Using Relative Utility Pricing to Explain Multibuy Prices in Supermarkets and on the Internet

doi:10.4236/ajibm.2013.38078

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American Journal of Industrial and Business Management, 2013, 3, 687-699

Published Online December 2013 (http://www.scirp.org/journal/ajibm)

http://dx.doi.org/10.4236/ajibm.2013.38078

Open Access AJIBM

687

Using Relative Utility Pricing to Explain Multibuy Prices

in Supermarkets and on the Internet

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

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

The Relative Utility Pricing (RUP) model is used to explain the prices for commodities being sold in supermarkets and

on the internet. Grocery prices offered by the supermarkets, Tesco, Sainsbury and Waitrose in December 2009 and Au-

gust 2013, are considered, as well as the prices of electronic items offered by Amazon on the internet at the same dates.

The obser v ed pr ice for a pack can be give n an ex planation in te rms of its size relative to the smallest pack considered by

the customer (the baseline pack), the price and variable cost associated with the baseline pack and the demand density.

The optimal price may be predicted using a profit maximising calculation if these data are available. Even if the ven-

dor’s knowledge of the demand density is poor or non-existent, it is still possible for a vendor knowing his unit variable

cost to calculate a useful approximation to the profit-maximising price by using a uniform or Rectangular demand den-

sity to represent customer demand. Alternatively, if there are no independent data on the demand density but the prices

of the packs are available, it is possible to determine the approximate shape of the demand density leading to those

prices. This demand density will then indicate whether the demand is soft or hard, with the Rectangular distribution

indicating a neutral market. We consider the vendor to be a large retailer, such as a supermarket, who can obtain the

product that he wishes to sell from a variety of suppliers at constant per-unit variable cost and hence marginal cost. Any

sales at a price above marginal cost will contribute to profit. The RUP model provides an approximate match to the

prices observed for supermarket milk and eggs by adjusting the demand density. A softening of the market for farm

retail commodities is revealed between December 2009 and August 2013, fully consistent with the co incident long pe-

riod of low growth and falling real wages in the UK economy. It is sh own how the vendor may use product differentia-

tion to buck this trend. The RUP model explains also the current prices of USB memory sticks, how those prices have

evolved and how they are likely to evolve over time.

Keywords: Relative Utility Pricing; RUP; Packs of Different Sizes; Pricing Strategies; Product Differentiation; Utility

Theory

1. Introduction

The Relative Utility Pricing (RUP) model, devised to

explain quantity promotions such as Buy One Get One

Free [1], will be used to predict the prices for commodi-

ties being sold in supermarkets and on the internet. The

paper will consider milk and egg prices offered by the

UK supermarkets, Tesco, Sainsbury and Waitrose in De-

cember 2009 and August 2013. The prices of electronic

components will also be analyzed, specifically universal

serial bus (USB) memory sticks of different capacity.

The method used to examine these prices will differ

from that presented in Section 8 of [1], which was con-

cerned with the “design” problem of choosing the size

and price of pack 2 from the three options: 1) “buy one

get one free”; 2) “2 for the price of 1” or 3) Golden Ratio.

Customers’ behaviour was then modelled using the Sin-

gle Comparison Relative Utility Pricing (SCRUP) model.

The task addressed in this paper is the “operational”

problem of deciding on the best prices for a set of packs

once their size has been fixed. This brings in the need to

respond with price changes to alterations in consumer

Using Relative Utility Pricing to Explain Multibuy Prices in Supermarkets and on the Internet

688

attitudes over time.

The structure of the RUP model allows a determina-

tion to be made of both the minimum price and the

maximum price, maxn, that potential purchasers may

assign to a pack of a given size, pack n. It is shown in [1],

Appendix B, that the MAP for every individual in the

cohort of consumers targeted by the retailer will fall

within the range to

p





Lnp, where n is the

price of pack n and is the “largeness” of pack n,

the ratio of its size to the size of the entry-lev el p ack 1, so

that .







11L

Once the range of possible prices has been identified,

as well as a mechanism for different individuals assign-

ing different values within that range to their maximum

acceptable price (MAP), as identified in Section 6 of [1],

the final unknown is the probability d istribution for MAP,

the “demand density” (see [2] and [3] for a discussion of

this concept). If the demand density for each pack size,

, can be found independently, perhaps by a cus-

tomer survey, then the RUP model will be able to give

good price predictions for all the pack sizes larger than

the entry-level pack, pack 1, g i ven the pri ce of pac k 1.

1n

In the absence of independent data on demand density,

it is nevertheless possible to calculate, at least approxi-

mately, the form of the demand den sity that is needed for

the observed prices to generate the optimal profit for the

vendor. The shape of the demand density curve then pro-

vides an easily understood explanation for the observed

prices—for example, is the market hard or soft? This

gives the RUP model the power to explain pack prices,

which is the principal aim of this paper.

The restriction on a probability distribution, namely

that the area under the curve must be unity, proves help-

ful in restricting the number of possible demand scenar-

ios needing exploration before a reasonable set of results

can be obtained. It is thus possible to vary the parameters

of the generalised Double Power demand density, the

properties of which are derived in [3], to find the profit-

maximising prices that give the best match to the ob-

served prices.

2. The Probability Distribution for MAP for

Pack n

As just noted, the upper and lower limits are defined for

the probability distribution for MAP for pack n, but the

form of the demand density within those limits remains

open, subject to the general but important condition that

its integral will be unity. This problem is addressed in

this paper throu gh the u se of two prob ability distribution s

with claims to generality. The first is the Uniform or

Rectangular distribution, often used in statistics wh en it is

known that some random distribution will be valid but

lack of data makes it impossible to specify the actual

form. The second is the generalised, Double Power den-

sity, discussed at length in [3], where the mode can be placed

anywhere within the range by the appropriate variation of

parameters. In fact, the Double Power density can be ar-

ranged to give a good approximation to the Rectangular

distribution by choosing its parameters appropriately.

