolation of the effect of price, i.e., treatment effect, four major
structural provisions were reckoned with in arriving at the
specific character of the experimental design: (1) the test was
to be limited to five prices including the current market price
and prices ranging both above and below this level, (2) the
prices were to be ordered in an approximate geometric progres
sion,2 (3) the pattern was to provide for the estimation of the
"carryover" effect of a change in price3 and (4) a multiplicity
of product brands was to be accommodated. While not altogether
a structural consideration, financial limitations served to modify
the conceptual design appreciably. A schematic representation
of the proposed datagenerating apparatus appears in Table 1.4
The position of the symbols in the twoway diagram indicates
the duration of the test prices and the distribution of these
prices among the sample stores. It should be noted that the
pricing mechanism, with respect to different brands, was per
fectly general in the sense that price differentials existing among
brands at the time the study was initiated were to be maintained
throughout the experimental period.
The success of the study was not only dependent upon the
appropriateness of the experimental model but was also con
tingent upon the skill exercised in handling problems posed by
the experimental performance. Perhaps the requisite most es
sential to the accomplishment of the experiment was assurance
of the full cooperation of store management. In this connection,
2 This requisite presumed an analysis linear in logarithms. In such event,
the resulting uniform dispersal of the observations along the derived de
mand curve would increase the "reliability" of estimates at the extremities
of the curve.
"Carryover" effect refers to the temporal adjustment in consumer
purchases following a price change. Conceivably, the purchase adjustment
pattern may be influenced by the storability of the product as well as the
vagaries of consumer price expectations.
SCredit for the development of the experimental design rests entirely
with Professors R. L. Anderson and H. L. Lucas of the Department of
Experimental Statistics, North Carolina State College.
10 Florida Agricultural Experiment Stations
it was necessary to persuade management that their relationship
with customers would not be jeopardized by the pricing arrange
ment. Furthermore, since food store management could not be
expected to bear the loss of revenue resulting from the sale of
orange concentrate at belowmarket prices, an acceptable plan
for reimbursement was required.
TABLE 1.PROPOSED EXPERIMENTAL DESIGN FOR STUDY OF CONSUMER
RESPONSE TO VARYING PRICES FOR FROZEN ORANGE CONCENTRATE.
___Stores
Period Dura I I
tion 1 2 3 4 5 6 7 8 9 10
(Test Prices)*
I week O 0 0 0 0 0 0 0 0
II weeks ++ + + 0 ++ 0
III 3 weeks 0 0 0 0 0 0 0 0 0 0
IV 3 weeks  0 0 ++ +  + ++ 
Price symbols are interpreted in descending order as follows: (+ +) represents the
highest test price; (+), the second highest price; (0), the market price; (), first test
price below the market price; and (), the lowest test price.
Prices were to be assigned to stores at random with the restriction that, for a given
store, the price assigned in weeks two, three and four could not be repeated in weeks eight,
nine and ten.
Because advertising could create undesired distortions in
customer purchases as well as present a real threat to customer
relations, consent on the part of management to refrain from
advertising orange concentrate during the course of the study
was vital.
In addition to the above operational problems, a few other
lesser refinements required notice. Measures to maintain the
normal atmosphere within each sample store clearly would be
desirable. Consequently, it was deemed advisable that enumer
ators assigned to the various stores should tactfully dissuade
store personnel from acquainting customers with the nature of
the pricing test. As a matter of fact, it was felt that the ex
periment would function more smoothly if even the sales per
sonnel were not informed of the details. Seemingly, such a
precaution would lessen the danger of store personnel having an
undue influence on customer actions. Enumerators were to be
charged with the additional responsibility of maintaining, at all
times, a nearnormal display of the test producta provision
Experimental Pricing As an Approach to Demand Analysis 11
designed to prevent any effect that a widely fluctuating stock
might exert upon purchases. Obviously, in view of the potentially
delicate nature of situations which might arise, enumerators
would need to be carefully screened on the basis of personality
and sound judgment.
Finally, there was the need for dealing with exogenous in
fluences not subject to withinstore experimental control. The
apparent advantage of operating in stores which pursued similar
merchandising procedures suggested that sample stores should
be members of the same chain. Moreover, the importance of
consumer income to the analysis led to the decision that the
study should be conducted in a moderately industrialized com
munity characterized by relatively stable income flows. With
this in mind, 10 large stores, located in the vicinity of Trenton,
New Jersey, were selected as the proper setting for the experi
ment. These stores were all members of a single major food
retailing organization.
TABLE 2.EXPERIMENTAL DESIGN FOR THE STUDY OF THE DEMAND FOR
FLORIDA FROZEN ORANGE CONCENTRATE, DELAWARE VALLEY AREA, JUNE 7
AUGUST 7, 1954.
Stores
Period Duration I I
S1 2 3 4 5 68 7 8 9 10
_______ ____________ 1
(Test Prices)*' 
I** 1 week ...... 2/33 2/33 2/33& 2/33 2/33 2/33 2/33 2/33 2/33 2/33
II 3 weeks .... 2/41 2/27 2/21 2/27 2/33 2/21 2/411 2/33 2/17 2/17
III 2 weeks .... 2/33 2/33 2/33 2/33 2/33 2/33 2/33 2/33 2/33 2/33
IV 3 weeks .... 2/17 2/33 2/33 2/41 2/27 2/17 2/27 2/21 2/41 2/21
Prices indicated apply to a 6ounce can of Brand B concentrate. The same absolute
price differentials were applied to the respective current or base prices of 6ounce containers
of other brands. However, the price of a 12ounce container of Brand B was varied pro
portionately to the price of the 6ounce can of the same brand.
** This price was in effect when the experiment was activated. Prices of other items
were as follows: Brand A2/37, Brand C2/29 and Brand B (12 oz.)2/55.
Because of the high cost of conducting the experiment, it was decided to reimpose
the market price for a period of only two, rather than three, weeks. The subsequent
analysis, however, suggests that a period of three weeks would have been highly preferable.
Upon initiating the experiment, it became apparent that the
proposed pricing arrangement would require some modification.
Because of the high price that concentrate was commanding at
12 Florida Agricultural Experiment Stations
the time, it was deemed unwise to run the risk of impairing
relations with company management by insisting upon testing
two prices higher than the current market price. The issue
was settled by testing one price above and three below the market
price. The actual pricing pattern imposed upon the sample stores
is presented in Table 2.
All stores included in the sample handled three brands of
frozen orange concentrate in 6ounce cans: a nationally adver
tised brand (Brand A) selling at the highest retail price; a private
label (Brand B), 2 cents lower; and a packer's label (Brand C),
4 cents lower than the advertised brand. In addition, the co
operating organization marketed Brand B concentrate in 12
ounce containers.5
THE STATISTICAL MODEL
Intuitive appeal, computational efficiency and a desire to be
consistent with ex ante economic theorization about the prob
lem led to the choice of a logarithmic "fixed unknown constants"
model for analyzing the data.6
Along with pricestores, store traffic, weeks and price "age"
seemed to be the logical sources of variation that could be given
meaningful interpretation. However, it was assumed that the
necessity of considering store traffic as a source of variation
could be avoided by expressing purchases of concentrate on a
per customer basis.7
"All frozen orange concentrate handled in the test stores was produced
in Florida.
Seemingly, economic intuition would call for a rejection of the arithmetic
"fixed unknown constants" model, which would require that the demand
curves for the various stores differ by a fixed absolute value, i.e., be equi
distant at all points. Suppose differences in concentrate sales among the
test stores could be attributed primarily to variations in the general income
status of the separate store clientele. Under these circumstances, the
assumption of a constant absolute difference in purchase rates among the
stores at any given price would imply that the income elasticity of demand
for concentrate could vary over an extreme range of values. For example,
Aq I
the formula for estimating income elasticity, & q, would become
c* Accordingly, income elasticity would vary inversely with
AIl q
quantity and directly with price along the demand curve.
