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1 Thomas, P.J. & Chrystal, A. (3). Retail rice otimisation from sarse demand data. American Journal of Industrial and Business Management, 3(3), doi:.436/ajibm City Research Online Original citation: Thomas, P.J. & Chrystal, A. (3). Retail rice otimisation from sarse demand data. American Journal of Industrial and Business Management, 3(3), doi:.436/ajibm Permanent City Research Online URL: htt://oenaccess.city.ac.uk/699/ Coyright & reuse City University London has develoed City Research Online so that its users may access the research oututs of City University London's staff. Coyright and Moral Rights for this aer are retained by the individual author(s) and/ or other coyright holders. All material in City Research Online is checked for eligibility for coyright before being made available in the live archive. URLs from City Research Online may be freely distributed and linked to from other web ages. Versions of research The version in City Research Online may differ from the final ublished version. Users are advised to check the Permanent City Research Online URL above for the status of the aer. Enquiries If you have any enquiries about any asect of City Research Online, or if you wish to make contact with the author(s) of this aer, lease the team at

2 American Journal of Industrial and Business Management, 3, 3, htt://dx.doi.org/.436/ajibm Published Online July 3 (htt:// 95 Phili Thomas, Alec Chrystal School of Engineering and Mathematical Sciences, City University London, London, UK; Cass Business School, City University London, London, UK. jt3.michaelmas@gmail.com Received March 4 th, 3; revised Aril 4 th, 3; acceted May 4 th, 3 Coyright 3 Phili Thomas, Alec Chrystal. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. ABSTRACT It will be shown how the retailer can use economic theory to exloit the sarse information available to him to set the rice of each item he is selling close to its rofit-maximizing level. The variability of the maximum rice accetable to each customer is modeled using a robability density for demand, which rovides an alternative to the conventional demand curve often emloyed. This alternative way of interreting retail demand data rovides insights into the otimal rice as a central measure of a demand distribution. Modeling individuals variability in their maximum accetable rice using a near-exhaustive set of demand densities, it will be established that the otimal rice will be close both to the mean of the underlying demand density and to the mean of the Rectangular distribution fitted to the underlying distribution. An algorithm will then be derived that roduces a near-otimal rice, whatever the market conditions revailing, monooly, oligooly, monoolistic cometition or, in the limiting case, erfect cometition, based on the minimum of market testing. The algorithm given for otimizing the retail rice, even when demand data are sarse, is shown in worked examles to be accurate and thus of ractical use to retail businesses. Keywords: Otimal Price; Monooly; Monoolistic Cometition; Oligooly; Sarse Demand Data; Retail. Introduction The markets faced by retailers in different sectors may san the range of market categories from erfect cometition, through oligooly, monoolistic cometition to monooly. But the roblem of setting the otimal rice faces retailers in all market categories. Selling to multile customers, they need to offer a rice that is common to all. This study aeals to standard economic theory to hel illuminate the osition of the retailer as he uses whatever information is available to him in order to set the rice of each roduct so as to maximize his rofit. The end oint will be an algorithm for setting the rice of a roduct that will be close to the otimum, whatever the market conditions revailing, based on the minimum of market testing. Finding the rice to maximize rofit would be relatively easy if the retailer knew the maximum rice accetable to each of his customers interested in buying the roduct. Any given rice would imly that the retailer would gain income from all those whose maximum accetable rice (MAP) lay above this level. His turnover having now been decided, the retailer could then calculate first his costs and then, by subtraction, his rofit. Alying the same rocedure to all or a selection of rices would quickly show the retailer what rice he should set to generate the highest rofit. Obtaining the necessary information on the MAP for each customer could be attemted via a market survey. Normalizing the results by dividing by the number taking art would give a robability distribution for MAP, which may be called the demand density for comactness. A demand density found in this way would, of course, be subject not only to the inaccuracies associated with any samling oeration but also to those associated with a samle that might be less than erfectly reresentative. Moreover the survey would need to be alied to each roduct and reeated at regular intervals. Such a large and ongoing data collection exercise, although feasible in rincile, would be imractical to imlement. The alternative, and the aroach used here, is to ostulate a wide range of ossible demand densities, h(), and then use standard economic theory to investigate the relationshi between the otimal rice and other central measures of MAP, which may lend themselves to use in a rice-otimization algorithm. The restriction on a robability distribution, namely that the area under the curve must be unity, viz. hudu, aids in this Coyright 3 SciRes.