One might reasonably expect that the fraction of the

potential consumers of pack n prepared to pay a price of





Lnp or close to it will become rather small as n and

hence





Ln increase. Moreover, the market for large

packs may lose its strong link to the market for small

packs. Hence when n is large, it may be sensible to as-

sume that the prospective purchaser of pack n will make

comparisons only with packs that are reasonably close to

pack n in size. This will involve rebasing, so that the

smallest non-empty pack with which the customer makes

a comparison is pack r. 1r



implies that pack n is be-

ing compared with each and every non-empty pack be-

low it, while implies that the prospective pur-

chaser is now ignoring the existence of all packs between

pack 0, the empty pack corresponding to no purchase,

and pack r, the first non-empty pack that he might be

interested in purchasing. Hence packs 1,

1r

2,,1r



 are

disregarded.

3. Setting the Price of Pack n so as to

Maximise Profit

By contrast with Section 9 of [1], which was concerned

with selecting both the size and the price of pack 2 given

the size and price of pack 1, now the sizes of the various

packs, , are fixed, and the problem is to select

optimal prices for the larger packs, given the price of

pack 1. The vendor will be faced with customers with

expectations conforming to the RUP model, and will

need to adjust his prices to take account of those expecta-

tions.

2,3, ,n

It is assumed that the total variable cost of the good

will be proportional to the amount of the good sold, irre-

spective of the size of the pack or packs in which the sale

is made. Let 1v be the variable cost per pack for pack 1,

assumed constant. Hence selling a single pack 1 will in-

cur a total variable cost, 1v, selling two packs 1 will

incur a total variable cost of 1 and so on. Moreover,

if pack 2 h as a largeness of 2, viz. , then it will

incur a total variable cost of 1v, while if it has a large-

ness of 3, viz.



22L





23L



, then pack 2 will incur a total

variable cost of 1, and so on. Hence, in general, the

variable cost per pack, , for pack n will be





1vn v

cLnc (1)

With the contents of pack 1 seen as the fundamental

unit of measurement, viz. , the variable cost per

pack, , for pack 1 may be seen as the variable cost



1L1

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incurred in selling each one of these fundamental units.

The achieved price, 1, of pack 1 is ex ogenous to the

RUP model, and this makes it convenient to express

in terms of :

11vvf

ccp1

(2)

Here 1vf is the fraction of the achieved price, 1, of

pack 1 absorbed by variable costs. Substituting from

Equation (2) into Equation (1) gives the variable cost per

pack for pack n, expressed in terms of pack 1 price, :



11vn vf

cLncp (3)

Let the probability density for the MAP, n, for pack

n be p



. Using the concept of the “uniconsumer”

introduced without loss of generality in [2] to character-

ise a person prepared to buy one but only one item if the

price is right, the fraction of the uniconsumers prepared

to pay n or more for a single pack n will be





Sp ,

given most generally by



 

d1 d

Spgv vgv v







(4)

Since the target group consists of consumers prepared

to pay between 1n and ma for pack n, pxn







for 1nn

(as well as for ). Hence Equa-

tion (4) may be rewritten:



maxnn

pp



Spgv v





d (5)

We may define an “offset MAP”, , for pack n by:

1nnn

qpp



 (6)

which will conform to a probability distribution,





hq ,

over the interval max , where, since

from Appendix B of [1], the maximum

value of the offset MAP, , for pack n, is given by

0nn

qq



max 1n

pLnp



max1 1n

qLnpp



 (7)

We may also note that differentiating Equation (6)

gives

p (8)

The probability dens ity for MAP,



p, and that for

offset MAP, , which may be termed “demand

densities”, are then linked by:



 

phqhq

 (9)

in which Equation (8) is used. Substituting from Equa-

tion (9) into Equation (5) gives:



Sphv vSq 

 (10)

Let there be N uniconsumers in the target group. The

vendor’s total pro fit,



, from selling packs n will be his

total income from packs n less both the total variable

costs and the fixed costs,

C, associated with packs n:





nnnvn F

NS ppNS pcC n

(11)

This may be converted using Equations (3), (6) and (10)

into the form:











111nn nnvfF

NSqqpNSqcLnpC



 n

(12)

The retailer will seek to maximise this profit, which,

for a constant size of target population, N, is equivalent to

choosing the offset MAP, , so as to maximise the av-

erage profit per consumer, n





111 0

nvf n

qcLnpp hvvN











 







(13)

in which Equation (10) has been used in the second step.

The term, Fn

CN, being a constant, is unaffected by

changes in n, and so, given a demand density, q





hq ,

it is possible by direct computation to find the value of

that will maximise





, where



111 0

nvf n

qcLnpp hvv







 





d

(14)

which will also maximise



Alternatively, the maximising value of MAP may be

found by differentiating Equation (13) with respect to the

offset MAP, , and then setting to zero the resultant

expression, n

dd 0qn





. This gives the optimal price

for pack n as 1nnn

pqp





, where is the solution

of n

 



111

nvfn n

hvvqc Lnpphq

0



 

 (15)

4. Explaining the Price of Pack n

Data are available in the supermarket or on the internet

for the achieved prices, 12 , of packs of vari-

ous sizes, and we selected for examination the pricing of

fresh milk and eggs in supermarkets and USB memory

sticks on the internet, specifically those offered for sale

on Amazon.co.uk.

,,,

pp p

The supermarkets considered were Tesco, Sainsbury

and Waitrose. Tesco is the UK’s leading supermarket,

with a market share of about 30%. Sainsbury has about

16% of the market, while Waitrose, a part of the pri-

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vately owned John Lewis Partnership, has about 4% of

the market. John Lewis, together with its subsidiary,

Waitrose, is widely regarded as an up-market retailer.

