American Journal of Industrial and Business Management, 2013, 3, 655673 Published Online December 2013 (http://www.scirp.org/journal/ajibm) http://dx.doi.org/10.4236/ajibm.2013.38075 Open Access AJIBM 655 Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool Philip Thomas1, Alec Chrystal2 1School of Engineering and Mathematical Sciences, City University London, London, UK; 2Cass Business School, City University London, London, UK. Email: pjt3.michaelmas@gmail.com Received October 20th, 2013; revised November 20th, 2013; accepted November 27th, 2013 Copyright © 2013 Philip Thomas, Alec Chrystal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In accordance of the Creative Commons Attribution License all Copyrights © 2013 are reserved for SCIRP and the owner of the intellectual property Philip Thomas, Alec Chrystal. All Copyright © 2013 are guarded by law and by SCIRP as a guardian. ABSTRACT Buyonegetonefree (BOGOF) promotions are a common feature of retail food markets, but why are they so wide spread? The theory of Relative Utility Pricing (RUP) developed in this paper provides an explanation not only for su permarket promotional offers but also for more general pricing of packs of different sizes in supermarkets and on the internet. A clear and simple explanation is given for the two most widely used quantity promotions: BOGOF and 3forthepriceof2. The RUP model may be linked to the theory of isoelastic utility functions, and this allows the re lationships amongst riskaversion, packsize ratio and demand elasticity to be explored. “Cautious consumers”, as de fined in the paper, are found to be the only sensible target for quantity promotions. It is argued that the needs of cau tious consumers of retail commodities will be best addressed if the vendor sets the ratio of successive pack sizes as the square of the Golden Ratio, namely 2.62, and the priceratio at the Golden Ratio, 1.62. Thus the Golden Ratio may be regarded as a marketing guide for vendors considering both their best interests and those of their customers. This proposition is supported by an analysis showing that higher profits are more likely to come from Golden Ratio sizing than from either BOGOF or 3for2 when variable costs lie in most of the upper half of the range that is required for any of these multibuy offers to generate profit. The paper’s theoretical predictions for both pack sizes and prices are sup ported by examples from the retail sector: grocery, paperback books and electronics. Keywords: Relative Utility Pricing; Golden Ratio Pricing; Buy One Get One Free; Multibuy Promotions; Utility Theory 1. Introduction Retailers often offer products in packs of different sizes, whether it is a food product, electronics or clothing. Moreover, multibuy promotions such as buyoneget onefree (BOGOF) are common, but why are they so widespread? Put formally, the research questions addressed by this paper are as follows. Suppose that a pack larger than the entrylevel pack is offered, what price will the vendor need to charge in order to convince the consumer that he or she should buy it? Then if further, larger packs are put on the market, what prices will the vendor need to set in order to sell these? What effect will the relative sizes of the packs have on the reaction of the consumers? Finally, how can the retailer maximise his profit given the atti tudes of the consumer. 2. Background In his 2008 article in The Daily Telegraph, the Consumer Affairs Editor revealed that “more than 80% of all pro motional activity within supermarkets is a bogof [buy onegetonefree] or threefortwo” [1]. So concerned is the UK Government about the success of such promo tions applied to food that it is considering banning mul tibuy deals in an effort to tackle obesity [2]. But there has long been puzzlement about vendors’ motivation in offering quantity promotions. Back in 1984, the Wall Street Journal carried a frontpage article stating that: “The dark secrets of packaging aren’t always easy to unwrap. While the issue of how much product a com
Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool 656 pany should put in a box, bottle or tube is about as basic as any in business, it also can be fraught with complexi ties” [3]; see also [4]. Tabarrok [5] in his response to Pressler [6] suggests that BOGOFs “generally increase total welfare because the price on the last unit sold is pushed closer to marginal cost and because of this output expands”. Certainly, it is relatively easy to understand how vendors will profit from the selling of larger packs of the same good when the marginal cost associated with the extra amount of good is small compared with the fixed costs the retail outlet has to bear (e.g. [7]). Nevertheless worries have been expressed for some time in academic journals that BOGOFtype promotions might lead people to endanger their health through over eating, e.g. [8]. While, according to Wallop, the distinc tion between perishable and nonperishable goods means that BOGOFs in supermarkets “are almost twice as popular in the toiletries aisles as in the meat or vegetable section”, the same restraint on food offerings does not carry over to fastfood chains. Here Dobson and Gerstner [8] confirm that “supersizing” food portions and offering the larger size at “only a few cents more” than the regular portion can be profitable for the vendor, but claim that this may be harmful to the health of the consumer and, moreover, create economic inefficiencies. However, no satisfactory economic explanation has been available until now for the reaction of the consumer to the multibuy promotions. It is thus necessary to pro vide a quantitative explanation for how the larger packs should be sized and priced if the consumer is to be in duced to buy a quantity greater than that contained in the standard size, and the logic of these promotions needs to be understood when restrictions are contemplated by the authorities. The problem is a general one, applying to retail outlets including supermarkets, restaurants, public houses, fast food stores and internet shopping. It also features in market surveys where the respondent is asked to set a price on different options. The first part of the paper (up to and including Section 8) will be devoted to developing a model to explain quantitatively the prices consumers will expect for dif ferent pack sizes. Utility theory will be used in Sections 3, 4 and 5 to develop the general Relative Utility Pricing (RUP) model, applicable where product differentiation is low and the goods may be regarded as commodities. The model developed provides, inter alia, a theoretical ex planation for quantity discounts. Section 6 develops the RUP model for the common case where, when deliberating whether or not to buy pack n, the customer chooses to compare its benefit with the benefit he would gain by buying pack 1n . This leads to the Single Comparison Relative Utility Pricing model (SCRUP). The results of this model are surprise ingly simple, and provide a complete explanation of why the retailer will find it necessary to offer buyoneget onefree (BOGOF) or 3forthepriceof2 promotions. Put colloquially, people will expect “to get n but pay 1n ”. The theoretical results are used in Section 7 to explain some actual prices achieved in supermarkets and on the internet, including in cases where the explanation for the pack size and price structure is nonobvious. Section 8 compares the price per unit produced by the SCRUP model with that produced by applying an iso elastic utility function. It is shown that the natural targets for multibuy promotions are cautious consumers, who have a riskaversion between 0.0 and 1.0. A link is de veloped between pack size and the average riskaversion amongst cautious consumers, which leads to the notion of Golden Ratio Pricing. Here the price quotient of suc cessive packs ratio is the golden ratio while the packsize ratio is the square of that figure. Then, in Section 9, an analysis will be made of how the retailer may design his pack 2 offering so as to maximise his profit while satisfying his customers’ re quirements on price and volume. It will be shown how the retailer may use the knowledge of his variable costs to maximise his profit in his choice of which multibuy offer to make: BOGOF, or 3for2 or the new promotion suggested in this paper, Golden Ratio Pricing. 