'It could be argued that the experiment would lead to biased estimates
of the demand parameters if customer choice of stores were affected by
the imposed variation in orange concentrate prices. However, a formal
test led to the acceptance of the hypothesis of zero correlation between
price of concentrate and store traffic (see Appendix II).
Experimental Pricing As an Approach to Demand Analysis 13
Price "age" was adopted as a source of variation because
customers could not reasonably be expected to make a onceand
forall, instantaneous adjustment to the new set of concentrate
prices imposed by the experiment. The result, reflected in cus
tomer purchases, of this delayed reaction to a price change has
been termed "carryover" effect. Given a basic (longrun) re
lationship between price and quantity, price "age," conceivably,
could affect demand by shifting the demand curve, changing the
slope of the curve or both. It was anticipated that the "age"
parameters would provide satisfactory estimates of shifts in the
demand function associated with price "age." By permitting
the fitting of separate demand functions to the various price
"ages," the proposed model also provided for the detection of
possible changes in the slope of the function.
Because the effect of price "age" on purchases would possibly
vary from store to store, the decision was made to introduce store
x age interaction as a possible source of variation.
The reason for considering stores and weeks as sources of
variation is fairly obvious from the discussion of the economic
model. Although storetostore variation in purchases per
customer unit could be isolated by ordinary analysis of variance
techniques, variation between weeks had to be partitioned into
"weeks within age" and "age." 8
Taking weekly purchases 9 in ounces of orange concentrate
per hundred customers as the predictive variable, the analytical
model assumed the following form: 10
Yi kj = + aA + pP + i W tk+ p ijpr
+ PX';iki + P2X2iki+ P3 ikj+ P4Xikj
+ PXiki + 6 ikj + iki
8 See Table 2 of Appendix III for the key to the planned analysis of
covariance.
Although concentrate purchases in each store were recorded on a daily
basis, sales were aggregated into weekly purchases because this measure
would seemingly conform more closely to customer shopping habits. Fur
thermore, from a computational standpoint, the model would have been
extremely unwieldy if daily purchases had been used.
An expression of the model in matrix notation appears in Table 1,
Appendix III.
14 Florida Agricultural Experiment Stations
where
vk = Logarithm of the quantity (in ounces per hundred cus
tomers) of orange concentrate purchased in store i,
week k, age j 11
i= 1...10
k=1...9
j 1, 2, 3
p; = Logarithm of regression constant
= Logarithm of effect of age j
Ai = 1 when r = j, zero otherwise 12
6, = Logarithm of effect of store i
so = 1 when p = i, zero otherwise
'k_i = Logarithm of effect of week k
w, = 1 when t = k, zero otherwise
aI = Logarithm of effect of interaction of store i and age j
Li P = 1 when p = i and r = j, zero otherwise
q = Regression coefficients
= Logarithm of price in age 1, zero otherwise 13
X;k.j Logarithm of price in age 2, zero otherwise
X ,ki_= Logarithm of price in age 3, zero otherwise
"k Random component.
An observation is completely "located" if i and k are given, but j is
carried to identify the "age" of price.
Variables A, S, W and L were introduced to allow each observation
to be expressed in terms of all of the parameters of the model. Writing
the model in this way is helpful in developing the leastsquares normal
equations for estimating purposes.
The weekly purchases of all brands and can sizes combined were
expressed as a function of the price of a 6ounce can of Brand B concentrate.
Experimental Pricing As an Approach to Demand Analysis 15
The following restrictions were imposed:
SM. : 0
"Mi= Number of observations in age j
S6; =0
"I 2 + '2 "51 '7 = 0, + '2+ 2= 0' 143+ 9 = 0
E M ii = 'i = 0
Further it was assumed that Ikj was distributed normally with
zero mean and constant variance.
More briefly, the model can be expressed in terms of loga
rithms by the polynomial form,
Yik = PO + + 6 ki + ;i+ PI IUkij +2Xkj
+ P3Xkj + P4Xkj + P5ki + 6X6iki + 'kj
or, in actual values by the monomial,
Yik,= (lkj) (Aikj) (Xliki likj (X21kV2kj (X3ii_)V3ikj
where
Aikj 0 ai 1 'ki '1
Viki' PI + 4ogXlikj
V2iki = 2+ 0lg X21ki
V31kj= P3 + P610 X31kl
In summary, the Pq's, i.e., regression coefficients of the model,
allow for possible differences in the slopes of the demand function
associated with price "age." The parameters, "i, 6S 'ki_nd A1,,
on the other hand, designate proportional shifts in the demand
function specified by the model.
16 Florida Agricultural Experiment Stations
ANALYTICAL PROCEDURE
The structure of the general model adopted to explain the
demand for concentrate was such that techniques of multiple
covariance could be readily applied as an analytical tool.14 In
fact, the model was constructed with covariance analysis in mind.
Consequent computational requirements of the chosen frame of
reference and statistical procedure involved deriving leastsquares
estimates of the model parameters and determining the "best"
specific model form that would summarize the demand for con
centrate for the given data. Decisions regarding the choice of a
specific model were to be made by testing certain hypotheses
about the model parameters. Composite hypotheses to be tested
were:
(1) P1 =2 =3 = 0, (2) P4 = 5 6 = o
i.e., do the regressions contain only a linear component or both
a linear and quadratic component; and, do the individual age
regressions differ significantly?
(3) ai = 0, 61 = 0, V"k = 0, 'ii 0,
i.e., do the specified "class constants" constitute significant
sources of variation?
ORGANIZATION OF THE DATA
Before the formal analysis could be undertaken, however,
the matter of performing necessary corrections and properly
organizing the data required consideration. Inspection of the
data and a knowledge of events which occurred during the course
of the study led to the rejection of the quantity data from one
store for the first two weeks.15 To obtain estimates of the omit
ted observations and preserve the computational simplicity of
the original model, "missing data" techniques were applied.16
"14 R. L. Anderson and T. A. Bancroft, Statistical Theory in Research
(New York: McGrawHill, 1952), Chapter 21.
W. G. Cochran and G. M. Cox, Experimental Designs, (New York: John
Wiley & Sons, 1950), pp. 75 ff.
A. M. Mood, Introduction to the Theory of Statistics (New York:
McGrawHill, 1950), pp. 350 ff.
M. G. Kendall, The Advanced Theory of Statistics (3rd ed.; London:
Charles Griffin and Company, Ltd., 1948), II, pp. 337 ff.
"1 For a portion of week one, in store nine only, the company's private
brand of concentrate was not available. It seemed reasonable to assume
that the missing brand might affect the data from this store for both
weeks one and two.
16 M. S. Bartlett, Some Examples of Research in Agriculture and Applied
Biology, Supplement to the Journal of the Royal Statistical Society, IV,
No. 2 (1937).
Experimental Pricing As an Approach to Demand Analysis 17
The particular technique utilized to estimate the missing
observations required the introduction of two new variables (and
two regression parameters) into the model. Quantities for store
nine, weeks one and two, were replaced by zeros and variables
defined as follows:
xki_= 1 for store nine, week one, 0 otherwise
xiki= 1 for store nine, week two, 0 otherwise
Upon introduction of these new variables, the general model
for testing and estimation assumed the revised form:
YIkj = 0 + a + 8i + k + i+ 1iikI + P2X2ki + P31iki
+ X4kkl + PXikl + P6X6ik + PTXiki + B8ki+ 'ikj
In organizing the data preparatory to analysis, a certain
amount of difficulty was encountered with the matter of price
"age" classification. To begin with, the market price of 16.5t
per 6ounce can of Brand B actually became effective one week
before the experiment began. From a temporal standpoint, then,
this price should have been identified with "age two." However,
because changes in the pricing unit, displays and availability of
the product were made at the beginning of the experiment, it
was decided to designate the first week of the experiment as a
component of "age one."