Data were available on the variable cost per pack for

pack 1, 1v, in the case of both milk and egg s, but not on

the demand density. No information was available on the

variable cost of a USB memory stick, nor on its demand

density.

The Rectangular demand density, if it were to be valid,

confers the advantage that it is defined by the RUP model

as soon as data are available on the pack largenesses and

the prices of packs below pack n, 12 1

,,,

pp p



:



 

1 if

0 otherwise

ppLn

Lnp p

gp 













(16)

Thus, if the variable cost per pack for pack 1, 1v, is

known (it can be presumed that the vendor will know this

figure), then it is possible to use the Uniform or Rectan-

gular demand density to calculate successively the opti-

mal prices for packs . This distribution de-

pends only on the prices of lower packs and on the

largeness of pack n. It can be applied successively as

soon as the price, 1, of the entry-level pack is known.

Hence it is useful in its own right for a vendor laun ching

a new product to be packaged in a variety of pack sizes at

a time when experience of the market is necessarily

lacking and when whatever market research is available

may be insufficient to establish reliable estimates of the

actual demand densities. More generally, it may provide

useful central estimates for quoted prices when it is not

known whether the market is soft, when the demand den-

sity is biased towards lower prices, or hard, when the

demand density is skewed towards higher prices.

2,3, ,n

However, the Rectangular demand density is just one

of the possible densities that might be valid. In this study,

therefore, the versatile Double Power demand density

was used to examine a large range of possible offset de-

mand densities. The Double Power demand density,

, characterising the offset MAP, , for pack n is

given by [3]:



hq n



max

for 0

0 for

nnn nn

hqaqbqqq

 



(17)

where a, b, c and d are non-negative constants. The mode

will be strictly interior if all of a, b, c and d are non-zero;

it will occur at the lower boundary, , when

q0c



while it will be located at the upper boundary,

maxnn

, when . The constants, a and b may be

defined in terms of c and d as detailed in [3].

qq0b

For any product sold in packs of different sizes, it has

been assumed that the parameters, c and d, defining the

demand densities will be the same for all of the packs

greater than the effective baseline (pack r + 1 and upwards).

The optimising results from [3] may be used after put-

ting

pq (18)

and





11vvf n

ccLnpp







(19)

which have the effect of converting Equation (13) into

Equation (3) of [2]. Thus, for example, if the mode is

strictly interior to the interval, , then the opti-

mal value, will be the solution of:



max

0, n

 

  

max max

111

maxmax max

12 12

vf nnn

cd dc

cLnp pqq





 

 

 

 



















(20)

(cf. Equation (67) of [3]).

5. Using the RUP Model to Examine the

Price of Milk in the Tesco Supermarket

Milk is retailed in the UK in 3 main categories: whole

milk, which has a butter-fat content of about 4% [4],

semi-skimmed milk (1.7% butter-fat) and skimmed milk,

with a fat content of about 0.1% [5]. Just about half of the

milk retailed in the UK is whole milk [6], with the re-

maining sales coming from reduced-fat milk. Price data

were taken from the My Supermarket website in De-

cember 2009 [7] and August 2013 [8].

It is usual for supermarkets to offer milk in four dif-

ferent sizes or “packs” in the terminology of this paper: 1

pint (= 0.57 litre), 2 pints, 4 pints and 6 pints. Tesco

prices packs of the same size the same for each category

of milk: whole, semi-skimmed and skimmed.

Estimates of the variable cost per pint may be made

from the information on farm gate prices released by

Tesco and covered in the farming press. These may be

checked using average farm gate prices [4] provided by

the UK Department of Environment, Farming and Rural

Affairs (DEFRA). The price paid to farmers supplying

Tesco was £0.1565 per pint in December 2009 [9] and

£0.1861 per pint in August 2013 [10]. Although super-

markets may offer special promotions on milk, they claim

that this does not affect the price paid to farmers, which

are for long-term contracts [6].

To this must be added the milk processing cost (col-

lection, pasteurization and packaging) estimated at

£0.0675 per pint for semi-skimmed milk [6]. In the ab-

sence of information on how the milk processors incur-

porate into their business model the benefit of selling the

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skimmed-off butter-fat, it has been assumed that the cost

of £0.0675 per pint applies to whole milk as well as to

semi-skimmed milk. Finally some further variable costs

must be added, such as distribution and stacking at the

different supermarkets, and these are assumed to increase

the total by 10%.

Hence the variable cost of Tesco milk was estimated at

£0.2464 per pint in December 2009 and £0.2790 per pint

in Augus t 2013.

5.1. Whole Milk, August 2013

Table 1 summarises the observed prices in August 2013,

and compares the actual prices with the optimal prices

generated by the RUP model with c = 0 and d = 0.29 (see

Figure 1). These parameters produced the best match to

the prices of packs as judged by the

minimisation of the weighted sum of the squared errors,

, between the observed and predicted prices for

packs :

1,2,,rr n



Wr 1,2,,rr

Table 1. Tesco whole milk and semi-skimmed milk, August

2013.