3. The Utility of Packs of Different Sizes Consider a basic good, B, that is offered for sale in a number of differently sized packs identified by index, j, with 0,1,2, ,, ,, ,jknN , where j = 0 signifies the empty pack – no purchase is made – and the index, j, increases with pack size. Thus indicates the smallest, entrylevel pack, while pack N is the largest pack on sale. The indices, k and n, with 1j kn , have been introduced to facilitate later comparisons between a general pack, n, and a pack, k, that is strictly smaller. The maximum ranges of the three indices, j, k and n are: 0 0 1 jN kN nN 1 (1) The amount of good, B, in the world may be very large indeed, but we shall be concerned normally with small amounts, b, measured in some unit of extent, for example mass or volume or an entirely different measure, such as gigabytes of memory on a USB memory stick. Let pack j contain an amount, b, of B, where, as noted above, 00b and 1 j for bb Nj 1. We will define the largeness, L(j), of pack j as its size relative to the entrylevel pack 1: Open Access AJIBM
Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool 657 1 for 0 j b Ljj N b (2) Thus the largeness of pack j is the number of standard units that pack j contains, where a standard unit com prises the contents of pack 1, namely . Hence and . 1 b 00L 11L Kahneman and Tversky [9] have argued that “people normally perceive outcomes as gains and losses [relative to the current asset position], rather than as final states of wealth or welfare”, a proposition supported by Thaler [10]. This approach of measuring the utility of the change in the stock level rather than the utility of the absolute value of the stock is adopted in this paper, and leads to a consideration of the utility of the contents of the pack. Thus the utility of the empty pack, with con tents 0 will be , the utility of the contents of pack 1, namely 1, will be 0b 0 ub b 1 ub , and, in general, the purchase of pack j will result in a utility, ub . Consider the case where the vendor is offering just two packs: pack 1, with contents, 1, and a bigger pack, let us call it pack x, with contents, 1. We will assume that the price of the entrylevel pack, pack 1, has been accepted by purchasers as 1, implying that these con sumers will have a maximum acceptable price (MAP) of at least 1. (See [11] and [12] for a discussion of the concept of maximum acceptable price and probability distributions.) b 16b p p Let us select one of these purchasers with a MAP of 1 or higher, who buys one standard pack. We may give him the name, “consumer X”. Meanwhile there may be another person, say consumer Y, who has a MAP of 1 or higher and also a need or strong desire for 1. Be cause no larger pack is offered, he will have no option but to buy two of the standard packs, which will cost him . p 2p p 2b 1 Suppose now that the vendor withdraws the existing pack 1 and replaces it with a new pack 1, which contains 1. Consumer X will be put in a quandary, but con sumer Y, whose MAP is at least 1, will be indifferent to this change, and will be content to continue paying 1 for the same amount of the good, B, now contained in the new pack 1. 2b 2p p Let us now transfer our attention to a person, let us call him consumer Z, with a MAP of 1 or more who has a need or strong desire for 1. Consumer Z would need initially to buy 4 of pack 1 at 1 each, then, after the vendor’s doubling of the size of pack 1, would be content to buy 2 of new pack 1 at 1 each, making his total outlay 1. Moreover, it would make no difference to him if new pack 1 were supplanted by a further revised pack 1 with contents, 1, and price . His total bill remains the same in all cases. p p 4b 2p 4p 4b1 4p Now suppose that the original pack 1 containing is reinstated, and moreover a pack 2 containing 1 is also offered. The pack containing 1 4 is retained but it should now be called pack 3, because there are now two nonempty, smaller packs. The presence of the two smaller packs has given the prospective purchaser more choices, and this makes the pricing of the pack contain ing 1 more complicated than it was before. Thus we can no longer say that the price of the pack containing 1 should be 1. Nevertheless the price, 1, retains the significance that it is the price that customer Z would prepared to pay for a pack of size 1 if no smaller packs were available and the only other option was to buy nothing – the empty pack or pack 0 in our terminal ogy. This limiting price for pack 3 may be denoted , where 1 b 2b 4p b 4b 4b4p 4b 30 p 3 301 1 The larger pack, pack x, has not figured in the analysis above, so that its existence or otherwise is not relevant. This will hold true for customers, X, Y and Z, as long as the price of pack x is not below that of the largest of the smaller packs. (This can happen on occasion, but rarely on a continuing basis, as it contradicts the economic principle that more will be preferred to less.) 4pLpp b p . Generalising, we may observe that if a purchaser, say consumer Ω, has a MAP of 1 or more for a pack con taining 1, has a need or strong desire for the quantity contained in pack n, namely , and he chooses to ignore any packs between pack 0 and pack n, then the retailer may continue offering the smaller packs without affecting the valuation, 0n, put on pack n by this pur chaser. Because he has decided to ignore smaller packs, person Ω will be prepared to pay for pack n, given by: p 1 Lnb n p0 n 0n pL1 p (3) The condition of ignoring packs smaller than pack n will be satisfied automatically when n = 1, since, by definition, no pack smaller than pack 1 is offered, apart from the empty pack, pack 0. Thus putting n = 1 in Equa tion (3) produces the price for pack 1: 110 1pL 1 pp (4) Now let us consider utility. The utility resulting from buying pack n will be n bu , and the difference, 0n u , between the utility of having purchased pack n and that of having bought pack 0, i.e. made no purchase, will be given by ub Ln 0 b0nn uu (5) Assume next that the purchase in question was made on behalf of a consortium of people, each of whom has a MAP of at least p; 1 has a need or strong desire for a quantity of the good, 1 b; Open Access AJIBM
Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool 658 is allotted 1 b from the contents, 1 Lnb, after pack n is purchased. If we make the assumption that the utility gain for each person in the consortium is the same, then the gain in the utility for each consortium member will be 0 . This will be the same as if that member had himself gone out and bought pack 1, as we may con firm by putting n = 1 in Equation (5) to give the total util ity gain from the purchase of pack 1 as: 1 ub ub 10 10 uubub (6) The total utility gain for the consortium of Ln people may be found by adding together all the compo nents identified in Equation (6): 01010 11 1 Ln Ln nii uuuLn 10 u (7) demonstrating that the utility gain from pack n will be proportional to the utility gain from pack 1, with a factor of proportionality, . For the utility gain from pack n to be proportional to the utility gain from pack 1, then the utility differences for the purchasers of pack n and pack 1 will obey a very restricted version of homogeneity, “singular homogeneity”, as explained in Appendix A. Ln An example of a consortium as described above would be 4 university students sharing a house and joining to gether to buy a 4pint pack of milk a day to be divided equally between them. Another consortium could be a mother buying the same pack of milk daily for her family. On the other hand, an individual living alone and con suming milk at the same rate as one of the students might buy a 4pack of milk every 4 days. In this last case, the consortium consists of the individual on day 1 and the same individual on days 2, 3 and 4. This generalization of the consortium so that it can include timedelayed ver sions of the same individual makes it reasonable to as sume that that every purchaser of pack n experiences the same utility gain, 0n, whether part of a multiperson consortium or not. This equality in the gain in utility is taken to be independent of the price paid for the pack, and to be valid in cases where the packsize ratio is non integer. u Combining Equations (3) and (7), we may write the ratio of the changes in utility in terms of the ratio of the limiting price for pack n to the achieved price for pack 1: 00 10 1 1 nn up nN up (8) But in general there will be packs intermediate be tween pack n and pack 0. In this case, when considering the purchase of pack n, in addition to the utility differ ence between pack n and pack 0, the consumer may well wish to take into account also the utility differences, , , that will arise from comparisons between pack n and packs in between pack n and pack 0: nk ukn 1 nk nk uubub kn (9) The change in utility between buying pack n and making no purchase may be then be expressed in terms of intermediate utility differences: 00 0 nn nkk nk k uubububububub uu 0 0 (10) where the last step has made use of Equation (5), replac ing the subscripts, n, and 0 with the subscript, k, as nec essary. Thus the change in utility between purchasing pack n and a smaller pack k may be written in terms of the utility gains from buying pack n and pack k, respect tively: 0nk nk uuu (11) Hence 0010 00 100 1 nk nn kk k uu uu uu uu 1 (12) Substituting from Equation (7) with the appropriate subscripts into Equation (12) gives 0 1 nk k Ln uk uLk 0 (13) Since Ln Lk, it follows that 0nk k uu will always be positive. 4. Comparing Packs of Different Sizes Because they are of the same good, packs may be char acterised by only two quantities, their size and their price. But while a single comparison involving multiple packs would be feasible if the packs could be characterised by just one quantity, when each pack has even as few as two characteristic features, it becomes difficult if not impos sible for the consumer to compare packs more than two at a time, even when both those characteristics are repre sented numerically. This is exactly the situation ad dressed by mathematics when comparisons are to be made between an array of properties, formalised mathe matically as a vector. Generally it is not meaningful to say that a vector, x, is bigger than another, y, of the same dimension, the only exception being when the vectors are linearly related: yx, where is a constant. In every other case, further mathematical operation is needed before it is possible to make a meaningful com parison, such as taking the modulus: 2 i i x, and then comparing the single number x with the single number, y. When the packs are characterized by size and price, it Open Access AJIBM
Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool 659 is, of course, possible to divide the price by the size to give the price per unit. This is normally done in UK su permarkets by the vendor, who is obliged to display the result to his customers. But while the prices per unit over all n are easily ordered and hence may be compared mul tiply, the unit price does not give a full characterization of the pack, for which the size and price are still needed. For example, suppose that the price of the standard pack containing one unit is £2 and the price of a pack contain ing 10 units is £19, then based on the unit price the larger pack is clearly preferable. But the number of units may be more than the purchaser needs or the amount of money required may be more than the consumer wants to lay out on this good, so that actually the consumer will prefer pack 1. Hence the role of the price per unit for pack n, n , is to add to the features characterizing the pack, meaning that the vector of characteristics has risen to three: n n Ln p strengthening the necessity for pairwise comparisons. Consumers may carry out comparisons over the whole range of packs, or they may make an outline decision early on in the selection process on the range of packs that they judge may meet their desire for the good, B, and their budgetary limitations. In each case, the pairwise comparison will be made by taking each pack, pack n, within the feasible range, and comparing it with a smaller pack, . A price, nk , will emerge from each comparison. The consumer is presumed eventually to eliminate from consideration all sizes greater than his ideal size, and base the price he is prepared to pay for this ideal size on the outcome of some or all of the com parisons he has made using this pack as his basis. Thus the final price for pack n will be a function of the com parisongenerated prices or a subset of them: :0kk n p 01 ,1 ,,,,, nnnnknn pgpp pp (14) Thus the set of feasible binary comparisons for pack n will be between pack n and the following packs, taken one at a time: pack 0 (= the empty pack or no purchase). pack 1. pack k. pack n – 1. At each comparison between pack n and pack k, the consumer is presumed to ask himself the question: “Suppose I am on the point of agreeing to buy pack k, what will it take to make me want to buy the greater quantity contained in pack n?” He may then reason that the best answer to this ques tion will be found by consulting a mentor with greater relevant experience, namely someone who has actually bought pack k. This strategy may be characterised as “phone a friend”. The mentor has something the con sumer does not: he knows what it feels like actually to own pack k. He has knowledge borne of experience, the knowledge of possession. Let us now consider the perspective of the mentor. As a purchaser of pack k, the mentor may be presumed to have most of his need satisfied by pack k, which contains lass="t m0 xb0 h2 y356 ff6 fs7 fc0 sc0 lscb wsd1"> , “ultracautious”. The utility, , of ultracautious individuals will approach an up per, asymptotic limit as b. Moreover, it is clear from Figure 1 that a riskaversion of greater than unity will call for a packsize ratio of less than 2. Therefore the limit contained in Equation (17) will always need to be invoked, and even then such ultracautious individuals will want to pay the same price for all packs, irrespective of size. ub While the vendor may face a market composed of cau tious and ultracautious individuals, only the cautious consumers will always want more and only the cautious, not the ultracautious, may be prepared to pay more for a larger pack. Therefore it is only the cautious consumers whom it is sensible for the vendors to target with quan tity promotions. In the absence of more precise information, it is rea sonable to assume a uniform probability distribution for riskaversion amongst the cautious consumers, over the open interval, 01 0.5 . The average value of this dis tribution is 0.5 , and, indeed, the same mean will result from any symmetrical distribution over the same interval. Thus, in the absence of evidence to the contrary, the figure, , may be regarded as the riskaversion for the average person likely to be interested in a sales promotion offering greater quantities. This value of riskaversion requires from Equation (46) that 21 (47) where is the golden ratio (see e.g. [21]). (The result is easily confirmed from Equation (46) by using Equation (47) to replace ln 1 with ln and substituting 2 ln2 llnn . Hence 1ln 2ln 0.5 .) The properties of mean that when the pack size ratio is set to its square, 2 , then the normalised difference in size between two successive packs will be equal to the golden ratio: 1 1 Ln Ln Ln (48) We may combine Equations (27) and (47) to show that the ratio of the price of pack n to the price of pack n – 1 will also be equal to the golden ratio: 1 n n p p (49) The value of , found from solving the quadratic equation of Equation (47), is approximately 1.62, so that, from the same equation, . 