3 96 rocess, allowing near-exhaustive testing of ossible demand scenarios using a finite number of candidate demand densities. Two sets of candidate demand densities are used. The first assumes that the MAP is roortional to the ability to ay as measured by ost-tax household annual income u to one of a set of ercentile levels. The second set consists of the Double Power demand density defined in Section 7, with different coefficients locating its mode anywhere between and the highest rice that anyone is reared to ay for the item (normalised to units in the examles considered). A third, generalized demand density is also considered, namely the Rectangular demand density. It is the demand density assumed by default by economists when they draw a straight-line, downward-sloing demand curve, and it has articular significance because the otimal rice and the mean rice are the same. The Rectangular is, moreover, the simlest demand density able to fit the mental model of a retailer who knows ) the lowest rice at which he would countenance selling the roduct and ) the highest rice he could get from his customers before sales became negligible. The Rectangular demand density may be fitted with good accuracy to demand densities in both of the candidate sets. The matched Rectangular demand density may then be regarded as a roxy for the underlying distribution, a fact that will be exloited in the rice otimization algorithm introduced later.. The Demand Density Underlying the Conventional Demand Curve The conventional demand curve may be regarded as a lot of rice,, against the fraction, S, of the target market reared to ay that rice or more [-3]. It is usual for the marginal revenue and the variable cost to be lotted also, resulting in a diagram similar to Figure, in which it is assumed that the variable comonent of cost is linear. While the conventional demand curve uses the cumulative robability,, as one of its axes, there are advantages in recasting S in terms of its fun- Price, m c v S Price (average revenue), Marginal revenue, + Sd/dS Marginal cost, cv Fraction, S(), of target oulation reared to ay a rice of at least Figure. Conventional demand curve. damental demand density, h, since this fraction is the integral of robability density above rice, : () S hudu hudu where the second develoment follows from the fact that hudu. Although the same information is carried by S() as by h(), the fact that the former is based on an integration of the latter means that it is a filtered version, so that the detail is more difficult to ick out. Thus the demand density curve of h() vs. offers finer discrimination than the conventional demand curve of vs. S(). A further advantage of the demand density curve is that, as exlained in the next Section, it allows the otimal rice to be found naturally. The lesser discrimination inherent in the cumulative robability, S(), exlains why a straight-line aroximation to the curve of vs. S() can be used routinely in economics text books, even though such a straight-line demand curve will hold true only when the demand density is Rectangular. For what shows u as a major discreancy from uniformity in the grah of h() vs. reduces to a minor deviation in the grah of S() vs.. For examle, in the case where a symmetrical underlying demand density is aroximated by a Rectangular demand density, it is clear from comaring Figures (a) and (b) that the difference between the two distributions becomes much less marked when the conventional demand curve is used. Monooly is the simlest of the selling situations where multile customers are involved, since the retailer need be concerned only with the reactions of customers and not of other suliers. While results as derived in Sections 3 to 9 may be understood in this light, it will be shown in Section that the results will aly equally to all the basic forms of interaction between the retailer and his customers: monooly oligooly monoolistic cometition as well as, in their limiting form, to erfect cometition. 3. Mathematico-Economic Model of Profit Maximization A retailer will, in general, face a differentiated market, with different eole being reared to ay a different maximum rice for the same good, as illustrated in the demand density curve. The term, uniconsumer, might be used to denote a consumer reared to buy one but only one item if the rice is right. Then a erson, a mul- Coyright 3 SciRes.