Pack identifier, n cv1, £/pin t 1 2 3 4

Size, pints 1 2 4 6

Largeness, L 1 2 4 6

Price, £ 0.49 0.89 1.3 9 1.99

RUP Double Power

price (£) c = 0,

d = 0.29 0.2790 0.49 0.69 1.38 2.07

RUP Double Power

price (£) (Rebased)

c = 0, d = 1.66 0.2790 - 0.89 1.35 2.02

RUP Rectangular

price (£) 0.2790 0.49 0.77 1.54 2.30

00.5 11.522.5 3

MAP (£)

Probability densit

Pack 4: 6 pints

Pack 3: 4 pints

Pack 2: 2 pints

Figure 1. Tesco milk, August 2013, c = 0, d = 0.29.







nirupi

ir i

Wr p





 (21)

It is assumed first that the 1-pint pack is the lowest

pack included in people’s comparisons with pack ,

so that r = 1, and is the prediction for the ob-

served price of pack i, i, based on the assumption that

profit is maximised, with the associated demand densities

as shown in Figure 1. Thus if the true demand densities

are as in Figure 1 and the figure for the variable cost per

pint is reflected accurately in the value, £0.2790, then the

model suggests that the vendor should chose pack prices,

,rup i, i = 2, 3 and 4, if he wishes to maximise his profit

when selling each of the higher packs.

:1ii

,rup i

The match between the observed prices and those pre-

dicted by the RUP model is reason able, but there ap pears

to be an anomaly in the pricing of pack 2, where the pre-

dicted price is about £0.20 lower than that actually

achieved.

The anomalous price for pack 2 suggests that for many

people the 2-pint pack constitutes the entry-level pack.

This may reflect a natural break point in the customer

base, where purchasers of a 1 pint carton may constitute a

market with only weak links to the main market of those

buying larger cartons. This proposition may be tested by

rebasing the smallest pack considered by the prospective

purchaser of pack n, in the manner described in Section 2.

Setting the rebasing integer, r, as 2

, means that what

was originally pack 2 becomes the new pack 1. This re-

basing improves the price match for packs 3 and 4 (see

the last but one line of Table 1), and results in demand

density parameters of c = 0 and d = 1.66, see Figure 2.

A set of calculations were also made using 1v



£0.2790 per pint, as before, but this time using a Rec-

tangular demand density. The discrepancy on the price of

the 2-pint pack 2 is observed once again, but the results,

given in the last line of Table 1, are rough approximations

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

0123

MAP (£)

Probability densit

Pack 4: 6 pints

Pack 3: 4 pints

Figure 2. Tesco milk, August 2013, rebased c = 0, d = 1.66.

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to the quoted prices. Interestingly, the prices from this

version of the RUP model are close to the prices for

semi-skimmed milk in December 2009, discussed next.

5.2. Semi-Skimmed Milk, December 2009

Table 2 summarises the data, and compares the actual

prices with the optimal prices generated by the RUP

model when c = 3 and d = 280. Examining Figure 3, it

can be seen that the distribution is weighted towards the

higher end of each interval.

Table 2 shows that the match between prices for packs

of 4 pints or more is good, but once again there is an

anomaly with the price of pack 2, containing 2 pints. Re-

basing to make pack 2 the entry p ack produces th e resu lts

shown in the last line of Table 2. Th e prices of the 4-p int

and 6-pint packs are reproduced well once again, while,

by the nature of the procedure, the anomaly associated

with the 2-pint pack is eliminated. The values of c and d

that minimise the sum of the weighted squared errors,

, are and . Demand densi- ties

favouring high-end prices for pack s 3 and 4 (4 pints and 6

pints) are found once again, as shown in Figure 4.



2W5c450d

Table 2. Tesco semi-skimmed milk, December 2009.

Pack identifier, n cv1, £/pin t 1 2 3 4

Size, pints 1 2 4 6

Largeness, L 1 2 4 6

Price, £ 0.45 0.86 1.53 2.25

RUP Double

Power price (£)

c = 3, d = 280 0.2464 0.45 0.76 1.50 2.30

RUP Double

Power price (£)

(Rebased)

c = 5, d = 450

0.2464 - 0.86 1.50 2.28

00.511.522.53

MAP (£)

Probability densit

Pack 4: 6 pints

Pack 3: 4 pints

Pack 2: 2 pints

Figure 3. Tesco semi-skimmed milk, December 2009, c = 3, d

= 280.

01234

MAP (£)

Probability densit

Pack 4: 6 pints

Pack 3: 4 pints

Figure 4. Tesco semi-skimmed milk, December 2009, re-

based, c = 5, d = 450.

5.3. The Change in the Market for

Semi-Skimmed Milk between December

2009 and August 2013

Because Tesco prices semi-skimmed milk and whole

milk identically, and it has been assumed that the unit

variable cost is the same for the two grades of milk, the

price analysis for semi-skimmed milk is the same as for

whole milk in August 2013, as summarised by the results

in Table 1 and in Figures 1 and 2.

Comparing Figures 1 and 2 for August 2013 with the

corresponding figures for December 2009, namely Fig-

ures 3 and 4, it is clear that the demand density curves

have switched between these dates from a weighting to-

wards higher prices to a weighting in favour of lower

prices. This change in the characterisation of customer

sentiment is striking, but it is quite credible given the

continuing period of austerity and low growth in the UK

from 2009 to 2013, during which period real wages fell

for many peop le (8.5% between 2009 and 2012 [11]).

The effects of “austerity Britain” may also be detected

in the observed behaviou r of Tesco, which, while passing

on the 3.26 pence rise in the variable cost of a single pint

pack, absorbed half the increased cost reaching the

two-pint carton by constraining the rise in the price of

pack 2 to only 3 pence. Moreover, despite facing a rise in

variable costs of about 13 pence for its 4-pint pack 3,

Tesco actually reduced the price by 14 pence. Similarly,

an increase of 19 pence in the variable cost of the 6-pint

pack 4 was absorbed in full and the price was dropped 26

pence besides. Such moves would seem to reflect Tesco

observing a significant softening in the market, fully con-

sistent with the change in the shape of the demand densi-

ties as illustrated by Figures 1-4.