22.62 On this basis, if the vendor is to increase pack sizes by a constant ratio, , within a range of quantities known to be desired by cohorts of his customers, the vendor can expect to be match his market most accurately if he sets the size ratio to the square of the Golden Ratio (Equation (47)) and his price ratios at the Golden Ratio (Equation (49)). This result will clearly apply to the case where just two packs are offered, when the size ratio of pack 2 to pack 1 should be the square of the golden ratio and the price ratio should equal the golden ratio. The matching of the pack size and its prices to the values desired by the average customer liable to be tempted by a multibuy of fer means that there is a strong case for regarding Golden Ratio sizing and pricing as delivering what the customer wants most. Figure 2 gives the normalised price per standard unit for packs of different sizes when the comparison is al ways made with the pack immediately below (the SCRUP model), and when the packsize ratio, , stays constant. The Figure illustrates the cases where this ratio takes the values, 2, 2.5, 2 and 3, and shows how the locus derived from isoelastic utility functions with cor responding riskaversions 1.0, 0.56, 0.5 and 0.37 respec tively will intersect the discrete data points. Also plotted is the locus corresponding to a pack size ratio of 50: the riskaversion, at 0.005, is now close to zero, implying an absolutely very large demand elasticity of −200, and a price density vs. quantity curve that is nearly horizontal. Figure 2 demonstrates that when the packsize ratio is 2, the SCRUP Model will predict a price equal to that produced by a logarithmic utility function. This corre sponds to the case where a pack containing double the quantity will attract the same price as the first pack, as Open Access AJIBM
Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool 665 may be seen from Equation (24). The demand elasticity is neutral under this condition. The retailer may move the market into a region of elastic demand by using a higher packsize ratio, and can expect to achieve better per unit prices as a result. However, as illustrated by Figure 2 and as a moment’s thought will confirm, these higher per unit prices will come at the expense of poorer coverage of the range of potential purchases. While higher size ratios (e.g. a ratio of 3) should give higher per unit prices, the square of the golden ratio (~2.62) should offer an optimal compromise between higher per unit price and coverage of the range of quantities required by consum ers. More detail on the prices achieved by major UK su permarkets will be given in [13]. Here we will content ourselves with a brief discussion of one result given there, concerning “value” eggs where the size of pack 1 was 6 eggs and the size of pack 2 was either 18 eggs (Sainsbury) or 15 eggs (Asda and Tesco) and where pack 1 was re tailing at the same price, £0.91, in all three supermarkets. The price per egg achieved by Sainsbury for its 18egg pack 2 was £1.85/18 = £0.103, which is within 2% of the theoretical figure of 101.0£1891.0£1618 . As noted in the previous paragraph, pack 2 for value eggs in Asda and Tesco contained 15 eggs, giving a packsize ratio of 2.5, which is significantly closer to the ideal ratio derived above of . By selling pack 2 at £1.50, these stores achieved a price per egg of £0.100 with their pack 2, which is 9% higher than the predicted price of 22.62 091.00£615 £1591. . The ideal number of eggs in pack 2 based on this theory would be or 16 after rounding. A pack 2 containing 16 eggs should command a price of 26 15.72 52.1£91.0£1616 , giving a price per egg of £0.095. The premium price per egg achieved by Tesco and Asda with their pack 2 may result from the pack size 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0123456789101112 Number of units in pack Normalised price per unit Pack size ratio = 2.0 Pack size ratio = 2.5 Pa ck s ize ratio = 2.62 Pack size ratio = 3.0 Riskaversion = 1. 0 Risk avers ion = 0. 56 Riskaversion = 0.5 Riskaversion = 0. 37 Risk avers ion = 0. 005 Figure 2. Effect of pack size ratio on the price per unit. satisfying better people’s sense of the most appropriate next size up. On the other hand, it may well be that they priced their pack 2 based on the price per egg that their competitor, Sainsbury, was achieving, and might have done better through higher sales by pricing their pack 2 at 37.1£91.0£1615 . The theory just developed suggests that they might have done better still by putting 16 eggs into their pack 2, which they would then have sold at £1.52, since this would have matched the re quirements of the average cautious customer best. Interestingly, by August 2013, Tesco was pricing its value eggs at £0.87 for a half dozen and £1.35 for 15 [13]. Using the SCRUP model, the predicted price for 15 eggs would be 31.1£87.01615 . The actual price had thus drawn significantly closer to the predicted value—a 3% difference as opposed to the earlier 9% difference. 9. The Retailer’s Problem: Choosing the Largeness of Pack 2 to Maximise Profit: BOGOF vs. 3for2 vs. Golden Ratio Pricing Fundamentally, although the customer will want quantity discounts, the retailer will be prepared to offer them if his variable costs are low enough for each sale to make a nonnegative contribution to covering his overall costs. The purpose of this Section is to examine the circum stances when the vendor can maximise his profit by set ting the largeness of pack 2 as , as in a BO GOF, when it is best to set it as , as in a 3for2 offer, and when the most profitable strategy might be to set the largeness of pack 2 at the square of the Golden Ratio, 22L 23L 2 22L .62. This Section is therefore con cerned with the overall design of pack 2, in terms of both size and price. 9.1. The Optimising Equation The exercise of choosing the largeness of pack 2 to maximise the vendor’s profits may be treated in a similar way to that presented in [11] by using a probability dis tribution to model the maximum acceptable price (MAP) amongst those making up the target market. The further complication in this case is that the MAP for pack 2 var ies with its largeness. For the cases that we want to con sider, where the packsize ratio is at least two: 2 , the relationship is given by putting n = 2 into Equation (27) and rearranging: 211 211 1 L ppp L 2 (50) where, since 11L , the largeness of pack 2 is identi cal with the packsize ratio, 2L . For convenience, let us define the price of pack 1 as our base money unit Open Access AJIBM
Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool 666 so that 1 and the price of pack 2, 2, is expressed as a multiple of the price of pack 1. Using this system of units, we may then simplify Equation (50) into the form: 1pp 21p2 (51) The variable cost of pack 1, v, may be expressed in terms of the same base unit, the price of pack 1. Assum ing moreover that the income from selling pack 1 will cover at least its variable cost, will lie in the defined range: c v c 01 v c.0 (52) Since pack 2 contains a factor, , times the contents of pack 1, then the variable cost associated with pack 2 will be v c , or, using Equation (51), . 1 2v cp dv v Let the probability density for the MAP for pack 2, 2, be . Using the concept of the “uniconsumer” introduced without loss of generality in [11] to charac terise a person prepared to buy one but only one item if the price is right, the fraction of the uniconsumers pre pared to pay 2 or more for a single pack 2 will be , given by p Sp 2 hp 2 Sp p p 2 22 20 d1 m pp p hv vh (53) where 2m is the most that anyone in the target market is prepared to pay for pack 2. Those prepared to pay more would ideally want a larger pack 2 for their money, but it is assumed that they would be content with the smaller pack 2, provided it cost less, if that was all that was offered. Let there be N uniconsumers in the target group. The vendor’s total profit, , from selling packs 2 will be his total income for pack 2 less both the total variable costs and the fixed costs associated with pack 2, C 1C : 222 2v ppNSpcp F NS (54) The retailer will seek to maximise this profit, which, for a constant size of target population, N, is equivalent to maximising the average profit per consumer, : 2 2 0 11d p vv C pcc h NN vv (55) where use has been made of Equation (53) in the second step. The maximum value of profit, , may be found by differentiating Equation (55) with respect to pack 2 price, 2, and then setting p2 dd 0p . This gives the opti mal price as the solution, , of 2 p 22 d 1 vv vvpchp 2 0 11 p ch 0 v c (56) which will be seen to depend on the variable cost associ ated with pack 1, v, and the probability distribution, c 2 hp , for pack 2 price, . 2 A limit in the range of riskaversions is implicit in the assignment of the maximum price that anyone would pay for a pack 2, 2m, in Equation (53), and it is appropriate now to consider the basis for this limit. The maximum price, 2m, that anyone is prepared to pay for pack 2, will impact on the maximum largeness, p p p max , via Equation (51): 2 2L 1 mm p where m is the maxi mum packsize ratio and equal to the maximum largeness for pack 2: max . The maximum pack size ratio, m 2L m , will imply a lower limit on the riskaversion via equation (46): ln 1 1ln m mm (57) When he is considering assigning a largeness for pack 2 of between 2 and 3, we can expect that the vendor will have an idea of the maximum size for pack 2 wanted by even the consumers possessed of the largest appetite for that good. The lowest maximum largeness we can take to be 3 (otherwise why would he be considering this as a possible size for pack 2?), and twice that figure, namely 6, would seem to constitute a reasonable estimate of the highest maximum largeness in the absence of more pre cise information. Putting 3 m into Equation (57) gives a minimum riskaversion of 0.369 m , while inserting 6 m gives m0.101 0.101 . In the latter case, about 90% of the possible range of riskaversion for cau tious consumers is covered: 1.0 m . 9.2. Distributions for RiskAversion The distribution of riskaversion has been modelled in the first instance as a uniform distribution over the re stricted range: 1.0 m , so that the probability den sity, g for riskaversion, , is: 11.0 1m m g (58) Meanwhile the probability density for the maximum acceptable price (MAP) among the target group for pack 2, , is given by 2 p 222 2 d1 dm hpgpp p p (59) where the relationship between pack 2 price, 2, and riskaversion, p , is found by combining Equation (46) and (51) to give: 2 2 2 ln1 ln ln 1 pp pp 2 (60) Differentiating Equation (60) with respect to 2 p Open Access AJIBM
Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool 667 gives 2 ddp : 22 22 2 222 2 ln1ln1 d d1ln 1 pp pp ppp p (61) The resultant probability distribution, 2 hp 6,0.10 , is shown in Figure 3, for the case when 1 m . It is clear that the probability density for MAP falls away quickly as MAP increases. The probability density for riskaversion, , has also been modelled using the generalised, Double Power den sity [12], which provides an analytically tractable model for a variety of smooth distributions with a wide range of modes. Let 1.0 mm (62) so that the minimum value of is zero, while the maximum value, m , of is 1 mm (63) The Double Power probability density for is then defined on 0, m by: 0 cd m fab (64) where, from [12] 1 11 c m cd adc (65) 1 11 d m cd bdc (66) where c and d are free parameters to be specified by the user. The corresponding probability density, g , for is then 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 012345 MAP for pack 2, p 2 ( standard units) Prob ab ility density, h(p 2 ) Figure 3. Probability distribution for maximum acceptable price for a uniform distribution of riskaversion with m 0.101 . d d cd m gfa b m (67) since dd 1 from differentiating Equation (62). Putting 1c and 2d produces a symmetrical distribution on m1.0 , see Figure 4, where the packsize ratio has been set at 6 m so that 0.101 m . Applying Equations (59) and (61) gives the probability density for MAP for pack 2 given in Figure 5, showing how the probability density falls away for high values of MAP in a way similar to when the distribution is uniform over the same interval. Skewed distributions for riskaversion were produced by setting c = 1.0 and varying the parameter, d, in the Double Power Equations (65) and (66). Setting 0.4d produced a distribution skewed towards less caution, 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 00.20.4 0.60.81 Risk aversion, Probability density, g( ) Figure 4. Double power probability distribution of risk aversion wh en , c = 1, d = 2. m 0.101 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0123456 MAP f or pack 2, p 2 Probability density, h(p 2 ) Figure 5. Probability distribution for maximum acceptable price for a symmetric double power distribution of risk aversion wit h , c = 1, d = 2. m 0.101 Open Access AJIBM
Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool 668 with a mode of mode 0.3 , as shown in Figure 6. A distribution skewed towards greater caution was found by putting , with a resultant mode of mode 8d0.77 . See Figure 7. The probability density falls away for high values of MAP in both these cases also. 9.3. The Optimal Largeness of Pack 2 To facilitate the discussion of the results, we shall intro duce the concept of the generosity of the offer, as meas ured by the price per unit of its contents. The price per unit for pack 2, 2 , is given by: 2 2 21 1 1 22 L p LL L 2 (68) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 00.2 0.4 0.6 0.81 Risk aversion, Pro bability d e nsity, g( ) Figure 6. Double power probability distribution of risk aversion skewed towards less caution; , c = 1, d = 0.4. m 0.101 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 00.2 0.4 0.6 0.81 Risk aversion, Probability density, g( ) Figure 7. Double power probability distribution of risk aversion skewed towards greater caution; m 0.101 , c = 1, d = 8. where the price of pack 1 provides the base unit for price once again. Hence 2 2 dd21 20LL , imply ing that increasing the largeness of pack 2 will raise the perunit price, thus decreasing the generosity of the offer. In these terms, BOGOF is the most generous offer of those under consideration and 3for2 is the least. We may also note that the vendor will wish to cover at least the variable costs in any longterm offer. The money brought in per sale of pack 2 will be 221pL , while the variable cost associated with pack 2 will be 2 v cL . Hence covering the variable cost of pack 2 requires that 212 0 v LcL or 1 21v Lc (69) Turning now to the problem of maximising profits, a solution may be found for the optimising Equation (56) in terms of v. The equation may be solved iteratively by first assigning a value to the optimal pack price, 2, and then finding the variable cost, v, associated with pack 1 that reduces the lefthand side of the equation to zero. (The reverse process of fixing v and finding the corresponding value of 2 is equally valid, but tends to be less well conditioned.) The optimal largeness may then