4 97 Probability density, h() Price, Maximum accetable rice, (a) Double Power Matched Rectangular Fraction, S(), of target oulation reared to ay a rice of at least (b) Figure. Aroximating a symmetrical robability density for demand by a Rectangular distribution. (a) Comarison of demand density curves; (b) Comarison of demand curves. ticonsumer, who will buy more than one item if the rice is right may be reresented, as far as his economic behaviour is concerned, as multile, identical uniconsumers. Suose that the retailer sets the rice at some value,. The good will be bought by the fraction of uniconsumers, S(), with a MAP at or above, as given by Equation (). In the rest of the aer we shall use the word, consumer, in lace of the more exact uniconsumer, simly to make it less cumbersome to read. The total number of sales will be NS(), where N is the size of the oulation of consumers under consideration. Given a common rice,, the retailer s income will be NS(). Following a standard economic model widely used in business, it will be assumed that the retailer s costs comrise a linear variable cost, cv ( er item), and a fixed cost indeendent of the number of items sold, CF ( ). His rofit,, will be the difference between income and costs: NS NS c v C F () The retailer will seek to maximize this rofit, which, for a constant size of target oulation, N, is equivalent to maximizing the average rofit er consumer, : CF cv hudu N (3) N where use has been made of Equation () in the second ste. Since no-one has infinite resources, everyone must have a MAP that lies beneath some maximum conceivable value, m, imlying that hu du. Hence, m we may rewrite Equation (3) as m C c h u du v F (4) N We may find the maximum value of rofit,, by differentiating Equation (4) with resect to rice,, and then setting d d. At this oint, *, the otimal selling rice, which it the rice that will generate the retailer the greatest rofit. Alying the rules of calculus: m d C F d cv hudu d d N m m d hudu cv hudu hudu d Differentiating the contents of the final bracket and setting the whole to zero gives the otimal rice as the solution,, of (5) m hudu cv h (6) 4. The Retailer s Estimate of Demand Density: Sarse Data The mathematics given in the revious Section rovide a way of calculating the otimal rice that may be slightly more sohisticated than the method exlained in the nd aragrah of the Introduction, but they do not get around the fact that solving Equation (6) still requires a knowledge of the demand density, h(). But in the absence of reeated market surveys, the data available to the retailer are likely to be sarse and fragmentary, in which case he will need to rely on a largely intuitive feel for the demand density characterizing his target market. Consider now the likely minimum information available to the retailer. We may assume that he will know his variable cost er item, c v. He will have no interest in selling the item at a rice below this level, since sales at a lower rice will not make a contribution to offsetting his fixed costs but will, on the contrary, increase his loss. Hence he will regard his target oulation of consumers as one containing only a negligible fraction for whom the MAP is less than c v. The variable cost er item may then be regarded as defining the lowest rice in the retailer s mental model, :. (It may be noted a a c v Coyright 3 SciRes.

5 98 that the variable cost, c v, is not a wholly exogeneous cost imosed on the retailer. Its value will reflect choices made by the retailer and those in his suly chain, all of whom will be influenced by ercetions of what the market will bear.) There are cases where the cost of the good is dominated by fixed costs, and the variable cost er item is essentially zero. Hence the retailer s mental model may include a as a limiting case. Meanwhile, a knowledgeable retailer should have an idea of the highest MAP, b, above which his total sales will be negligible: almost nobody will ay more than b. These two rice levels, a and b, are sufficient on their own to generate for the retailer a simle mental model, g(), of the true demand density, h(). The aroximating robability density, g(), will be uniform between a and b and zero elsewhere a Rectangular distribution. The simle mental model just ascribed to the retailer working with sarse data will generate a straight-line demand curve, as noted in Section. Thus it coincides with the default model frequently used by economists when considering the roblem of demand. 5. Proerties of the Rectangular Demand Density 5.. Mean, Median and Mode The Rectangular demand density, g for a g for a b a for b The mean value of the MAP, average: g d a b, has the form: b (7), is then the weighted b a b d (8) a a Because the Rectangular distribution is symmetrical, the mean and median are equal. By mathematical convention regarded as unimodal, the Rectangular distribution may be seen as having a mode anywhere in the range, (, ). b 5.. The Otimal Price, * Alying Equation (6) with the Rectangular robability density, g, relacing h and using the equality of the variable cost er unit and the retailer s lowest MAP of interest, : a b d a b a d d b a b a b a b a (9) So that, denoting the otimal rice by *: a b * () Thus, using the Rectangular demand density likely to be used initially by the retailer as well as by economists in their first consideration of demand, the imortant result emerges that the otimal rice and the mean rice will coincide: * () Since the mean and median are equal in a symmetrical distribution, it follows also that * med () where med is the median of the retailer s Rectangular demand density. It must be considered unlikely, however, that the true demand curve will be exactly straight, nor, by the same token, will the demand density curve be recisely Rectangular. Hence, while the value, a b, may be close to otimal, it will not be the true otimum. To find out how near to the true otimum the mean of the Rectangular distribution is likely to be, we may examine how far it can be made reresentative of a wide variety of underlying demand densities Matching the Rectangular Demand Density to the Underlying Demand Density If the true, underlying distribution were known, it would be ossible to model the retailer s intuitive identification of the lower and uer limits of his Rectangular distribution by the mathematical rocedure of minimizing the integrated square error between the Rectangular distribution and the underlying distribution. In ractice, the underlying distribution is not likely to be known, but a near-exhaustive survey may be made using candidate distributions with characteristics sanning the ossible robability sace. Once the Rectangular demand density, g(), has been matched to the underlying demand density, h(), the mean of g(), which we shall call the matched Rectangular mean, becomes a roerty of h(). 6. Demand Density: Candidate, When the MAP Is Proortional to the Ability to Pay Prima facie, one reasonable assumtion is that the MAP is conditioned by, and, in the simlest case, roortional to the ability to ay. Demand density may then be related to ost-tax household annual income [4]. It is further suosed that the rice of commodities that are needed and obtained by almost everyone in the oulation will be determined by the attitudes and decisions of those with incomes u to some ercentile level, θ. Those with Coyright 3 SciRes.