Unfortunately, the pricing design was the source of addi
tional difficulty in delineating price "age" categories in that the
pricing arrangement prevented the specification of age classes
that were completely free from ambiguity.17 Despite this di
lemma, the following "age" classifications appeared to be the
most logical: weeks one, two, five and seven were assigned to
"age one"; weeks three, six and eight, to "age two"; and weeks
four and nine, to "age three." is
Aside from the problem of "age" classification, the identifica
tion of other categorical (or class) sources of variation was
"1 Hindsight allows the conclusion that the price effective in weeks one,
five and six should not appear in the pattern for weeks two, three and four,
nor in the pattern for weeks seven, eight and nine.
Apparently, notwithstanding the difficulties cited, this classification
is consistent with the notion that the demand function might possibly have
rotated about the existing market price as purchasers adjusted to the
induced price changes. It is only when demand curves are thought of as
shifting under the impact of price changes that the ambiguity of the "age"
classes becomes obvious.
18 Florida Agricultural Experiment Stations
straightforward. It may be recalled that these categorical vari
ables consisted of stores, weeks (within age) and the interaction
of stores and age. Price was considered as a continuous co
variable, i.e., a logarithmic function containing both linear and
quadratic components was intended to summarize the relationship
between price and quantity.
ALTERNATIVE FORMS OF THE GENERAL MODEL
The general model purposely was constructed broad enough
to accommodate several visualized sets of possible demand cir
cumstances. Below are some of the adaptations that might ap
propriately describe the demand relationship in question and, at
the same time, include the parameters and variables required to
estimate the missing observations. For instance, if
P1 = P2 P3 and P4 = P5 = 6 ='
the model would reduce to
Y'ki = ikj+ P7X7k + PXBk + 9X9ik + 'kj (A)
where Pp PI = 2 =
In the event that P 2 0 P3 but 4 = 5 = P6 = 0, it would become
Yki = Ikj likj+ P2X k + Pa kj (B)
+ P$X7ki+ PB iki + 'Iki
Should p1 = P2 = P3 and 4 = P5 = 6. the model could be written
Y;kj = Akj + 7X71ki + BX8ki
(C)
+ PX91ikj + P1B lki + e;kj
where P9= P1= 02= P3 and Pl0 = P4 = 5 = P6
Or if P0 / 22, P3 and If P4 P5 / P6, the model would revert to the original general
form,
Ylk = Ak + PIXlkj + P2X'2kj+ P3X3kj + 4X4k (D)
+ PB5Xilkj + P6XAik + P7X71kj + P8X81k + 'kij
It should be noted that in all of the model forms,
"Aki = PO + ai + 6 + k + A;l. Furthermore, it should be apparent
that these expressions are all particular forms of the general
model. Other forms of the model could be written by systemati
cally omitting in turn each of the categorical variables.
Experimental Pricing As an Approach to Demand Analysis 19
EXAMINATION OF HYPOTHESES FOR SELECTION OF
SPECIFIC MODEL
As previously implied, the choice of a specific model to sum
marize the experimental demand relationships for orange con
centrate was to be decided by the application of formal statistical
tests to certain adaptations of the general model. It will develop
that, from these tests, conclusions could be drawn respecting the
particular form, i.e., degree of the logarithmic function relating
quantity and price. Moreover, reasonably justifiable decisions
could be made concerning the retention or omission of certain
class constants. Procedural stages leading to the development
of the final model are dealt with subsequently.
Linear Component.Attention initially was directed toward
choosing a particular model form. A procedure of first testing
the least complex model and then the progressively more com
plex alternatives, outlined in the preceding section, appeared to
constitute a logical approach to the problem.
Considering the linear form (A) and allowing the age,
store, weeks and interaction constants to take on any (least
squares estimated) finite value, the relevant question to be
answered related to the existence of a linear component. Making
use of statistical notation, the hypothesis to be tested was
H1 : 09= 0 against Ql: P9,/ 0.
The method used to choose between the two hypotheses
amounted to testing the reduction in error sum of squares due
to linear regression. Consequently, the testing of H1, given
form (A), required the computation of two new error sums of
squares. These consisted of the remainder after fitting X7 and
Xs and the remainder after fitting X7, Xs and X'9.
Solution of the following matrix equations was required to
obtain the sum of squares (SSRs) for the regression on
X7 and Xs:19
[ 2 1x
[b be] x7 8I = [yx7 lyx8]
S0.675000 0.225000
[ J b7 0.225000 0.675000 .879816 0.932323 ,
The necessary sums of squares and sums of cross products were ob
tained from line E, Table 2 of Appendix III.
20 Florida Agricultural Experiment Stations
from which
b7 = 1.984317
bs = 2.042658
The additional reduction in sum of squares due to adjusted re
gression on X7 and Xs was given by
ASSR8 = [b7 68] [Eyx7 Eyx8]
ASSR7,8 = 3.650251 .
Sum of squares for regression (SSR7ys), when X7, Xs and X'9
were used as regression variates, was obtained by solving the
following matrix equations: 20
E x7 1 x7x8 Zx7x
1X7X9 1X X8X9
[''7 ^s ^l ^a7x 82 8 9 1 yx7 yx8 ryx9
I x7x lx8x9 TEx
ASSR7,8,9 b7 b8 bs ] [yx7 Lyx8 Eyx9 ]
The solution gave
b7 = 2.071092
bs = 2.305635
bg = 0.823312
ASSR7,8,9 = 4.160755
Table 3 illustrates the significant reduction in error sum of
squares associated with the regression of adjusted quantity on
adjusted price. That is to say, after adjustment for the vari
ation associated with the categorical variables, there existed a
significant regression of quantity on price.
TABLE 3.TEST OF REDUCTION IN ERROR SUM OF SQUARES DUE TO LINEAR
REGRESSION OF ADJUSTED LOGARITHM OF QUANTITY ON ADJUSTED LOGA
RITHM OF PRICE.
Source D.F. SS MS F
Remainder (1) ............... 54 4.315089
Reduction due to
fitting X, and Xs ..... 2 3.650251
Additional reduction due
to fitting X'9 ............... 1 0.510504 0.510504 168.71
Remainder (2) .............. 51 0.154334 0.003026
" Sums of squares and sums of cross products were taken from the
remainder line of Table 2, Appendix III.
Experimental Pricing As an Approach to Demand Analysis 21
Quadratic Component.The large reduction in the error sum
of squares, accounted for by the linear component, suggested the
possibility of extending the model to form (C). That is, the
rejection of H1 directed attention to the tentative inclusion of a
quadratic component in the model. Upon the postulation of
form (C), the hypothesis to be tested became H2: B0 = 0 against
"2 : 010 0.
The regression coefficients and sum of squares for regression
resulting from the introduction of the quadratic component of
price into the model were obtained by solving
Erx Zy7x8 y7x9 Ey7XI0
Sy7x, xgx, Exx, 2xOxl
x7xIO xx9O zX9XIO 8xlO
txixi0 x8xo0 Ex9x, o xo J
ASSR7,8,9,10 [b7 8 b bp "ld EYx7 yx8 E Yx9 'YO]
The solution gave
b7 = 2.050910
bs = 2.341879
b9 = 6.261668
blo = 2.409690
ASSR7,s,9,1o = 4.207578.
The F ratio in Table 4 indicates the significant reduction in
the remainder term when X'io was included in the model. The
outcome of the tests of H1 and H2 led to the decision that form
(C) was preferable to form (A) as a model for estimation. That
is, after price and quantity were adjusted for variation in the
categorical variables, a quadratic appeared more satisfactory
than a linear regression of the logarithm of quantity on the
logarithm of price. Stated in another way, the addition of
Xkiand Xj0iki to the model, along with associated parameters,
significantly improved the fit of the model to the data.