be found from the golden ratio, Equation (48), not ing that cp c c p 2L . Figure 8 shows the optimal largeness, , of pack 2 versus the variable cost, v, associated with pack 1 for four distributions for riskaversion. It plots additionally the largeness of pack 2 needed to cover the variable cost (Equation (69)), which provides a base line. Clearly when 2L c 0.5 v c , selling 2 for the price of 1 (BOGOF) will cost 1 standard unit and bring in 1 standard unit, so that the variable cost of pack 2 is only just covered. When 0.66 v c7 , selling 3 for the price of 2 (BOGOF) will cost 2 standard units and bring in 2 standard units, just covering the variable cost of pack 2 again. Similar calculations will apply for all offers in between. Let us consider the profitmaximising offer based on the assumption that the population follows a symmetrical Double Power probability density for riskaversion with a lower level of 0.369 m by the assumption that max 2L3 . From Figure 8, the largeness offered when 0.5 v c is just less than 2.3, which is less generous than the largeness of 2.0 which would just cover the variable costs. A largeness of 2.3 can sustain a profit because there are sufficient consumers in the target population prepared to pay 1.3 standard units or more for an appro priately sized pack 2. When , the optimal largeness has gone up to 2.65, which may be compared with the largeness of 2.5 needed to cover variable costs. A largeness of 2.65 can be sustained because there are still enough consumers “left in” prepared to pay 1.65 standard units for the appropriately sized pack 2. The 0.6 v c Open Access AJIBM
Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool 669 margin between the optimal largeness and the variable costcovering largeness has now reduced, however. This margin disappears altogether when the variable cost rises to , when the optimal largeness of 3 will bring in only enough money to cover the variable costs. Figure 8 demonstrates how the optimal largeness asso ciated with the Uniform distribution converges to the same point as that associated with the Double Power distribution. This convergence will be independent of the precise form of the distribution of riskaversion above the lower limit of 0.667 v c 0.369 m that is consequential on max largeness to a less generous value above 3, there will be no consumer prepared to pay the correspondingly higher price, which would be more than 2 base units. 2L 3. While the vendor might want to set his The vendor’s plight will be eased if there are more people with lower riskaversions in the target population. Thus when max and so 26L0.101 m , the ven dor will be able to set his largeness at less generous, higher values throughout the range. Thus he will be able to assign pack 2 a largeness of 3 for a variable cost, v, for the symmetrical Double Power density and for a variable cost, , for a Uniform distri bution over the same range. 0.59 c 0.50 v c Based on Figure 8, we may state generally that, if the vendor expects to be able to sell a 3for2 offer, so that max , and requires that at least his variable costs should be covered, then the variable cost, v, associated with pack 1 must be at or below 66.7% of its selling price. More generally, a multibuy offer that is at least as gen erous as 3for2 cannot be profitable unless: 2L3 67 c 00.6 v c (70) 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 0 0.10.40.50.60.7 Variable dard units) Optimal largeness of pack 2, L(2) 20.30. cost, c v (stan DP, L(2)max = 3 DP, L(2)max = 6 Uniform, L(2)max = 3 Uniform, L(2)max = 6 Just covers cv Golden Rat io s qua red Figure 8. Optimal largeness of pack 2 vs. variable cost for Uniform and Double Power (DP) distributions for risk aversion, with L max 23 and L max 26. Also shown is the locus of the lowest largeness needed to cover the variable cost of pack 2. Figure 9 compares the effects of skewing the symmet rical Double Power distribution for riskaversion towards lower caution and towards greater caution. The minimum riskaversion is 0.101 m in all three cases. The effect of a more cautious population of consumers is to lower the optimal largeness at any given v, while the optimal largeness of pack 2 increases when the population is less cautious. c Pinpointing the optimal largeness for pack 2 requires a knowledge of both v and the distribution of riskaver sion in the target population. While v may be deter mined accurately, the vendor is unlikely to know the ex act form of the latter. It is argued in [19] that riskaver sion stays constant during any decision (whether to buy or not, in this instance), but nevertheless riskaversion will vary with the importance of the decision. See also [22]. It is also likely to vary from person to person, de pendent on both temperament and personal wealth. In the absence of more precise information, it is necessary to employ a range of credible distributions, leading to a plausible range of optimal largeness for a given variable cost, . c c c v The interval for v for which it is sensible to consider multibuy offers in the range, is given by condition (70). If the retailer is to restrict for simplicity his selection of the size and hence price of pack 2 to a ternary choice between BOGOF, Golden Ratio Pricing and 3for2, then, based on Figures 8 and 9, the follow ing approximate v ranges are appropriate for the three possible values of largeness for pack 2: c 22L3 c 2 0.00.4,22.0 0.4 0.6,22.62 0.60.667,23 v v v cL cL cL (71) 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 00.10.2 0.3 0.4 0.5 0.6 0.7 Variable cost, c v (standard units) Optimal largeness of pack 2, L(2) Less c autious Symmetric riskaversion More caut ious Just covers variable cost Golden Ratio squared Figure 9. Optimal largeness of pack 2 vs. variable cost for Double Power d istrib utions for risk aversion, with L max 26 with c = 1 and d = 0.4 (less cautious), d = 2 (symmetric), and d = 8 (more cautious). Open Access AJIBM
Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool 670 The curves suggest that the BOGOF provides the closest approach to optimality over roughly the lower three fifths of the range of variable cost where any of the three multibuy offers is able to generate a profit. The Golden Ratio will then be the best of the three offers in roughly the next three tenths of the profitable range of v, with 3for2 becoming the optimal selection in roughly the highest decile. The boundaries suggested here are all approximate, as is clear from Figures 8 and 9, and should be regarded as indicative only. c Clearly if the entry pack contains a single, discrete item (a white shirt, for example), then fractional quanti ties, as implied by the Golden Ratio offer, will be impos sible. Hence the vendor will need to adopt either a BO GOF or a 3for2 as an approximation. The analysis suggests that the generous BOGOF can be justified on pure economic grounds only when the variable cost is below about 40% of the selling price of pack 1. On the other hand, BOGOF will be attractive to the most cautious of the target market of cautious con sumers, namely those with a riskaversion, 1 , and an offer that attracts the most cautious of the cautious con sumers will also be attractive to rest. The BOGOF is thus ideal for eyecatching promotions that will appeal to all consumers who can be tempted by a multibuy offer. However, it should be offered on a temporary basis only unless the variable costs are low. 10. Conclusions At a time when legal restrictions on quantity promotions are being given serious consideration in the UK, it is vital to have a proper understanding of multibuy promotions such as BOGOF. The theory of Relative Utility Pricing (RUP) developed in this paper takes account of the reac tion of the customers to provide a quantitative economic explanation not only for supermarket promotional offers, but also