6 99 incomes above the th ercentile are then considered to be rice-takers for these goods. (For examle, very wealthy eole may have their shoing done for them, and their less wealthy agents may tend to aly their own ersonal judgements on what constitutes value for money). The ercentage of eole,, determining the rice of each commodity may vary according to commodity, and moreover, that ercentage may not be known with any recision. To coe with this situation, we have allowed for to take a range of ossible ercentages, from 5% to 99%. Table gives cumulative robabilities for income; it may be noted that the last column, where %, is incomlete due to lack of IFS data beyond the 99 th ercentile. Figure 3 shows the case for the 85 th ercentile cohort. (The highest MAP that anyone in the cohort will assign, m, is set to units in each case, a convention that will be followed throughout the aer.) It may be remarked immediately that the distribution shown in Figure 3 is interior unimodal, in the sense that the mode lies strictly within the interval mode m. Figure 4 lots the normalised mode, median and mean as well as the otimal rice for the underlying distribution, h(), as well as the mean of the retailer s matched Rectangular distribution, g(), versus ercentile for all the ercentiles listed in Table. Clearly the otimal rice, based on the underlying distribution, h(), is distinct from all the other measures. However, both the median and the mean of the underlying distribution, h(), are reasonable aroximations to the otimal rice, the mean erforming better for lower ercentiles, the median Probability densities, h(), g() Maximum accetable rice, Figure 3. Demand density, h(), for 85 th ercentile cohort. Also shown is the matched Rectangular demand density, g(). Table. UK ost-tax household income 9: Cumulative robability, F y,, u to the th ercentile income (equivalised, based on a coule with no children). House-hold income, y (.a.) Cumulative robability, F y, θ = 5% θ = 59% θ = 67% θ = 78% θ = 85% θ = 93% θ = 96% θ = 99% θ = % , , , , , , , , , , , , , Coyright 3 SciRes.

7 3 Mode, otimal, median and mean for underlying distribution; Rectangular mean. (Normalised) Mode Otimal Median Mean Rectangular mean Percentile Figure 4. Mode, otimal, median and mean of underlying demand density, h(); mean of Rectangular demand density, g(). Plotted versus income ercentile. doing better for ercentiles above about 75%. The rootmean-squared error is about 5% for the underlying median and 6% for the underlying mean. The matched Rectangular mean is a slightly worse aroximation to the otimal rice for the underlying distribution, being an average of about % too high over the range considered, rising to % in the worst case of the 85 th ercentile. The mode gives a relatively oor aroximation to the otimal rice. However, it rovides a useful way of characterizing the demand density. Figure 5 shows the otimal, mean, and matched Rectangular mean lotted against the mode of the underlying distribution. Larger deviations between the mean and the otimal rice are evident when the mode is located centrally in the range. 7. Demand Density: Candidate, the Double Power Demand Density The Double Power demand density introduced here is defined on non-negative values of MAP,, by: h c d a b for for m m (3) where the coefficients, a, b, c and d are non-negative. The Double Power demand density has the desirable roerty that, through suitable selection of its arameters, a, b, c and d, its mode may be located anywhere between zero and the maximum conceivable value: mode m, thus ensuring the necessary coverage of the robability sace. 7.. Mode at the Zero Boundary The mode lies at the zero boundary,, when c =. The fact that the matched Rectangular demand density has a imlies a zero variable cost: c v. The curve of h is strictly convex when d, linear when d and strictly concave when d. See Figures 6(a) and (b). 7.. Mode at the Maximum Boundary, P m The mode occurs at the maximum rice, m, when b = (excet for the limiting case where c is also zero: if b c, the robability distribution becomes uniform Otimal, mean, median and Rectangular mean Otimal Mean Median Rectangular mean Mode of underlying distribution Figure 5. Otimal, mean, median and rectangular mean versus the mode of the underlying distribution. Probability density, h() Probability density, h() Maximum accetable rice, (a) Maximum accetable rice, (b) Figure 6. Matching a Rectangular demand density to convex and concave Double Power demand densities when the mode is.. (a) Convex Double Power demand density: c =, d =.5; (b) Concave Double Power demand density: c =, d = 8.. Coyright 3 SciRes.