Distinctness of Age Regression.The choice of form (C)
over form (A) precluded the necessity of testing the adequacy
of form (B) because of the linear property of this model. How
ever, the particular form of the model was yet undecided, be
cause the distinctness of age regression in relation to the quad
ratic form remained to be examined. Form (D), of which form
22 Florida Agricultural Experiment Stations
(C) is a special case, provides a representation of a quadratic
model which would allow the regression parameters to vary with
price "age." In contrast, it will be noticed that form (C) as
sumed the regression of quantity on price to be independent of
price "age."
TABLE 4.TEST OF H : /10 = 0.
Source D.F. SS MS F
Remainder (1) .............. 54 4.315089
Reduction due to
fitting X, and Xs ........ 2 3.650251
Additional reduction due
to fitting X', ........... 1 0.510504
Additional reduction due
to fitting X'o .............. 1 0.046823 0.046823 21.778
Remainder (3) ............... 50 0.107511 0.002150
To determine if a distinct regression were associated with
each price "age," the hypothesis
H3 : P = 2 = P3
P4= P5= P6
was tested against
"3 : < iki' P1' P2' P3' P4' P5' 6' P7' PS<8
As with the preceding tests, the procedure involved a statisti
cal appraisal of the additional reduction in the remainder sum
of squares resulting from the fitting of form (D) rather than
form (C).
The sum of squares for regression (SSR1,2,3,4,5,6,7,8) for form
(D) was obtained by solving the matrix equations 21
[B]' [G] = 1,2,3,4,5,6,7,8
[B]' [. =[G]
[B]' bl . b8]
xl2 ......... IXlX8
A =
Zxlx8 ........ Zx2
[G] '=[yx .. yx8]
"The elements of [A] and [G] were obtained from line E, Table 2,
Appendix III.
Experimental Pricing As an Approach to Demand Analysis 23
The solution gave
bl = 4.792780
b2 = 5.821589
b3 = 9.040260
b4 = 1.728837
b5 = 2.202856
b6 = 3.682643
b7 = 2.063749
b, = 2.353265
ASSR1,2,3,4,5,6,7,8 = 4.216103.
The outcome of the test of Table 5 led to the decision that
the parameters of the quadratic function relating the logarithm
of the quantity to the logarithm of the price were not affected
by the "age" of the price. Therefore, form (C) of the model
was tentatively chosen.
TABLE 5.TEST OF H3.
Source D.F. SS MS F
Remainder (1) .............. 54 4.315089
Reduction due to fitting
X,, Xs, X'., and X'lo .... 4 4.207578
Additional reduction due I
to fitting X'1, X'2, X'3,
X'4, X'5, X'0, X,, X... 4 0.008525 .002131 <1
Remainder (4) .............. 46 0.098986 .002152
Significance of Class Constants.Although the acceptance of
form (C) settled the problem relating to the choice of a particu
lar model form (i.e., function of a given degree), the prior as
sumption that an appropriate model would contain all of the
previously specified class constants remained open to justifica
tion. Hence, at this juncture, there arose the need to test
separately each of a series of hypotheses,22
H4 : = 0
"H5 : = 0
"H6 :k =0
"H7 il = 0
"2 More precisely, the test is for equality among the 68 s (likewise forai's
" *s and i's).
24 Florida Agricultural Experiment Stations
against
"4 =  < p 81, ai' kj' Aij, Pq i . 8 <"
Individual tests of hypotheses H4 through H7 were performed
by fitting a quadratic regression model of form (C), from which
was omitted, in turn, the set of class constants relevant to the
hypothesis being tested. Then the remainder sum of squares
in each case was compared with the remainder sum of squares
from fitting the original model of form (C), i.e., the model which
included all sets of specified class constants.
A brief explanation of steps involved in testing the signifi
cance of store constants (H4 : 6, = ) is used to demonstrate the
general procedure. To obtain the remainder sum of squares for
model (C) with the 6; omitted, the zy2 [remainder (1) +
stores]23 was adjusted for the regression of Yikj on xw9k "ad XlOkj.
This required sum of squares of regression was supplied through
the solution of the matrix equations
S7,8,9,10 = [bi7 b8 b9 bl] [ y Tlyx. yx9 Lyxi0 ]
Ex72 1XXg 1KyX tx
7X7 X9 rxgx, rx9 Zx9x10
The solution gave
b, = 2.065742
b, = 2.304440
bs = 4.413188
blo = 1.662648
ASSR7,s,9,1o = 5.992420
Subtraction of ASSR7,8,9,10 from the 2y2, denoted by [Remain
der (1) + Stores], gave the desired remainder sum of squares
[Remainder (5)] for the special form of model (C), from which
the store constants were omitted. If the remainder sum of
squares from fitting the original model of form (C) were identi
fied as Remainder (3), obviously, the additional reduction in
sum of squares due to fitting store constants would be the
Line J of Table 2, Appendix III contains the sums of squares and cross
products for all the variables of the model "adjusted" for class variables
other than the 1 "s.
Experimental Pricing As an Approach to Demand Analysis 25
difference between Remainder (5) and Remainder (3). The me
chanics of the test are given in Table 6.
TABLE 6. Testof H4 : 6 = 0.
Source D.F. SS MS F
Remainder (5)
(omitting 6: )  59 0.927910
Remainder (3)
(including s) .......... 50 0.107511 0.002150
Additional reduction due
to fitting store
constants ............... 9 0.820399 0.091155 42.398
Similar tests for each set of class constants are presented
in Tables 7 through 9.
An examination of F ratios beginning with Table 6 indicates
that the inclusion of stores, weeks and ages in each case resulted
in a significant reduction in the remainder term. On the other
hand, as shown by Table 9, the additional reduction from fitting
store x age constants was nonsignificant.
Since the store x age constants did not materially contribute
to an explanation of the variation in concentrate purchases,
this set of constants was removed from the model. Adoption of
the revised model [form (C) with x i omitted] necessitated the
computation of new estimates of the regression coefficients.24
"2 From the standpoint of formal completeness, the exclusion of the inter
action constants called for a reexamination of the model form. In
particular, it became necessary to reopen the question concerning the
distinctness of the separate age regressions, i.e., to retest the hypothesis,
I = P2 = 3, P4 = P5 = P6. This amounted to testing the appropriateness
of a modified version of form (D), consisting of form (D) with the A;',
omitted.
Modified form (D) was rejected as a satisfactory model on the basis
of the outcome of the following test:
Factor D.F. Sum Squares Mean Sq. F
Remainder (1)
+ Store X Age
(Line G, Table 2, App. III) .. 72 6.555333
Additional reduction
due to modified (C) ............... 4 6.406250
Additional reduction
due to modified (D) ............... 4 0.007919 0.001980 <1
Remainder (9) .......................... 64 0.141162 0.002206
26 Florida Agricultural Experiment Stations
TABLE 7. Testof H5 : a = 0.
Source D.F. SS MS F
Remainder (6)
(omitting ).............. 52 0.133631
Remainder (3)
(including a) ............... 50 0.107511 0.002150
Additional reduction due
to fitting age
constants .................. 2 0.026120 0.013060 6.074
TABLE 8. Test of H6 : = o0.
Source D.F. SS MS F
Remainder (7)
(omitting ) .... 56 0.144622
Remainder (3)
(including w ) . 50 0.107511 0.002150
Additional reduction due
to fitting week
constants ....................... 6 0.037111 0.006185 2.877
TABLE 9. Test of H7 : = 0.