6. Using the RUP Model to Explain the Price

of Eggs in UK Supermarkets

Data were taken from the My Supermarket website in

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Using Relative Utility Pricing to Explain Multibuy Prices in Supermarkets and on the Internet 693

December 2009 [7] and in August 2013 [8] for “value”

eggs at Tesco, barn and free-range eggs at Sainsbury’s

and organic free range eggs at Waitrose.

Estimates of the variable costs per dozen eggs were

made from the average price per dozen paid to egg pro-

ducers, as provided by DEFRA [12] on a quarterly basis.

Data are supplied for eggs categorized as intensive, barn,

free range and organic, with an overall average also given.

The data were incomplete for barn and organic eggs from

the first quarter of 2012 onwards, and so extrapolations

were made using the price of free range eggs as a guide:

free range eggs were intermediate in price between barn

eggs and organic eggs for the years for which data were

essentially complete, from 2007 to 2011 inclusive. In

addition, data for the 3rd quarter of 2013 were not avail-

able at the time of the calculations, and so 2nd quarter

data were used instead.

It is important to make clear that the benefits of bulk

sales are not linked to any production or distribution

economies of scale (or scope). Indeed, we consider the

vendor to be a retailer who is faced with something close

to a constant marginal cost of the product (and thus also a

constant per-unit variable cost and a constant average

variable cost). The retailer is not greatly affected by any

possible economies or diseconomies in production costs

as he can gain access to supplies from many different

producers each of whom is operating close to his opti-

mum scale but all of whom can supply the product at the

competitive (or contracted) wholesale price. The retailer

will thus increase profit by increasing sales whenever he

can do so at a price that is above his marginal costs.

The variable costs associated with egg packaging were

judged to be less than those associated with milk process-

sors, so that the additional variable costs were repre-

sented simply by adding 25% to the price paid to the

producer, roughly half the mark-up on milk. The esti-

mated variable costs per half dozen for the various egg

types are shown in Table 3.

Table 3. Estimated variable cost per half dozen eggs (all

prices in £).

Date Intensive Barn Free RangeOrganic

Paid to

producer per

dozen 0.5630.74 0.901 1.373

December

2009 Estimated

variable cost per

½ dozen 0.35190.4625 0.563 0.858

Paid to producer

per dozen 0.7540.904 1.055 1.737

August

2013 Estimated

variable

cost per

½ dozen

0.47130.5652 0.6594 1.086

6.1. Tesco Value Mixed Eggs

The Double Power demand density matches exactly the

observed price for pack 2 (15 eggs) of Tesco’s Value

Mixed Eggs in December 2009 given in Tabl e 4 when c

= 0 and d = 7.8. See Figure 5. However, the Double

Power demand density is unable to match the observed

price for pack 2 in August 2013. It gets closest when c =

0 and d = 0.01, see Figure 6, when the profit-maximizing

price for pack 2 is £1.48 rather than £1.35. This may be a

result of the inexact nature of the data. It is necessary to

reduce the variable cost per half dozen given in Table 3

by about 7 pence to £0.4019 to enable the model to re-

produce the observed price of £1.35 for pack 2 with c = 0

and d = 0.01.

The general trend of market tightening observed in the

milk market would seem to be mirrored in the case of

these commodity eggs.

Comparing Figures 5 and 6, it is clear that people in

the target market are taking a much less relaxed view of

price, with significant clustering at the lower end of the

price range.

Table 4. “Value” eggs at Tesco and “Basics” eggs at Sains-

bury (December 2009 and August 2013).

Supermarket Tesco Sainsbury

Product name Tesco Value

Mixed Eggs Sainsbury’s Basics

Barn Eggs

Pack identifier, n1 2 1 2

Number in pack6 15 6 18

Largeness, L 1 2.5 1 3

Price, £,

Dec. 2009 0.91 1.50 0.91 1.85

Price, £,

Aug. 2013 0.87 1.35 1.00 2.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

00.511.522.53

MAP

Pro bability dens ity

Figure 5. Tesco value mixed eggs, December 2009, c = 0, d =

7.8.

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00.511.522.53

MAP

Probability density

Figure 6. Tesco value mixed eggs, August 2013, c = 0, d =

0.01.

6.2. Sainsbury Basics Barn Eggs

The Double Power demand density matches exactly the

observed prices given in Table 4 for Sainsbury’s Basics

Barn Eggs, both in December 2009 when c = 0 and d =

1.49 and in August 2013 when c = 0 and d = 0.01. See

Figures 7 and 8. It is clear from comparing these figures

that the market has tightened for Sainsbury’s Basics eggs

in the same way as for Tesco’s Value eggs, again with a

preponderance of people limiting their MAP to a low

level.

A further interpretation of the degree of market sof-

tening may be found by using the demand density char-

acteristic of December 2009 (c = 0 and d = 1.49) to pre-

dict the price of pack 2 of Sainsbury’s Basics Barn Eggs

in August 2013. Given the pack 1 price of £1.00, the pre-

dicted price of pack 2 is £2.14. Thus this manifestation of

the softening of the market has cost Sainsbury 4p in

profit on its 18-egg pack of Basics Barn Eggs.

6.3. Sainsbury’s Woodland Free Ran g e Large

Eggs

These eggs constitute Sainsbury’s top of the range offer-

ing. The Double Power demand density matches the ob-

served prices given in Table 5 for Sainsbury’s Large

Woodland Free Range Eggs in December 2009 when c =

34.6 and d = 100. Meanwhile, the observed prices listed

in Table 6 for Sainsbury’s Woodland Free Range Large

Eggs (there seems to have been a minor rewording of the

name in the years between 2009 and 2013) in August

2013 are matched when c = 6 and d = 65. See Figures 9

and 10.

