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- Permanent Link:
- https://ufdc.ufl.edu/UF00026878/00001
## Material Information- Title:
- Experimental pricing as an approach to demand analysis a technical study of the retail demand for frozen orange concentrate
- Alternate title:
- Bulletin 592 ; University of Florida. Agricultural Experiment Station
- Creator:
- Powell, Levi A.
O'Regan, William G. Godwin, Marshall R ( Marshall Reid ), 1922- - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida Agricultural Experiment Station
- Publication Date:
- March, 1958
- Copyright Date:
- 1958
- Language:
- English
- Physical Description:
- 43, 2 p. : ill. ; 23 cm.
## Subjects- Subjects / Keywords:
- Demand (Economic theory) -- Mathematical models ( lcsh )
Frozen concentrated orange juice -- Prices ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references.
- General Note:
- Cover title.
- Statement of Responsibility:
- by L.A. Powell, Sr., William G. O'Regan and Marshall R. Godwin.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- AEN7506 ( ltuf )
18283837 ( oclc ) 027106259 ( alephbibnum )
## UFDC Membership |

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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 semi-controlled 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 6-ounce 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 carry-over 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 6-ounce 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- spect-the basic data were generated under semi-controlled 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 data-generating 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 neo-classical 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 non-existent 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 curves-one 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 case-assumed to be a function of time-related 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- short-run and short-run 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 non-homogeneous 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, data-gener- 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 "carry-over" 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 data-generating apparatus appears in Table 1.4 The position of the symbols in the two-way 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. "Carry-over" 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 below-market 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 near-normal display of the test product-a 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 within-store 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 6-ounce can of Brand B concentrate. The same absolute price differentials were applied to the respective current or base prices of 6-ounce containers of other brands. However, the price of a 12-ounce container of Brand B was varied pro- portionately to the price of the 6-ounce can of the same brand. ** This price was in effect when the experiment was activated. Prices of other items were as follows: Brand A-2/37, Brand C-2/29 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 6-ounce 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 price-stores, 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 once-and- for-all, 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 "carry-over" effect. Given a basic (long-run) 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 store-to-store 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 k-j- = + aA + pP + i -W tk+ p ijpr + PX';ik-i + P2X2ik-i-+ P3 ik-j-+ P4Xik-j- + PXik-i- + 6 ik-j- + ik-i- 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 least-squares 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 6-ounce 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 Ik-j- 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 k-i- + ;i+ PI IUk-ij- +2Xk-j- + P3Xk-j- + P4Xk-j- + P5k-i- + 6X6ik-i- + 'k-j- or, in actual values by the monomial, Yik,-= (lk-j-) (Aik-j-) (Xlik-i- lik-j- (X21k--V2k-j- (X3ii_)V3ik-j- where Aik-j- 0 ai 1 'k-i- '1 Vik-i-' PI + 4ogXlik-j- V2ik-i- = 2+ 0lg X21k-i- V31k-j-= P3 + P610 X31k-l- 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 'k-i-_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 least-squares 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: McGraw-Hill, 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: McGraw-Hill, 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: xk-i_= -1 for store nine, week one, 0 otherwise xik-i= -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: YIk-j- = 0 + a + 8i + k + i+ 1iik-I- + P2X2k-i- + P31ik-i- + X4kk-l- + PXik-l- + P6X6ik-- + PTXik-i- + B8ki-+ 'ik-j- 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 6-ounce 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'k-i- = ik-j-+ P7X7k-- + PXBk-- + 9X9ik-- + 'k-j- (A) where Pp PI = 2 = In the event that P 2 0 P3 but 4 = 5 = P6 = 0, it would become Yki- = Ik-j- lik-j-+ P2X k-- + Pa k-j- (B) + P$X7k-i-+ PB ik-i- + 'Ik-i- Should p1 = P2 = P3 and 4 = P5 = 6. the model could be written Y;k-j- = Ak-j- + 7X71k-i- + BX8k-i- (C) + PX91ik-j- + P1B lk-i- + e;k-j- 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-- + PIXlk-j- + P2X'2k-j-+ P3X3k-j- + 4X4k-- (D) + PB5Xilk-j- + P6XAik-- + P7X71k-j- + P8X81k-- + 'k-ij- It should be noted that in all of the model forms, "Ak-i- = 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 8-I = [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 Xk-i-and Xj0ik-i 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 : < ik-i-' 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' k-j-' 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 Yik-j- on xw9k-- "ad XlOk-j-. 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 non-significant. 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 re-examination of the model form. In particular, it became necessary to re-open the question concerning the distinctness of the separate age regressions, i.e., to re-test 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 Yik-i- = 0 + Si + i + 'kj-- + 2.00 2.288554XODk-j- 5.584685 Xi_ + 2.120746X;O1k-j- + .;k-j-. 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 2-1- = -0.02874 a2 = 0.02513 Ai Ai A, 2 = 0.04973 3-2- = -0.01451 a3 = -0.00292 3 = -0.21705 = 0.02505 64 = 0.15449 5-1- = 0.03158 5 = 0.12897 62- = 0.02374 6 = 0.01189 7-1- = 0.00757 A, Ag 67 = -0.01709 w8-2- = -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 three-way 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 Yik-i- [;Y;kij i+ k ;I-Y. u kk -'-i F l + I [ Yiki-] + r[ bp : Yk-i-pk-i-" Independent computations of these three sums of squares yielded SY(Yik-i k-j-)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 6-ounce 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 market-wide 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 re-allocative 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, because-it will be recalled-the 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 ^k-j-' (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 "carry-over 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 "carry-over 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 under-adjust- ing their purchases, then over-adjusting, 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 6-ounce 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 Y-1- 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'2-1- 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'3-2- 1 0 0 0 00 0. 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 Y'4-3- 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'19-3- 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 Y21-1- 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 Y29-3- 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 Y9.9-3- 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'092-1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0-1 .... . . . . . . . . . . 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 1-1- )2 0 . X02-1- O 0 (X'2-1-)2 0 0 6 e12-1- 0 X132- 0 0 (Xi3-2)2 0 62 '13-2- o o x44.3. o 0 (X14-3-J2 63 "14-3- .64 65 0 0 X'19-3- 0 0 (Xi9-3-2 6g8 -19-3- X'21-1- 0 0 (X21-1-2 0 0 4 '21- 510 'i 0 0 X29-3- O 0 (X29-3_-2 '3 '29-3- -4 15 + '6 X'gl-l- O 0 (X_91-1J2 0 0 9 91-1- 92-1- 0 0 (X2-1 2 0 0 a1 e92-1- a2 P7 X'O, l-- 0 0 (X o,1 -1-)2 0 0 p '10,1-)- P2 P3 P4 *PS 0 0 X0,9-3- 0 0 (X;0o,9-3 _2 P6 '10,9-3- As-suption 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 0-10 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 |