Source D.F. SS MS F
Remainder (8)
(omitting ) .. 68 0.149083
Remainder (3)
(including A;i) ............... 50 0.107511 0.002150
Additional reduction due
to fitting store
x age constants .............. 18 0.041572 0.002310 1.074
Experimental Pricing As an Approach to Demand Analysis 27
These estimates turned out to be
b7 = 2.009675
bs = 2.288554
bg = 5.584685
blo = 2.120746
ASSR7,8,9,10 = 6.406250
The Determinate Model.From outcomes of the preceding
tests, modified form (C) was inferred to be the best fitting
model.25 Accordingly, the final version of the model could be
appropriately written
Yiki = 0 + Si + i + 'kj + 2.00 2.288554XODkj
5.584685 Xi_ + 2.120746X;O1kj + .;kj.
Estimates of the regression constant (p) and of the class
constants (s6i, .is and .ji's), designated below, supplied the finish
ing touches to the specific model used to summarize the demand
for orange concentrate for the given experimental data.26
57 A, ^A
0 = 5.71766 1_ = 0.01042 a = 0.01740
As ^ A,
81 = 0.02494 21 = 0.02874 a2 = 0.02513
Ai Ai A,
2 = 0.04973 32 = 0.01451 a3 = 0.00292
3 = 0.21705 = 0.02505
64 = 0.15449 51 = 0.03158
5 = 0.12897 62 = 0.02374
6 = 0.01189 71 = 0.00757
A, Ag
67 = 0.01709 w82 = 0.00924
68 = 0.01502 e_ = 0.02502
69 = 0.02444
610 = 0.04662
"2 The reduction in sum of squares due to fitting modified form (C) was
10.368225, representing a proportional reduction in variance of 0.986. How
ever, the method of estimating the missing observations greatly increased
28 Florida Agricultural Experiment Stations
As a precautionary measure, a threeway check was performed
to establish the accuracy of computations leading to estimates
of the model parameters. Estimated values and residuals were
computed for the 90 observations of concentrate purchases.27
The following algebraic identities were then used to verify com
putations:
f (Yik jIki) = REMAINDER (8) = ik j (TOT)SSR,
where
(TOT)SSR = 4o Yiki [;Y;kij i+ k ;IY.
u kk 'i F l
+ I [ Yiki] + r[ bp : Ykipki"
Independent computations of these three sums of squares yielded
SY(Yiki kj)2 = 0.149
REMAINDER (8) =0.149
,2
f ZYik (TOT)SSR= 0.153
The close agreement among the three sums strongly sug
gested that calculations performed in the analysis were free of
important computational errors.
ECONOMIC IMPLICATIONS OF THE ESTIMATING MODEL
Although interesting from the standpoint of methodology,
the study was primarily motivated by the need for information
essential to a clearer insight into economic problems of citrus
marketing. As a consequence, final attention was directed toward
translating the statistical results into economic concepts. How
ever, because of the limited scope of the study, no claim is made
that the results have general application to "real world" problems.
the proportional reduction in variation due to regression. A more realistic as
sessment of the fit of the model might be given by the following computation.
variation among original 88 observations = 1.668241
error sum of squares from model = 0.149083
reduction in sum of squares due to model = 1.519158
proportional reduction = 0.911
" Estimates of these parameters were obtained by resorting to the nor
mal equations of Appendix IV and the restrictions of Table 1, Appendix III.
"2 It should be noted that the residuals for the observations with missing
Y's were zero in each case.
Experimental Pricing As an Approach to Demand Analysis 29
THE GENERAL DEMAND FUNCTION
Reference to the final form of the statistical model shows
that the estimated general demand function evolving from the
analysis was given by:
Y' = 5.71766 5.584685X', + 2.120746X'io.
Since price elasticity of demand for this function is expressed by
Np = 5.584685 + 4.241492X%9,28 estimated demand elasticity
was clearly a function of price.
It may further be observed that, beginning with the lowest
test price, demand elasticity decreased as price increased over
the entire range of prices tested. Moving from lower to higher
prices, demand changed from an elastic to an inelastic relation
ship at the pivotal price of about 12.050 per 6ounce can of con
centrate. That is, the demand function was found to have uni
tary elasticity at an estimated price of about 12.05.29 Estimated
demand elasticity at the various test prices obviously would be
either elastic or inelastic depending upon whether a particular
test price was higher or lower than the estimated price associated
with unitary elasticity.
A demand relationship of the foregoing nature is character
ized by a revenue function convex to the origin with minimum
revenue occurring at the price corresponding to the point of
unitary elasticity on the demand curve.30
If Y = 5.71766 5.584685X + 2.120746X'0,
then dY Y 5.584685 + 4.241492X'
dY y[ 5.584685 + 4.241492X9
dX X
From the definition of price elasticity of demand, Np = X. ,the
SdX Y
equation for estimating demand elasticity becomes
N [5.584685 + 4.241492X X, or Np = 5.584685 + 4.241492X9.
:X Y
"2The slight discrepancy between the estimated price associated with
unitary demand elasticity as given in Florida Experiment Station Bulletin
589 and the estimate appearing in this report arises because in the former the
iis were included in the model.
If the demand function in terms of logarithms is given by
Y' = 5.71766 5.584685X's + 2.120746X'1o, then the revenue function
expressed in logarithms would be equivalent to
R' = 5.71766 4.584685X', + 2.120746X'io.
Because R' is a monotonic function of R (total revenue), maximum or
30 Florida Agricultural Experiment Stations
If such a demand relationship could be considered as having
marketwide applicability, important implications might readily
be suggested regarding an organized marketing program for the
citrus industry.31 Existence of the postulated demand situation
would imply that, as far as gross revenue from concentrate is
concerned, a policy of stabilizing sales over time would have a
tendency to reduce total returns to the citrus industry.32 How
ever, since single strength juice and fresh citrus are also im
portant sources of industry revenue, an evaluation of the annual
reallocative potential of the citrus crop among the three outlets
would be necessary before a positive statement could be made
concerning how a marketing policy for concentrate alone might
affect total gross revenue. With the present lack of knowledge
of production costs and costs incurred in storing concentrate for
extended periods, any assertion regarding expected net revenue
to the industry would possess far more speculative content than
the foregoing discussion of gross revenue.
DEMAND SHIFTING PARAMETERS
From an economic point of view, the categorical variables of
the statistical model may be regarded as demand shifters. Hence,
for this model, the effect of each store, week and age was mani
fested by individual upward or downward proportional shifts of
the general demand function.
Store Effect.From general impressions formed about the
communities in which the various stores were located and the
patronage of each store, the estimated store parameters (60')
would appear to largely reflect the composite effect of income
and preference differences among consumer groups associated
with the several stores. While the size of the normal display
minimum revenue would be defined by the maximum or minimum of the
logarithmic revenue function.
For dR'= 4.584685 + 4.241492X'9 = 0, X', = 4.584685 = 1.0809133, or
dX', 4.241492
the estimate of the logarithm of the price at which revenue would be mini
mum. Substituting in the equation for demand elasticity, it is seen that
NP = 5.584685 + 4.241492 (1.0809133), or NP = 5.584685 + 4.584685 = 1.
Hence, minimum revenue would occur at the point of unitary elasticity on
the demand curve.
"1 The term, citrus industry, is used in a broad sense to include both
processors and primary producers.
"32 This statement was cautiously phrased, deliberately, because the
truth of the contention would hinge upon the validity of the assumption
that the derived gross revenue function facing the citrus industry, after
accounting for transportation, wholesaling, retailing and other marketing
costs, would have characteristics similar to the revenue function at the
retail level.
Experimental Pricing As an Approach to Demand Analysis 31
of concentrate in each store may have exerted some influence
upon sales differences, intuitively, this effect would seem negli
gible. Presumably, differences in store traffic did not contribute
to the store effect, becauseit will be recalledthe analysis was
performed using per customer unit data.