8 3 on (, m ), with no unique mode). The curve of h is strictly concave when c, linear when c and strictly convex when c (Figures 7(a) and (b)) Mode Strictly Interior The mode will be strictly interior when the coefficients, a, b, c and d are all ositive. See Figure 8. Once c has been set, a suitable selection of d allows the mode to be located anywhere in the range between and m, with the mode increasing as d increases. Figure 8(c), where c = 8 and d = 8, shows also how a high value of the ower, c, used in conjunction with a high value of the ower, d, can simulate aroximately the situation where the effective lowest MAP is above zero roughly = 5 in this case. A large majority of oulation is reared to ay 5 units or more for the good. 8. Results Using the Double Power Demand Density 8.. Mode at the Zero Boundary Figure 9 shows the behaviour of the otimal rice, the mean rice, the median and the matched Rectangular mean rice, as the ower, d, is varied from 3 to 3. While the otimal is distinct from the other central mea- Probability density, h() Probability density, h() Maximum accetable rice, (a) Maximum accetable rice, (b) Figure 7. Matching a Rectangular demand density to concave and convex Double Power demand densities when the mode takes the maximum value, m. (a) Concave Double Power demand density: b =, c =.5; (b) Convex Double Power demand density: b =, c = 4.. Probability density, h() Probability density, h() Probability density, h() Maximum accetable rice, (a) Maximum accetable rice, (b) Maximum accetable rice, (c) Figure 8. Matching a Rectangular demand density to a Double Power demand density with a strictly interior mode. (a) Double Power demand density. c =, d =.5; (b) Double Power demand density. c =, d = 4; (c) Double Power demand density. c = 8, d = 8. Otimal, mean and Rectangular mean, normalised to (,) E-3.E-.E-.E+.E+.E+.E+3 Power, d Otimal Mean Rectangular mean Figure 9. Boundary mode at =. Varying the ower, d. Coyright 3 SciRes.

9 3 sures, the mean and the matched Rectangular mean are reasonable aroximations to it over the whole range, with a maximum error of about %. 8.. Mode at the Maximum Boundary Figure shows the behaviour of the otimal rice, the mean rice, the median and the matched Rectangular mean rice, as the ower, c, is varied from -3 to 3. The mean and the matched Rectangular mean show a good corresondence with the otimal over the whole range. The maximum error is about 4% Mode Strictly Interior Figure shows the otimal rice, the mean rice and the matched Rectangular mean rice against the mode when the arameter, c, is equal to. The otimal rice is distinct, but each of the other central measures acts as a reasonable aroximation to it over the whole range of modes for all the values of c examined. The maximum discreancy, of about 9%, occurs when the mode is mid-range. Otimal, mean and Rectangular mean Power, c Otimal Mean Rectangular mean Figure. Boundary mode at = m. Varying the ower, c. Otimal, mean and Rectangular mean Otimal Mean Rectangular mean Mode of underlying Double Power distribution, c = Figure. Otimal, mean and Rectangular mean vs. underlying mode. Distribution with interior mode, c = As an examle, for the case shown in Figure 8(c), where just about everyone is reared to ay 5 units or more for the good, the otimum rice is 9.6 units, the mean is 8.93 while the matched Rectangular mean is 9.4 units. Clearly the matched Rectangular mean can give a very good aroximation to the otimal rice Summary A very large number of Double Power demand densities have been examined, with modes sanning the full range of MAP: m. While the otimal rice is distinct from each of the mean, the median and the matched Rectangular mean, both the mean and the matched Rectangular mean offer good aroximations to the otimal. The discreancy is less than % in almost all cases, with 5% being more tyical. 9. Matching of the Rectangular Demand Distribution to the Two Candidate Demand Distributions It is reasonable to suose that a Rectangular distribution may be fitted to any conceivable, unimodal demand density. Since some form of demand density must be valid in all market situations, it is reasonable to ostulate, under the mild restriction that it must be unimodal, that the true demand density may be aroximated by the Rectangular demand density matched to it. The mean of the matched Rectangular demand density may then serve as an aroximation to the otimal rice of the underlying demand density. The analysis using the two sets of demand densities, the first based on the ability to ay and the second on the general, Double Power distribution, suggests that the degree of aroximation is likely to be relatively small, tyically of the order of 5%. These results rovide, inter alia, a degree of validation for the straight-line demand curve conventionally cited by economists, since this is equivalent to a Rectangular demand density. The results may be exloited further to use market testing data to identify the matched Rectangular demand density rather than the underlying demand density, which would be more difficult to do. A simle algorithm can then be develoed that is able to give a good aroximation to the otimal rice after only a single erturbation of rice.. Extension of the Results to Situations Other than Monooly.. Monoolistic Cometition Monoolistic cometition is held to occur when there are many firms roducing different brands of a similar roduct, when those firms may enter and leave the market freely and when a new market entrant will take sales Coyright 3 SciRes.