Demand density remained biased towards the upper

end of the price interval in both December 2009 and Au-

gust 2013. However there appears some degree of market

softening, with a greater proportion of the target market

having a MAP in the lower range of the interval more

recently. It would appear that the potential purchasers of

Sainsbury’s Woodland Free Range Large Eggs were

highly price-insensitive in December 2009 but have be-

come less so after the years of “austerity” following 2009

(by mid 2013, UK GDP was still down by 3.3% on what it

had been 5 years earlier). Even in 2013 , however, customers

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.5 1 1.5 2 2.5 3

MAP

Probability density

Figure 7. Sainsbury’s basics barn eggs, December 2009, c =

0, d = 1.49.

00.511.522.53

MAP

Probability den s ity

Figure 8. Sainsbury’s basics barn eggs, August 2013, c = 0, d

= 0.01.

Table 5. Top of the price-range eggs: Sainsbury and Wait-

rose, December 2009.

Supermarket Sainsbury Waitrose

Product name Sainsbury’s Large

Woodland Free

Range Eggs

Waitrose Organic Large

Free Range Columbian

Blacktail Hen Eggs

Pack identifier, n1 2 1 2

Number in pack6 12 6 12

Largeness, L 1 2 1 2

Price, £ 1.58 2.98 2.28 4.30

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Table 6. Top of the price-range eggs: Sainsbury and Wait-

rose, August 2013.

Supermarket Sainsbury Waitrose

Product name Sainsbury’s

Woodland Free

Range Large Eggs

Duchy Originals from

Waitrose: Organic

West Country Large

Free Range Eggs

Pack identifier, n 1 2 1 2

Number in pack 6 12 6 12

Largeness, L 1 2 1 2

Price, £ 1.70 2.90 2.36 4.70

0123

MAP

Probability densit

Figure 9. Sainsbury’s large woodland free range eggs, De-

cember 2009, c = 34.6, d = 100.

0.5

1.5

2.5

3.5

0123

MAP

Probability densit

Figure 10. Sainsbury’s woodland free range large eggs, Au-

gust 2013, c = 6, d = 65.

for these eggs are tending to see the packs before them

not so much as bundles of commodities (when it would

be difficult to raise the price for a dozen much above the

price of half a dozen as explained in [1]) but rather as

containing valuable, differentiated items to be valued

closer to their maximum worth, the highest MAP:

2max 201

2ppp



. The more evocative branding,

“Woodland”, as opposed to “Basics”, has, no dou bt, been

chosen to help in this regard.

The achieved price for pack 2 in December 2009 was

2£2.98p



, within 6% of the highest MAP possible,

namely 20£3.16p



. However, while 21

in Au-

gust 2013, at 2

pp

£2p.70



, the achieved price was more

than 20% off the highest MAP, 20 , indicating a

softening even at the top end of Sainsbury’s egg market.

£3.40p

6.4. Waitrose Organic, Free Range, La rge Eggs

Waitrose’s highest priced eggs in December 2009 were

“Waitrose Organic Large Free Range Columbian Black-

tail Hen Eggs”, but by August 2013 their position had

been taken by “Duchy Originals from Waitrose Organic

West Country Large Free Range Eggs”. The Duchy

Originals company, named after the Duchy of Cornwall

estates held by the Prince of Wales, was set up by the

Prince in 1990 as a supplier of organic foo d. However, 20

years into its development, the company reached an

agreement with Waitrose whereby the latter would invest

and pay a royalty to Duchy, set initially at £1 million. In

return, Waitrose won the exclusive right to sell and dis-

tribute the products, which from August 2010 would be

rebranded “Duchy Originals from Waitrose”.

The Double Power demand density matches the ob-

served prices given in Table 5 for Waitrose Organic

Large Free Range Columbian Blacktail Hen Eggs in De-

cember 2009 when c = 34.2 and d = 100. These parame-

ters are almost exactly the same as those that generated a

match with the prices of Sainsbury’s Large Woodland

Free Range Eggs in December 2009. The distribution is

heavily weighted towards the upper end of the price in-

terval, as shown in Figure 11.

Interestingly, the observed prices listed in Table 6 for

Duchy Originals from Waitrose Organic West Country

Large Free Range Eggs in August 2013 required the de-

mand density to be almost a spike at the very top of the

price range, produced when c = 810 and d = 820. See

Figure 12.

This hardening of the market for Waitrose’s most ex-

pensive eggs illustrates a divergence of behaviour from

Sainsbury’s top of the range eggs. While the achieved

price for Waitrose’s pack 2 in December 2009 was

2£4.30p



, close to 20 £4.56p



, the achieved price,

2£4.70p



, for a dozen eggs in August 2013 was hardly

different from the highest possible MAP, 20 £4.72p



Waitrose may be benefitting here not only from the pos-

sible greater resilience of its customer base in the face of

economic austerity but also from clever marketing in

allying its own brand with Duchy and its royal links.

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012345

MAP

Probability density

Figure 11. Waitrose organic large free range Columbian

Blacktail hen eggs, December 2009, c = 34.2, d = 100.

012345

MAP

Pro bability density

Figure 12. Duchy originals from Waitrose organic west

country large free range eggs, August 2013, c = 810, d =

820.

7. Using the RUP Model to Examine the

Price of USB Memory Sticks Offered on

the Internet

USB memory sticks are the most convenient way of

storing large amounts of computer data for exchange be-

tween PCs and laptop computers, for example. Their ca-

pacity has grown enormously over recent years. The en-

try-level capacity, tens of Megabytes in the mid 2000s,

had reached 1 gigabyte (Gb) of memory in December

2009, a size that was already on the point of being su-

perseded by the 2 Gb memory stick, with the top of the

easily available commercial range being 16 Gb. By Au-

gust 2013, the entry-level memory stick was 4 Gb, a size

about to be replaced by 8 Gb, while the top of the easily

available commercial range had reached 128 Gb.