In reference to the general demand function, the store effects
(6,,), which are expressed in terms of logarithms, may be easily
translated into percentage shifts in demand.33 The positive and
negative parameter estimates, of course, are to be interpreted
as upward and downward shifts in demand, respectively.
Week Effect.The week effects (kI,.,s) designate proportional
shifts in demand attributable to differing demand conditions
among weeks within a particular price "age." As implied by
the signs of the week parameters, for a given price "age," condi
tions prevailing in certain weeks served to shift demand upward
in some instances and downward in others. Naturally, both
positive and negative shifts within each age were to be expected,
because of the restriction imposed upon the analytical model
that the ^kj' (as well as the other categorical variables) sum
to zero.
Effect of Price "Age".It will be remembered from the sta
tistical analysis that the age of price had no significant effect
upon the slope of the demand function. However, price "age"
presumably did effect shifts in demand as reflected by the .
Since there were perceptible, though not significant, differences
in the slopes of demand functions fitted to the separate price
"ages" [see form (D)], perhaps a more sensitive model would
have yielded a demand curve for each price "age"a result
apparently more in keeping with economic intuition.
Because slope differences in the demand function were not
manifested by the analysis, the .,s constituted the sole measure
of the "carryover effect" of a change in price.34 Measures of
Percentage shifts in the general demand function may be determined
readily by use of the following simple formulae:
For positive 6j's: antilog(6 + 2) 100 = percentage upward shift in de
mand,
For negative si's: 100 antilog ( 6 + 2) = percentage downward shift in
demand.
3 Technically, there would seem to be no difference between the two
concepts, age of a price change and age of a price, since one implies the
other.
32 Florida Agricultural Experiment Stations
this influence upon the general demand function can be expressed
in percentage terms by resorting to the formulae of footnote 33.
Perhaps a more meaningful interpretation of the impact of
"carryover effect" could be achieved by relating demand changes
associated with price "age," say, to the first price "age." For
instance, weekly purchases of concentrate for the second week
following a price change were about 10.3 percent larger than for
newly established prices one week old. But for prices three
weeks old, purchases were only about 3.4 percent larger than
for prices one week of age. This phenomenon suggests that
consumers reacted to a new price situation by first underadjust
ing their purchases, then overadjusting, but finally settling to
ward some equilibrium, intermediate, purchase rate.
Experimental Pricing As an Approach to Demand Analysis 33
APPENDIX I
BASIC DATA
TABLE 1.QUANTITIES OF FROZEN ORANGE CONCENTRATE PURCHASED PER
100 CUSTOMERS, BY STORES AND WEEKS.
Store Week Beginning
Num June June I June June July ] July July July I Aug.
ber 7 14 I 21 28 5 12 19 26 1 2
1 114.4 88.6 96.2 94.2 118.6 104.1 214.0 260.2 230.0
2 89.5 124.6 152.9 162.6 139.4 147.9 121.5 144.3 145.3
3 61.3 88.8 96.4 107.0 75.5 81.9 63.7 78.9 71.6
4 177.5 151.3 204.1 199.4 162.9 175.1 133.2 163.3 163.1
5 162.4 144.9 156.5 178.8 162.8 164.3 185.5 157.3 146.4
6 113.8 179.2 198.4 172.6 104.0 127.6 238.9 259.4 194.8
7 110.0 104.6 110.3 126.5 121.7 121.1 110.8 123.2 97.8
8 98.2 97.7 121.9 126.7 111.6 136.9 142.4 169.0 150.9
9 99.1 135.7 222.9 198.0 125.6 136.0 108.0 94.0 96.5
10 96.4 182.5 191.5 210.0 98.2 122.8 193.3 150.7 133.9
TABLE 2.INDIVIDUAL CUSTOMER COUNTS, BY STORES AND WEEKS.
Store _Week Beginning
Num June June June June July July I July July Aug.
ber 7 14 21 28 5 12 19 26 2
1 3,904 3,907 3,917 4,026 3,284 3,888 3,920 3,558 3,590
2 6,423 6,797 6,077 6,631 5,865 6,488 7,917 6,770 6,145
3 6,446 6,975 6,884 7,780 5,914 6,857 7,241 7,196 7,062
4 7,104 8,215 6,820 8,016 6,550 7,482 8,305 7,579 6,878
5 6,032 6,826 6,433 6,586 5,464 6,174 5,882 4,791 5,320
6 4,058 4,566 4,197 4,857 3,899 4,343 4,912 4,971 4,898
7 5,561 5,576 5,490 5,511 4,767 4,976 5,598 5,421 4,999
8 5,780 6,158 5,799 5,874 5,414 5,732 6,317 5,638 5,581
9 5,443 5,150 4,414 5,051 3,686 4,500 4,505 4,787 4,141
10 4,187 4,047 4,092 4,214 3,597 4,162 4,191 4,022 3,966
34 Florida Agricultural Experiment Stations
APPENDIX II
EFFECT OF EXPERIMENTAL PRICES ON STORE TRAFFIC
Since purchases of orange concentrate for a particular period
in a certain store undoubtedly would be affected by store traffic
as well as price, it could be argued that the model used to esti
mate sales should contain traffic as a variable. However, the
validity of the argument would appear to rest upon the exist
ence of a dependency relationship between store traffic and price.
If store traffic were independent of prices imposed by the experi
ment, apparently, satisfactory adjustment for the variability
in purchases associated with store traffic could be effected by
reducing sales to some per customer unit basis. That is, store
traffic could be considered as a scalar rather than a variable in
the model used to explain total concentrate sales in a given store.
The problem of ascertaining whether or not store traffic was
affected by the pricing experiment was approached by assuming
that such relationship would be manifested in a logarithmic
regression of traffic on price. It was inferred from this assump
tion that the existence of a functional relationship would hinge
upon the significance of the regression coefficient.
For the proposed analysis, variables were defined as follows:
z ki_ = Logarithm of customer count, store i, week k, age j.
Xi_i = Logarithm of the price of a 6ounce can of Brand B
orange concentrate, store i, week k, age j.
An analysis of covariance was made and the reduction in sum
of squares due to regression of adjusted Z;k.j on adjusted xb9k
was tested. It was concluded that there was no significant re
gression of traffic on price and that age of price was not sig
nificant.
Experimental Pricing As an Approach to Demand Analysis 35
TABLE 1.BASIC TABLE FOR ANALYSIS OF COVARIANCE.
Factor I D.F. SSZ' SSX'9 SCPZ'X'g
Stores .............. ........... ...... 9 0.817668 0.272253 0.294918
Age .... ........................... 2 0.000740 0.030366 0.003292
Weeks (within age) .............. 6 0.054196 0.151832 0.048523
Stores x Age ............ ... 18 0.008781 0.022688 0.007932
Remainder (1) ............. ......... 54 0.041282 0.814347 0.012155
Total ...................... ....... .. 89 0.922667 1.291486 0.247326
Remainder (1) + Age ......... 56 0.042022 0.844713 0.008863
TABLE 2.TEST OF REDUCTION IN SUM OF SQUARES DUE TO FITTING X',.
II
Factor D.F. SS MS F
Reduction due to fitting X'9 ...... 1 0.000181 0.000181 <1
Remainder (2) ............... 53 0.041101 0.000775
TABLE 3.TEST OF REDUCTION IN SUM OF SQUARES DUE TO FITTING
AGE CONSTANTS.