10 33 from existing retailers in roortion to their current market share []. A firm, let us call it firm, will set its otimal rice without taking into account the individual reactions of its cometitors. However, their resence means that the downward sloe of the demand curve it faces is exected to be gentler than if it had a monooly. This is because the uer rice ertaining at S will be lower than in the monooly situation, while the rice at S will be unchanged at cv (assuming that firms will not sell at less than variable cost). Hence the average sloe between the two oints must be less stee. In terms of the demand density curve, the maximum conceivable rice, m, valid in the monooly situation, will have been reduced to a lower value, m. But the results set out above were for a general value of m, and will therefore aly equally when m is relaced by. m.. Oligooly This is a common situation in a modern economy, where, for examle, food shoing is dominated by a small number of large suermarket chains. Whatever the details of the oligoolistic interaction, it is reasonable to suose that some sort of demand curve will aly, with a sloe that we can exect to be generally downward sloing even if we might have difficulty secifying its recise shae. In terms of the demand density, we can exect the robability density, h(), to exist over some finite, non-zero range of rices. The results of Sections to 9 suggest that it will be ossible to aroximate such a demand density curve reasonably well by a Rectangular distribution, resulting in a straight-line demand curve. Cournot analysed, in 838, the situation of a duooly where each firm chose the size of its outut based on the assumtion that the other firm would hold its throughut constant [5]. The effect on the monooly demand curve, that is to say the curve that each firm would see if it held a monooly, is simly to shift it to the left by the fraction of the market held by the other firm, rovided the overall market size stays constant. For examle, Figure shows the effective demand curve facing firm if firm is sulying 5% of the market. Firm will now see a lower maximum feasible rice, m, and will have to work with only 75% of the total market; Figure can be seen to be fully analogous to Figure. In 883 Bertrand claimed that the behaviour of an oligooly could be understood better on the assumtion that firm would see firm as keeing its rice constant, rather than its outut [6]. However, this would lead to destructive cometition, driving rice to the level of short-term marginal cost, so that fixed costs would not be covered. More recent research has suggested that firms avoid this loss-making situation by curbing their ambi- Price, m 8 ' m 6 4 c v Market demand Marginal cost, cv, firm Effective demand, firm Marginal revenue, firm..4.6 S'.8 Fraction, S(), of target oulation reared to ay a rice of at least Figure. The demand curve of firm in the Cournot model. tion to suly the whole market by deliberately limiting their maximum feasible outut. Under these conditions, Cournot s model offers a useful insight into oligooly behaviour, while yielding a clearly defined demand curve of the shae we have considered reviously. Half a century later, Hall and Hitch [7] and Sweezy [8] came u indeendently with similar secifications for the general form for an oligoolistic demand curve. Sweezy suggested that, in the case of oligooly, the effective demand curve would be concave, with a kink linking two downwardly inclined lines he drew as essentially straight, but with the second line having a more negative sloe. The kinked demand curve may be seen as an asymmetric combination of the Bertrand and Cournot assumtions and suggested that oligoolistic rices would tend to be sticky. This made the construct controversial amongst some economists such as Stigler [9], who considered the raid adjustment of rices a fundamental economic tenet. The authors give no oinion either way, but include the kinked demand curve for the sake of argument and comleteness. The iecewise linear, kinked demand curve, shown in Figure 3(a), has a corresonding demand density curve that exhibits a ste, with the ratio of the robability densities before and after the ste being the ratio, r, of the sloes of the demand curve before and after the kink. See Figure 3(b). Figure 4 shows a contour lot for a constant ratio of the mean rice to the otimal rice for a kinked demand curve. It is clear that the mean rice lies within % of the otimal rice for a wide range of values of, rovided the ost-kink sloe multilier, r, is or less. Equally, the mean rice will be within % of the otimal for a wide range of values of r, rovided the ratio of uer and lower rices,, is or less. If both and r are below, then the discreancy between the mean rice and the otimal rice must be be- Coyright 3 SciRes.