for more general pricing of packs of different sizes in supermarkets and on the internet. Grounded in standard economic thinking, the RUP model allows the derivation of a general formula that explains clearly and simply for the first time the two most widely used quantity promotions: BOGOF and 3forthepriceof2. It also provides a general explana tion for why the price of a pack 2 that contains twice the amount of a commodity as the entry level pack, pack 1, may need to be priced exactly the same to sell. The paper has linked the RUP model to the theory of isoelastic utility functions, allowing the relationship between riskaversion and packsize ratio to be stated. The same theoretical development also allows a simple relationship to be set down between riskaversion and the demand elasticity for the product in question. Successive doubling of pack sizes has been shown to correspond to a logarithmic utility function, where the riskaversion is unity. This is associated with a neutral demand elasticity, viz. 1 . Finite packsize ratios greater than 2.0 are suitable for individuals with a riskaversion value that lies above zero but no higher than unity – cautious consumers. They are also associ ated with a lower (more negative) values of demand elas ticity: 1 , which implies elastic demand. Cautious consumers are shown to be the only sensible target for quantity promotions. There is no asymptotic limit to their gain in utility from additional quantities of a good, and they may be prepared to pay more for larger packs. Satisfying the desires of the average cautious consumer will result in a ratio of successive pack sizes equal to the square of the golden ratio, namely 2.62, while the priceratio will be the golden ratio, 1.62. It is arguable that this pair of ratios is necessary in order to best satisfy the needs of consumers. Thus the golden ra tio may be regarded as a marketing tool that vendors should consider using in the interests of their customers. Golden Ratio Pricing can also be in the best interests of the vendor, as shown by an analysis of profit maximi sation in the presence of the consumer attitudes reported in the paper. Promotions ranging from BOGOF through Golden Ratio Pricing to 3forthepriceof2 can be prof itable only if the variable cost of pack 1 is less than two thirds of the price of that pack. But higher profits are likely to come from Golden Ratio Pricing than from ei ther BOGOF or 3for2 when variable costs as a fraction of pack 1 price lie in most of the upper half (~0.4 to ~0.6) of the range required to generate profit from any of these multibuy offers. The RUP model has been applied to the prices of gro ceries, where it has been able to illuminate the pricing structure for “value” eggs in supermarkets. A significant degree of validation of the model comes from the close ness of its predictions both for pack size and price. The application of the RUP model to the electronics sector has explained the nonobvious prices of USB memory sticks of increasing capacity. Linked to the bi nary system at the heart of every computer, the ingrained practice of the silicon chip industry to double the capac ity of its products each year is enshrined in the influential Moore’s Law [23]. This has the beneficial sideeffect for the consumer that the price of the new product that is twice as fast or has twice the storage capacity as the old will settle down at the old price. Undoubtedly this has produced great benefits to the consumer. Although prima facie the practice may not be ideal for the vendor, it will be a driver for the exceptionally high rate of technologi cal progress observed in the memory industry. Manufac turers are pressured to create higher capacity products due to the previous generation’s prices quickly descend ing to commodity rates and thus returning low profits. Open Access AJIBM
Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool Open Access AJIBM 671 Hence, the RUP model has explained the structure of quantity promotions, in terms of both the sizes of packs and the prices. Different packs should be able to com mand. This new theory should be of interest and value to vendors, consumers and regulators. 11. Acknowledgements The authors would like to acknowledge numerous fruitful discussions during the preparation of the paper with Mr. Roger Jones, Honorary Fellow, City University London, and with Mr. Edward Ross and Dr Ian Waddington, both of Ross Technologies Ltd. REFERENCES [1] H. Wallop, “BuyOneGetOneFree Offers Are One of the Most Effective Marketing Tools in the Supermarket Industry,” 2008. http://www.telegraph.co.uk/news/uknews/2263645/Food wasteWhysupermarketswillneversaybogoftobuyon egetonefree.html [2] T. Evans, “The End of Buy One Get One Free? Govern ment Considers Plans to Scrap Multibuy Deals to Tackle Obesity,” This is Money.co.uk, 2013. http://www.thisismoney.co.uk/money/news/article23545 83/TheendbuyonefreeGovernmentconsidersplanssc rapmultibuydealstackleobesity.html [3] J. Koten, “Why Do Hot Dogs Come in Packs of 10 and Buns in 8s or 12s?” Wall Street Journal (Western Editio n), 1984, p. 1. [4] E. Gerstner and J. D. Hess, “Why Do Hot Dogs Come in Packs of 10 and Buns in 8s or 12s? A DemandSide In vestigation,” The Journal of Business, Vol. 60, No. 4, 1987, pp. 491517. http://dx.doi.org/10.1086/296410 [5] A. Tabarrok, “Buy One Get One Free,” 2004. http://www.marginalrevolution.com/marginalrevolution/2 004/03/buy_one_get_one.html [6] M. W. Pressler, “Something Racy in the Male. Spilling Victoria’s Secret, and Other Reader Correspondence,” Washington Post, 2004, p. F05. [7] T. Harford, “The Undercover Economist,” Abacus, Lon don, 2011. [8] P. W. Dobson and E. Gerstner, “For a Few Cents More: Why Supersize Unhealthy Food,” Marketing Science, Vol. 29, No. 4, 2010, pp. 770778. http://dx.doi.org/10.1287/mksc.1100.0558 [9] D. Kahneman and A. Tversky, “Prospect Theory: An Analysis of Decision Under Risk,” Econometrica, Vol. 47, No. 2, 1979, pp. 263291. http://dx.doi.org/10.2307/1914185 [10] R. H. Thaler, “Mental Accounting and Consumer Choice,” Market ing Scienc e, Vol. 4, No. 3, 1985, pp. 199214. Re published 2008 in Marketing Science, Vol. 27, No. 1, 1985, pp. 1525. [11] P. Thomas and A. Chrystal, “Generalized Demand Densi ties for Retail Price Investigation,” American Journal of Industrial and Business Management, Vol. 3, No. 3, 2013, pp. 279294. http://dx.doi.org/10.4236/ajibm.2013.33034 [12] P. Thomas and A. Chrystal, “Retail Price Optimization from Sparse Demand Data,” American Journal of Indus trial and Business Management, Vol. 3, No. 3, 2013, pp. 295306. http://dx.doi.org/10.4236/ajibm.2013.33035 [13] P. Thomas and A. Chrystal, “Using Relative Utility Pric ing to Explain Prices in Supermarkets and on the Inter net,” American Journal of Industrial and Business Man agement, in Press. [14] R. G. Lipsey and K. A. Chrystal, “An Introduction to Positive Economics,” 8th Edition, Oxford University Press, Oxford, 1995. [15] Amazon.co.uk, 2009. http://www.Amazon.co.uk/tag/usb%20memory%20stick/ products/ref=tag_tdp_sv_istp#page=2:sort=relevant [16] Mysupermarket.co.uk, 2009. http://www.mysupermarket.co.uk/Shopping/FindProducts .aspx?Query=Eggs [17] J. W. Pratt, “Risk Aversion in the Small and in the Large,” Econometrica, Vol. 32, No. 12, 1964, pp. 122 136. [18] K. J. Arrow, “Social Choice and Individual Values,” John Wiley and Sons, New York, 1951. [19] P. J. Thomas, “An Absolute Scale for Measuring the Util ity of Money,” Journal of Physics: Conference Series, Vol. 250, No. 1, 2010, Article ID: 012045. http://dx.doi.org/10.1088/17426596/250/1/012045 [20] H. M. Treasury, “Green Book,” 2009. http://greenbook.treasury.gov.uk/annex05.htm [21] T. Crilly, “50 Mathematical Ideas You Really Need to Know,” Quercus, London, 2008. [22] P. J. Thomas, R. D. Jones and W. J. O. Boyle, “The Lim its to Risk Aversion. Part 2: The Permission Point and Examples,” Process Safety and Environmental Protection, Vol. 88, No. 6, 2010, pp. 396406. http://dx.doi.org/10.1016/j.psep.2010.07.001 [23] G. E. Moore, “Cramming More Components onto Inte grated Circuits,” Electronics, Vol. 38, No. 8, 1965, pp. 114117. Reprinted Proceedings of the IEEE, Vol. 86, No. 1, 1998, pp. 8285. http://en.wikipedia.org/wiki/Moore%27s_law
Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool 672 Appendix A. Singular Homogeneity Consider the following transformation between two func tions: 21 for all numbers, mxmHxm (A.1) The function, 1 x 2 , may be seen to be transformed into the function, mx , by multiplying both its input, x, and its output, 1 x, by the constant, m. Now sup pose that is a homogeneous function of degree one. In this case, by the property of firstdegree homoge neity: 1.H 11 for all numbers, mxmH xm (A.2) Linear operators possess this property of homogeneity of degree one. Comparing Equations (A.1) and (A.2), it is clear that if is a homogeneous function of degree one, then the second operator will be the same as the first: . 1.H ..H H 21 Now consider Equation (7), which may be written in terms of a continuous variable for extent, x, as: 010 0 n uLnxLnux xb 1 (A.3) By Equation (A.3), the function, , may be transformed into the function, , by multiplying both the input, x, and the output, , by the con stant, . Clearly Equation (A.3) bears a similarity to Equation (A.1), but it differs in the fact that Equation (A.3) applies only for the single value of m: 10 .u . 10 ux 0n u Ln mLn. Hence we may describe Equation (A.3) as having the property of “singular homogeneity”. Under singular homogeneity, and 0. n u 10 .u will be different, nonlinear functions, except for the lim iting case when is linear in its argument. Such a situation will occur only when riskaversion is zero, when the riskneutral utility function emerges. (This contention becomes evident after substituting 10 .u 0 into equations (A.5) and (A.6) below. At this point, the function, , has become homogeneous of degree 1, because the utility function is linear in its argument.) 10 .u An alternative but equivalent formulation of Equation (A.3) arises after replacing the extent variable, x, by a new variable of extent, b, where so that bLnx bLn. Then equation (A.3) becomes: 010 0 nb ub LnubLnb Ln 1 (A.4) As an example, let us assume a packsize ratio, 3 , as used by Sainsbury for its value eggs. From Equation (46), the matching riskaversion is 0.369 . Using the utility function given in Equation (37) to represent the utility that the person will gain from using the contents of pack 1 containing an amount, , of good, B, we achieve, after noting that : 1 b 11L 11 0.631 10 1 11 1.5850 111 bb ubb bb (A.5) Meanwhile the utility of that the person or consortium of people will gain from using the contents of pack n may be found by combining Equations (A.4) and (A.5) to give 11 0 0.369 0.631 1 1 11 1.585 0 nbb u bLnLn Ln Lnbb Lnb (A.6) Figure 10 shows the utility gain from pack 1, 10 u , and that from pack 2, 20 u , plotted against the amount of the good, B, measured in standard units of the contents of back 1, so that 11b and 2. These graphs might represent the case of one person buying pack 1 containing 6 eggs and a family unit of three buying pack 2 containing 18 eggs. The final utility gain for the indi vidual, at b = 1, will be 1.585, while the final utility gain for the family of three will be 3 × 1.585 = 4.755, at b = 3 standard units. The righthand arrows on the graph show how this result may be derived by multiplying by the factor, 3b 23L , both the x and the y coordinates asso ciated with the complete consumption of pack 1. Because Equation (A.3) has been formulated in con tinuous terms, we may apply the same transformation within the contents of pack 1. Hence we may take the case where the individual has used 1/6 of his pack 1, one egg in this case, and multiply each of the coordinates of 10 u , namely (0.167, 0.51) by a factor of three to give the corresponding coordinates (0.5, 1.53) of 20 u . These show that the utility gained by the family of three in consuming half a standard unit, that is to say 3 eggs, is three times greater than that achieved by the single per son eating one egg. The smaller arrows on the lefthand side of the graph indicate the graphical process. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.51 1.5 2 2.53 3.5 Pack contents in standard units Utility gain, u 10 , u 20 Figure 10. Graphical analysis of singular homoge ne ity. Open Access AJIBM
Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool Open Access AJIBM 673 21 Appendix B. The Range of Maximum Acceptable Price (MAP) for Pack n But, by Equation (3), 020n , so that we may sim plify the righthand side of condition (B.7) to: pp 20 0200 max, n pp pp B.1 The Highest MAP for Pack n n (B.8) Comparing Equations (3) and (17), it is clear that the price, 1n, resulting from a comparison of pack n and pack 1 will be no greater than the price arising by com paring pack n with pack 0: pSubstituting from condition (B.8) into condition (B.6), the price arising from the comparison of pack n with pack 2 will obey: 20 for 2 nn pp n (B.9) 10 for 1 nn pp n (B.1) If we put n = 3 into Equation (22), the MAP for pack 3 emerges as It should be emphasised here that the comparison price, nk , arising from comparing pack n with smaller packs will depend on the value of pack largeness and the achieved prices of for the lower packs, but not on the opinion of the individual, allowance for which comes from the weightings, . p i k w 30 301312 32 iii i pwpwpwp (B.10) Since by condition (B.1), 3130 , and by condition (B.7), pp 32 30 pp , the highest value of any individual’s MAP, , will occur when , 3 i p0 w110w and 20w , so that Putting n = 2 into Equation (22) gives the individual’s MAP for pack 2 as 330 for all i ppi 30 (B.11) 20201 iii pwpwp (B.2) Moreover, since , it follows that the achieved price for pack 3, , will obey 33 i pp 3 p By condition (B.1), 2120 , and so, from equation (B.2), the highest value of the MAP, , for any individual will occur when that person sets pp 2 i pp 0 i w 20 1 and . Hence we may write: 10 i w 3 pp (B.12) where, from equation (3), 30 1 3pLp i p (B.13) 220 for all i ppi 0 (B.3) It is clear that this process may be continued indefi nitely, so that for any pack, , the upper limit of MAP will be defined by 1n The achieved price, 2, must be less than or equal to the highest MAP in the customer cohort, and so it follows that p 0for all i nn pp (B.14) 22 pp (B.4) where, from Equation (3), In other words the upper limit, , for MAP for pack n is given by: maxn p 20 1 2pLp (B.5) max 01nn ppLn (B.15) We may now consider the highest MAP for pack 3. Putting k = 2 into Equation (20) and then using condition (B.4) gives the price of pack n coming from a comparison with pack 2 as: B.2 The Lowest MAP for Pack n 22 1 11 20 020 max 1.0,1 2 max 1.0,12 2 max2 ,2 max ,2 n n Ln pp L Ln Lp L LpLnL p pp pn A price for pack n, n, equal to the price of the next lowest pack, 1n p p , is possible, as demonstrated by the buyonegetonefree promotion and as explained above. However, bearing in mind the economic principle that more will be preferred to less, we may presume that the vendor will not allow the price, n, of pack n to fall below the price, p (B.6) , of pack n – 1. 1n p The lower price that the vendor would countenance, 1nn pp , will have the effect that any potential cus tomer for pack n who has a MAP lower than 1n p will exclude himself from the cohort of customers of interest to the retailer. Thus the lowest MAP for pack n will be simply the price, 1n p , of the pack next down in size: where Equation (B.5) has been used twice in the devel opment. Clearly, for any strictly positive values, : 20 0 ,n pp min 1n pp n (B.16) 20 02020 0 max ,max , nn pp ppp (B.7)