II I I
Factor D.F. SS MS F
Adjusted (Remainder + Age).. 55 0.041929
Remainder (2) ...... ............53 0.041101 0.000775
Adjusted Age ................. 2 0.000828 0.000414 <1
36 Florida Agricultural Experiment Stations
APPENDIX III
TABLE I
MATRIX REPRESENTATION OF THE GENERAL ANALYTICAL MODEL, A;i s OMITTED
Y1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
Y'21 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0
11000000.0000010000 001000
Y'32 1 0 0 0 00 0. 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0
Y'43 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0
Y'193 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
Y211 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
Y293 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0
Y9.93 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0
Y'0921 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 01
.... . . . . . . . . . .
iS99 = 0 0 + 0\ +5 + 0 =
ErMiai = 0 + 6 +'8 = 0
M = number of weeks in age j lT + T = 0
Experimental Pricing As an Approach to Demand Analysis 37
TABLE 1: Continued
X11_1 0 0 (Xe 11 )2 0 .
X021 O 0 (X'21)2 0 0 6 e121
0 X132 0 0 (Xi32)2 0 62 '132
o o x44.3. o 0 (X143J2 63 "143
.64
65
0 0 X'193 0 0 (Xi932 6g8 193
X'211 0 0 (X2112 0 0 4 '21
510
'i
0 0 X293 O 0 (X293_2 '3 '293
4
15 +
'6
X'gll O 0 (X_911J2 0 0 9 911
921 0 0 (X21 2 0 0 a1 e921
a2
P7
X'O, l 0 0 (X o,1 1)2 0 0 p '10,1)
P2
P3
P4
*PS
0 0 X0,93 0 0 (X;0o,93 _2 P6 '10,93
Assuption
edk is distributed normally and independently
with mean zero and constant variance.
38 Florida Agricultural Experiment Stations
TABLE 2.BASIC TABLE OF SUMS OF SQUARES AND SUMS OF CROSSPRODUCTS
FOR ANALYSIS OF COVARIANCE.
Line Factor D.F. SSX, SSX_ SSX3
A Stores .................. 9 0.03025029 0.03025029 0.03025029
B Age .................... 2 30.40736531 26.62736568 19.58382684
C Weeks (within
age) ........... 6 0.09109939 0.06073293 0
D Stores x Age .... 18 0.03781286 0.06050057 0.10587600
E Remainder (1) 54 0.30169920 0.27901149 0.23363607
F Total .................. 89 30.86822705 27.05786096 19.95358920
G Remainder (1) +
Stores x Age .. 72
H Remainder (1) +
Weeks (within
age) ............. .... 60 .. .............
I Remainder (1) +
Age ....... ......... 56 ... ..... ... ....
J Remainder (1) +
Stores ............. 63 ............. .. .....
TABLE 2.(Continued).
Line SSXSSX SSX, SSX SSX8
A 0.14681368 0.14681369 0.14681369 0.10000000 0.10000000
B 42.31110713 36.21924637 25.38477671 0.01388889 0.01388889
C 0.41947136 0.27964758 0 0.07500000 0.07500000
D 0.18351711 0.29362736 0.51384791 0.12500000 0.12500000
E 1.52784999 1.41773973 1.19751918 0.67500000 0.67500000
F 44.58875927 38.35707473 27.24295749 0.98888889 0.98888889
G ............. .............. .............. 0.80000000 0.80000000
H ....... ... .............. ................ 0.75000000 0.75000000
I ........ ............... 0.68888889 0.68888889
J ............... ................ ............. 0.77500000 0.77500000
Experimental Pricing As an Approach to Demand Analysis 39
TABLE 2.(Continued).
Line SSX9 SSX1o SSY SCPYXI SCPYX2
A 0.27225258 1.32132317 2.60524086 0.02481332 0.02481332
B 0.03036646 0.13982379 0.63034404 4.37703574 2.52305311
C 0.15183232 0.69911894 0.72639035 0.06070460 0.04015063
D 0.02268766 0.11011027 2.24024367 0.01432078 0.00657723
E 0.81434682 4.14310890 4.31508922 0.17657480 0.24025047
F 1.29148584 6.41348507 10.51730814 4.27165808 2.21126146
G 0.83703448 4.25321917 6.55533289 ....... .......
H 0.96617914 4.84222784 5.04147957 ......... .....
I 0.84471328 4.28293269 4.94543326 .......... ..
J 1.08659940 5.46443207 6.92033008 ......... ..
TABLE 2.(Continued).
Line ISCPYX3 SCPYX, SCPYX5 SCPYX6 SCPYX,
A 0.02481332 0.06917101 0.06917101 0.06917101 0.42791100
B 1.74497980 5.16318743 2.94260518 1.98668172 0.09354583
C 0 0.13026121 0.08615605 0 0.15328650
D 0.01877574 0.01128172 0.00177437 0.02522499 0.52539500
E 0.16585900 0.39706363 0.53458826 0.36772983 0.87981600
F 1.53553174 4.95427430 2.25446423 1.52455589 2.07995433
G .......... ................ ............. ................ 1.40521100
H ................ ........... .. ............... ..... ...... 1.03310250
I ............... ................. 0 6................ 0.97336183
J ............... ....... ..... ............... ............... 1.30772700
40 Florida Agricultural Experiment Stations
TABLE 2.(Continued).
Line SCPYXs SCPYX9 SCPYXIo SCPXiX2 SCPX1XS
A 0.42791100 0.07443997 0.20751303 0.03025029 0.03025029
B 0.09354583 0.10900284 0.23390054 17.99631112 11.66673000
C 0.10077950 0.10085523 0.21641725 0 0
D 0.52539500 0.01103217 0.01216890 0.03025029 0.03025029
E 0.93232300 0.22953467 0.50525447 0 0
F 2.07995433 0.52486488 1.17525419 17.99631112 11.66673000
G 1.45771800 0.24056684 0.51742337 .............. ................
H 1.03310250 0.33038990 0.72167172 ................ ................
I 1.02586883 0.33853751 0.73915501 ............... ............
J 1.36023400 0.30397464 0.71276750 ................ .........
TABLE 2.(Continued).
Line SCPX1X, SCPX1Xs SCPXiX. SCPX1X, SCPXIXs
A 0.06652973 0.06652973 0.06652974 0.00032200 0.00032200
B 35.86877877 20.98887187 13.28272071 0.64986500 0.64986500
C 0.19548296 0 0 0.04772300 0.04772300
D 0.08316220 0.06652973 0.06652974 0.00040250 0.00040250
E 0.67819056 0 0 0.00072450 0.19188950
F 36.89214422 20.98887187 13.28272071 0.69758800 0.40952800
G
G  ...................... . .. ... .... ......... ... ...... ......
H .. ... ..  ...... ... ... ....... ....................
Experimental Pricing As an Approach to Demand Analysis 41
TABLE 2.(Continued).
Line SCPX2X3 SCPXa2 SCPX2X5 SCPX2XO SCPX2X7
A 0.03025029 0.06652974 0.06652974 0.06652974 0.00032200
B 8.63105455 21.22859696 31.05516249 9.82656554 0.38461644
C 0 0 0.13032197 0 0
D 0.03025029 0.06652974 0.13305949 0.06652974 0.00032200
E 0 0 0.62829326 0 0
F 8.63105455 21.22859696 32.01336695 9.82656554 0.38461644
G ......  ........ .... ................ ........... ......... .. .
H ............... ................ ..I....... ..... ................ ............
I .. .......... ... .... .... .. ................ ........... ................
TABLE 2.(Continued).
Line SCPX2Xs SCPXaX, SCPXaX5 SCPX3X. SCPXJX.
A 0.00032200 0.06652974 0.06652974 0.06652974 0.00032200
B 0.38461644 13.76217089 10.06629063 22.29643629 0.24934089
C 0 0 0 0 0
D 0.00032200 0.06652974 0.06652974 0.23285410 0.00032200
E 0 0 0 0.52849866 0
F 0.38461644 13.76217089 10.06629063 23.12431879 0.24934089
J I
G I  ................................. ................ ................
S ...... .. .. ....... ... .. ............ ................ ................