11 34 Price, Probability density, h() MAP Marginal revenue cv Fraction, S(), of target oulation reared to ay a rice of at least k r.k (a) Maximum accetable rice, (b) k Figure 3. The kinked demand curve and its corresonding demand density. k, r, α and β are defined in Figure 3(b). (a) Demand curve when: r, 6,.5 ; (b) Corresonding demand density. / Ratio of demand curve sloes, r Figure 4. Contour lot of constant ratio of the mean rice to the otimal rice for kinked demand curve. low 5%. For the examle shown in Figure 3, where.5 and r =, then the mean is just 3.% below the otimal rice..3. Perfect Cometition In the case of erfect cometition, the demand curve is horizontal, which imlies a single rice set by the market, not subject to change by the retailer. The corresonding demand density curve will be an imulse at the market rice a rectangular ulse of unit area with a width aroaching zero and a height aroaching infinity, a function known to hysicists as a Dirac delta function. Self evidently all central measures, such as the mean and median of the distribution and the mean of the matched Rectangular distribution, will converge to a single value under these conditions, which value will also constitute the otimal rice..4. Summary of the Results for Situations Other than Monooly Demand curves that are continuous and downward sloing will characterize both monoolistic cometition and oligooly in the case where Cournot s theory gives an adequate characterization. These will imly a demand density, h(), that will exist over some finite, non-zero range of rices. It will be ossible to aroximate such a demand density by a Rectangular demand density, the mean of which (the matched Rectangular mean) can be exected to aroximate the otimal rice reasonably well. Another ossibility has been examined in the case of oligooly, namely the kinked demand curve. This has been shown to corresond to a steed demand density curve. It is found that the mean of the kinked robability distribution will be similar to the otimal rice for a lausible range of its rincial arameters. In the case of erfect cometition, the roosition that the mean and the matched Rectangular mean will aroximate the otimal rice is satisfied exactly, if trivially: all the central measures converge to the single market rice in this case. Thus the mean of the matched Rectangular demand density can be exected to rovide a good aroximation to the otimal rice in all market situations.. Estimating the Otimal Retail Price with Minimum Market Testing The fact that the matched Rectangular mean gives a good aroximation to the otimal rice for a near-exhaustive range of demand densities means that a simle method may be advanced to estimate the otimal rice. The method is aimed at overcoming the difficulty the retailer may have in roviding an accurate estimate of the highest rice in the retailer s mental model, b. It is assumed that the retailer will be able to determine his variable cost er item,, to good accuracy, thus fixing the lowest c v Coyright 3 SciRes.