There is limited scope for product differentiation in the

field of USB memory sticks since small size is essential

for the device and the need for functionality makes it

difficult to customise to any great extent, although at-

tempts are now being made by offering the same capacity

stick in different colours. Competition is fierce, with

many manufacturers competing, and a strong need

amongst all manufacturers to keep up with the technol-

ogy and increase the size of the largest available storage

capacity. Industry practice is for pack size to increase by

a factor of 2 at each step.

Prices were taken from Amazon’s UK web site in De-

cember 2009 [13] and August 2013 [14], with less ex ten-

sive data obtained from the same website in August 2010.

7.1. December 2009

Table 7 shows the prices for the Integral USB memory

sticks of capacity: 1, 2, 4, 8 and 16 Gb. Given the ab-

sence of information on the unit variable cost, it was de-

cided to specify the shape of the demand density curve

and then vary the variable cost of pack r so as to mini-

mise the sum of the weighted errors, , with the

rebasing integer, r, set to unity in the first instance. Set-

ting c = 0 and d = 2 produces a distribution weighted

moderately towards the lower end of the price interval,

but with a significant fraction of the target population

prepared to pay higher prices. These parameters were

chosen to bring the shape of curve (Figure 13) roughly

into line with the curves found to give good matches to

the commodity egg prices at the same date (see Figures 5

and 7).



It was not found possible to reproduce the prices of

packs 2, 3 and 4 satisfactorily using a single RUP model.

Instead three different versions were required to generate

prices for pack 2, packs 3 and 4 and packs 4 and 5. This

suggests a segmented market, with purchasers interested

in buying pack 4 (8 Gb) having no interest in pack 1 (1

Gb), and purchasers interested in pack 5 (16 Gb) disre-

garding the existence of packs 1 and 2 (2 Gb).

The reduction in the quoted price for a 2 Gb below that

of a 1 Gb memory stick reveals an economic anomaly

that the RUP model is unable to reproduce. In its closest

Table 7. USB memory sticks of the integral brand offered

for sale on Amazon.co.uk, December 2009.

Size (Gb) cv1, £1 2 4 8 16

Pack

identifier, n 1 2 3 4 5

Largeness, L 1 2 4 8 16

Price, £ 6.956.44 7.88 12.7316.95

<1.176.956.96 - - -

0.81- 6.44 7.00 13.54-

RUP

Double

Power

price, £,

c = 0, d = 20.71- - 7.88 9.5 18.66

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0.05

0.1

0.15

0.2

0.25

0481216

MAP, £

Probability density

Figure 13. USB sticks demand density for pack 2: 2 Gb,

December 2009, c = 0, d = 2.

approach, the price of pack 2 is only marginally greater

than that of pack 1, an outcome that will occur when 1v

takes any value less than or equal to £1.17 per Gb. The

model prices and values for are shown in Table 7.

The three versions of the RUP model are in agreement

in predicting a low variable cost per gigabyte of memory.

Although uncertainty over the shape of the demand den-

sity curves rules out a confident prediction, the results of

this modelling exercise suggest that the variable cost in

2009 might have been around £0.75 per gigabyte.

7.2. August 2013

Table 8 shows the prices for the Integral USB memory

sticks of capacity: 4, 8, 16, 32, 64 and 128 Gb. Tables 7

and 8 show striking ly similar price structures, except th at

the capacity of each of the memory sticks has increased

by a factor of 4 by August 2013.

As with Section 7.1, the procedure for modelling was

to adjust the unit variable cost to find the best match to

the data, while keeping the parameters defining the de-

mand densities the same as their December 2009 values,

namely c = 0 and d = 2. Figure 14 shows the demand

densities for packs 3 to 6.

Two versions of the RUP model were required, one to

generate the price for pack 2, and the other to generate

the prices successively of packs 3, 4, 5 and 6. The

model-generated prices and associated values for are

included in Table 8. Once again the unit variable cost

was low, with the best matches to the data occurring

when 1 per 4 Gb, that is to say £0.31 per Gb.

This would suggest that the variable cost per Gb had

dropped by a bout 6 0% o ver the 4 year perio d.

£1.25

c

7.3. The Evolution of the Price Structure of USB

Sticks

It is pointed out in [1] that the consumer has a natural

Table 8. USB memory sticks of the DataTraveler brand

offered for sale on Amazon.co.uk, August 2013.

Size (Gb) cv1, £ 4 8 16 32 64 128

Pack

identifier, n 1 2 3 4 5 6

Largeness, L 1 2 4 8 16 32

Price, £ 5.796.03 8.06 14.40 27.0160.62

1.215.796.03 - - - -

RUP Double

Power price, £,

c = 0, d = 21.28- 6.03 7.72 15.24 30.4857.78

0.05

0.1

0.15

0.2

0.25

0.3

0 102030405060708090100

MAP, £

Probability density

Pack 3: 16 Gb

Pack 4: 32 Gb

Pack 5: 64 Gb

Pack 6: 128 Gb

Figure 14. USB sticks demand densities for packs 3 to 6,

August 2013, c = 0, d = 2.

tendency to value pack 2 of a commodity the same as

pack 1 whenever pack 2 contains double the quantity held

in pack 1. A similar process will be in play when pack 3

is twice the size of pack 2. On this basis one would ex-

pect the prices of packs 1, 2 and 3 to be very similar. And,

indeed, it is clear that this is the case in both December

2009 and August 2013.

The vendor may dispute and resist this valuation

through product differentiation and branding, as demon-

strated by Sainsbury with its “Sainsbury’s Woodland

Free Range Large Eggs”. The same process has been

used even more successfully by Waitrose with its “Duchy

Originals from Waitrose Organic West Country Large

Free Range Eggs”, a product description that combines

product differentiation with complexity, two features

identified at the end of Section 3 of [1] as causing the

consumer to value the good as other than a commodity.