S........ ...... ... .... .. ................ ......... ........ ...........
42 Florida Agricultural Experiment Stations
TABLE 2.(Continued).
Line SCPXaXs SCPXX5 SCPX4X. SCPX5sX SCPXiX
A 0.00032200 0.14681369 0.14681368 0.14681368 0.00329000
B 0.24934089 24.75864627 15.66840684 11.46060009 0.76658611
C 0 0 0 0 0.10240500
D 0.00032200 0.14681369 0.14681368 0.14681368 0.00411250
E 0 0 0 0 0.00740250
F 0.24934089 24.75864627 15.66840684 11.46060009 0.86899111
G .............. .... ...... ..... ................ ................ ...............
H ....... ..... ...... .......... ................ ................ ................
I ................ ................ ................ ................ ................
J   ................ ................
TABLE 2.(Continued).
Line SCPXX5 SCPXXO SCPX7Xs SCPXXio SCPXsXi
A 0.00329000 0.00329000 0.10000000 0.00987000 0.00329000
B 0.44857333 0.28387778 0.01388889 0.03413500 0.76658611
C 0 0 0.02500000 0.10240500 0.10240500
D 0.00329000 0.00329000 0.12500000 0.00246750 0.00411250
E 0 0 0.22500000 0.00740250 0.42103250
F 0.44857333 0.28387778 0.01111111 0.13654000 0.25055111
G ....... ..... ................ 0.10000000 ............. ..........
H ................ ............... 0.25000000 ..............
I ......... ... ................ 0.21111111 ............... .........
J ................ ................ 0.12500000 ... .........
Experimental Pricing As an Approach to Demand Analysis 43
TABLE 2.(Continued).
Line SCPXsX5 SCPXX. SCPXS.XX SCPXsX7 SCPXeX
A 0.00329000 0.00329000 0.00987000 0.00096600 0.00096600
B 0.44857333 0.28387778 0.03413500 0.01590767 0.01590767
C 0 0 0.10240500 0.04772300 0.04772300
D 0.00329000 0.00329000 0.00246750 0.00024150 0.00024150
E 0 0 0.42103250 0.00072450 0.19188950
F 0.44857333 0.28387778 0.48190000 0.06363067 0.22442933
G ....... ... ............. ................ 0.00096600 0.19164800
H ............. ............ ........ ... 0.04844750 0.23961250
I .......... .... ....... ..... ................ 0.01663217 0.17598183
J .... .... ....... ... ....... ... 0.00024150 0.19285550
TABLE 2.(Continued).
Line SCPX9Xlo
A 0.59876769
B 0.06516097
C 0.32580494
D 0.04989731
E 1.83498247
F 2.87461338
G 1.88487978
H 2.16078741
I 1.90014344
J 2.43375016
APPENDIX IV
NORMAL EQUATIONS USED IN ESTIMATING THE REGRESSION, STORE, WEEK AND AGE CONSTANTS FOR THE FINAL MODEL.
B 6; S6 86 J 6 B B66 6  6; 6o 10,  ; ~
90 9 9 9 9 9 9 9 9 9 9 40 30 20
9 9 0 0 0 0 0 0 0 0 0 4 3 2
9 0 9 0 0 0 0 0 0 0 0 4 3 2
9 0 0 9 0 0 0 0 0 0 0 4 3 2
9 0 0 0 9 0 0 0 0 0 0 4 3 2
9 0 0 0 0 9 0 0 0 0 0 4 3 2
9 0 0 0 0 0 9 0 0 0 0 4 3 2
9 0 0 0 0 0 0 9 0 0 0 4 3 2
9 0 0 0 0 0 0 0 9 0 0 4 3 2
9 0 0 0 0 0 0 0 0 9 0 4 3 2
9 0 0 0 0 0 0 0 0 0 9 4 3 2
40 4 4 4 4 4 4 4 4 4 4 40 0 0
30 3 3 3 33 3 33 3 3 3 0 30 0
20 2 2 2 2 2 2 2 2 2 0 0 20
10 1 1 1 1 1 1 1 1 1 1 10 0 0
10 1 1 1 1 1 1 1 1 1 1 10 0 0
10 1 1 1 1 1 1 1 1 1 1 0 10 0
10 1 1 1 1 1 1 1 1 0 0 10
10 1 1 1 1 1 1 1 1 1 1 10 0 0
10 1 1 1 1 1 1 1 1 1 1 0 10 0
10 1 1 1 1 1 10 0 0
10 1 1 1 1 1 1 1 1 1 1 0 10 0
10 1 1 1 1 1 1 1 1 1 1 0 0 10
1 0 000 0 0 0 0 0 0 010 1 0 0
1 0 0 0 0 0 0 0 0 1 010 0
103.84644 10.37595 10.69587 10.36845 10.97868 10.69587 9.504271 10.97868 10.36845 10.37595 9.5042 46.79028 34.61548 22.44068
121.11480 12.20031 12.72651 12.022051 13.44180 12.72651 10.16673 13.44180 12.02205 12.20031 10.16673 55.19420 40.37160 25.54900
________ ____ I I I I I ____ ________
APPENDIX IV. (Continued).
b" b8 I b b0 I
10 10 10 10 10 10 10 10 10 1 1 103.84644 121.11480 187.19589
1 1 1 1 1 1 1 1 1 0 0 10.37595 12.20031 19.16207
1 1 1 1 1 1 1 1 1 0 0 10.69587 12.72651 19.16322
1 1 1 1 1 1 1 1 1 0 0 10.36845 12.02205 17.09669
1 1 1 1 1 1 1 1 1 0 0 10.97868 13.44180 20.04376
1 1 1 1 1 1 1 1 1 0 0 10.69587 12.72651 19.87644
1 1 1 1 1 1 1 1 1 0 0 9.50427 10.16673 20.04872
1 1 1 1 1 1 1 1 1 0 0 10.97868 13.44180 18.49951
1 1 1 1 1 1 1 1 1 0 0 10.36845 12.02205 18.91493
1 1 1 1 1 1 1 1 1 1 1 10.37595 12.20031 14.86839
1 1 1 1 1 1 1 1 1 0 0 9.50427 10.16673 19.52216
10 10 0 0 10 0 10 0 0 1 1 46.79028 55.19420 79.45634
0 0 10 0 0 10 0 10 0 0 0 34.61548 40.37160 64.58527
0 0 0 10 0 0 0 0 10 0 0 22.44068 25.54900 43.15428
10 0 0 0 0 0 0 0 0 1 0 12.17480 14.82260 18.33122
0 10 0 0 0 0 0 0 0 0 1 11.22034 12.77450 18.85629
0 0 10 0 0 0 0 0 0 0 0 11.22034 12.77450 21.71242
0 0 0 10 0 0 0 0 0 0 0 11.22034 12.77450 21.82745
0 0 0 0 10 0 0 0 0 0 0 12.17480 14.82260 20.76094
0 0 0 0 0 10 0 0 0 0 0 12.17480 14.82260 21.10776
0 0 0 0 0 0 10 0 0 0 0 11.22034 12.77450 21.50789
0 0 0 0 0 0 0 10 0 0 0 11.22034 12.77450 21.76509
0 0 0 0 0 0 0 0 10 0 0 11.22034 12.77450 21.32683
1 0 0 0 0 0 0 0 0 1 0 1.21748 1.48226 0
0 1 0 0 0 0 0 0 0 0 1 1.21748 1.48226 0
12.17480 11.2203411.2203 11.2203 12.174811 0 11.22034 11.22034 11.22034 1.21748 1.21748 121.11464 142.62284 215.47099
14.82260 12.77450 12.77450 12.77450 14.82260 14.82260 12.77450 12.77450 12.77450 1.48226 1.48226 142.62284 169.40010 250.73800