12 35 MAP of interest, a : a cv. The algorithm is based on the minimum amount of market testing, using just two rice levels, i, i =,, where a i b. These will lead to two different numbers, n i, i =,, of consumers buying the roduct where: n NS i b hu du gu du g i i (4) Meanwhile, from Equation () the fraction of the market reared to ay at least will be: S i i i b i (5) where g is the rectangular distribution given in Equation (7), and g b a. Hence, combining Equations (4) and (5), i b i n Ng (6) Thus the ratio, n n, of the numbers of customers buying at rices, and, will be: n n b (7) so that the estimate of the highest rice in the retailer s mental model,, is then b b n n b (8) n n The estimate of the otimal rice is then simly the mean of the Rectangular distribution: cv b n n * cv cv n n (9) This algorithm has been tested against a number of Double Power demand densities, and roved to be highly accurate when the initial rice level,, is already close to the underlying otimum (which should be so even if the retailer had available only the sarse information on rice discussed in Section 4), and the rice erturbation,, is of the order of % or less. Hence, for the demand density shown in Figure 7(a), when the rice level is set first at 5 and second at 5.5, the algorithm gives an estimated otimal rice of 5.35 units, comared with the true otimum calculated for the underlying demand density of 5.34 units. Meanwhile, for the demand density of Figure 8(b), setting the rice levels at 6 then 6.5 leads the algorithm to redict an otimal rice of 6.6 units, comared with the true otimum calculated for the underlying demand density of 6.7 units. It may be noted that the extra information coming from the simulated market testing has imroved uon the first aroximations to the otimal rices coming from the matched Rectangular mean, which were 5.3 units and 6.3 units resectively.. Conclusions A erfect knowledge of the distribution of MAP, or demand density, would enable the retailer to extract the maximum rofit, but using market surveys in an effort to obtain such comrehensive and accurate information would be exensive and roblematical even if the number of items researched were small. Recognizing the imracticality of reeated market surveys for each and every retail good, the study has tested a near-exhaustive range of ossible demand density curves, as embodied in candidate sets and. The evidence of the study is that the otimal rice may be aroximated reasonably well by the mean of the underlying distribution for all the candidate demand densities. It has been found that all the demand densities considered may be matched well with a Rectangular demand density, which yields an otimum rice equal to the mean rice. The mean of the matched Rectangular demand density has been found to lie close to the otimum rice of the underlying distribution for all the candidate demand densities considered. This is in itself an imortant result, since the matched Rectangular demand density is likely to corresond to the retailer s initial mental model of demand density. It suggests that the retailer will be able to make a reasonable initial estimate of the otimal rice to charge based on rather sarse rice data. An imroved estimate may be found if the retailer is reared to use the minimum of market testing, using a single erturbation from his initial rice. An algorithm has been develoed using the assumtion that the underlying demand distribution may be aroximated well by a Rectangular demand distribution. Worked examles show that the rice estimated by the algorithm is an imrovement on the mean of the matched Rectangular demand density and is very close to the actual otimal rice based on the true, underlying distribution, whatever it is. The results aly to all the basic forms of interaction between the retailer and his customers, monooly, monoolistic cometition and oligooly, as well as, in their limiting form, to erfect cometition. 3. Acknowledgements The authors are grateful to Sir John Kingman and Mr Roger Jones for their helful and useful comments on earlier drafts. REFERENCES [] R. G. Lisey and K. A. Chrystal, An Introduction to Posi- Coyright 3 SciRes.

13 36 tive Economics, 8th Edition, Oxford University Press, Oxford, 995. [] D. Begg, S. Fischer and R. Dornbusch, Economics, 3rd Edition, McGraw-Hill, London, 99. [3] G. F. Stanlake, Introductory Economics, 5th Edition, Longman, Harlow, Essex, 989. [4] Institute of Fiscal Studies (IFS),. htt:// [5] A. A. Cournot, Recherches sur les Princies Mathématiques de la Richesse, Chez L. Hachette, Paris, 838. htt://books.google.co.uk/books?id=kvhaaaayaaj &rintsec=frontcover&source=gbs_ge_summary_r&cad= #v=oneage&q& [6] J. Bertrand, Review of Théorie Mathématique de la Richesse Sociale and Recherches sur les Princies Mathématiques de la Richesse, Journal des Savants, 883, [7] R. L. Hall and C. J. Hitch, Price Theory and Business Behaviour, Oxford Economic Paers, No., 939, [8] P. M. Sweezy, Demand under Conditions of Oligooly, Journal of Political Economy, Vol. 47, No. 4, 939, doi:.86/554 [9] G. Stigler, Kinky Oligooly Demand and Rigid Prices, Journal of Political Economy, Vol. 55, No. 5, 947, doi:.86/5658 Coyright 3 SciRes.