The main route open to the vendor of USB memory

sticks is to differentiate his product through novelty, as

manifested by a capacity that will be judged unusually

high by those in his customer base. This will allow the

memory sticks with higher capacity to command higher

prices on a temporary basis. But as the capacity of the

memory stick becomes less unusual, as the novelty and

differentiation wear off, so the price of the stick will fall

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ever closer to the price of the memory stick below it,

which will, in turn, be declining towards the price of the

next lowest memory stick, and so on. There will be, as a

result, a decline towards the price of the lowest capacity

stick. The sales of the stick with the lowest capacity will

dry up at the point when the prices of the two lowest

sticks are equal, and the vendor will drop it from his

catalogue.

Thus the 1 Gb memory stick ceased to be offered for

sale on Amazon shortly after December 2009, leaving the

2 Gb memory stick as the entry level pack, priced at

£6.44, no higher than the 1 Gb pack it replaced. Less than

a year later, in August 2010, the Kingston Datatraveler

101 4 Gb USB 2.0 Flash Drive was being offered on

Amazon at £6.19, so that the days of the 2 Gb memory

stick were clearly numbered. The 4 Gb memory stick,

priced at £5.79, is the entry level pack in August 2013,

but it is clearly about to be supplanted by a 8 Gb stick,

priced at £6.03, almost as low. Clearly, the 16 Gb mem-

ory stick is set to become the entry-level pack in the near

future, and the 32 Gb memory stick a little later. It is the

RUP mechanism that lies behind the similarity of the

pack-price structures revealed by Tab l es 7 and 8.

It is thus clear that the RUP model has been able to ex-

plain not only the current pricing structure of the market

in USB memory sticks but also how it has evolved and is

likely to evolve over time.

8. Discussion

The figures for milk and egg prices have built on the

available farm gate and egg producer prices that were

used to estimate per-unit variable costs, 1v. Neverthe-

less it is clear that the resulting figures are only approxi-

mations. These per-unit variable costs feed through to the

demand densities derived to match the observed prices,

assumed optimal for the vendor. Thus the resulting de-

mand densities will be approximate also.

That said, the per-unit variable costs for milk are ex-

pected to be reasonably close to the true values, and,

subject to a greater tolerance, the same may be said for

the variable costs of eggs. Moreover, it is very helpful

that the demand distributions are bounded by the re-

quirement that the area under the curve must be unity, so

that, in this case, , where



max

up huu





1





hp is

the demand density for MAP for pack n. This inherent

property allows the versatile Double Power density to

provide a good coverage of plausible demand scenarios,

and one of its manifestations can be expected to provide a

reasonable approximation to the true demand density.

Finding the appropriate realisation (setting the c and d

parameters) becomes theoretically possible when the

variable cost, 1v, is available, and this is what has been

done for milk and egg prices. The ou tputs of the analysis

are then the demand density curves, as shown in Figures

1-12. Even with avowedly approximate data, these de-

mand densities provide not only a reasonable explana-

tion for the observed prices but also, via their changes

over time, a credible interpretation of consumers’ reac-

tions to changed economic circumstances. Amongst other

findings, the curves point to the advantages to th e vendor

of product differentiation.

Less information was available on USB memory sticks,

and it was necessary to assume a plausible demand den-

sity, based on those found for Tesco’s and Sainsbury’s

commodity eggs in December 2009, and back-calculate

the corresponding variable cost per Gb. While no quanti-

tative claim on the accuracy of the figures can be made,

the deduction that the calculated cost per Gb in Decem-

ber 2009, already low, falls to an even lower figure in

August 2013 seems intuitively appealing.

Finkelstein has pointed out that models may have a va-

riety of uses, from descriptive through explanatory to

predictive [15]. It has been demonstrated that the RUP

model can have useful explanatory power even when

available data are limited.

Full predictive power for the RUP model requires an

accurate knowledge of 1) the price of pack r, the effec-

tive entry-level pack, 2) the per-unit variable cost, 3) the

size of the pack in question and 4) the form of the de-

mand density. The vendor should have a good grasp of

the first three items, even if he may not know the fourth.

Given this admittedly imperfect knowledge, the vendor

may still use the Uniform or Rectangular demand density

to make an initial estimate of his profit-maximising

prices. This is because the rectangular demand density

depends ultimately only on items 1), 2) and 3) and may

be applied successively to build up the prices of packs

1,2, ,rr n



.

In using the Uniform or Rectangular demand density,

the vendor will be employing the assumption that the

market is neutral, neither soft nor hard. The procedure

has been shown to produce starting prices that are close

enough to the optimal prices for them to be used by the

retailer as the basis for market testing and hence optimi-

sation.

9. Conclusions

The theory of Relative Utility Pricing (RUP) has been

developed to understand the pricing of packs of different

sizes in supermarkets and on the internet. It has been

shown that the observed price for a pack can be ex-

plained in terms of its size relative to the smallest pack

considered for purchase by the customer (his baseline

pack), the price and per-unit variable cost associated with

the baseline pack and the probability density for maxi-

mum acceptable price – the demand density. Given this

full set of data, the price emerges as the outpu t of a profit

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699

maximising calculation.

If knowledge of the demand density is poor or non-

existent, it is still possible for a vendor with a good

knowledge of his unit variable cost to calculate an ap-

proximation to the profit-maximising price by using a

Uniform or Rectangular demand density to represent

customer demand.

The RUP model has been successful in generating

prices that provide an approximate match to those ob-

served for supermarket milk and eggs by means of ad-

justing the demand density. The resulting demand densi-

ties for milk and value eggs reveal a softening of the

market for farm retail commodities between December

2009 and August 2013 , which is fully consistent with the

long period of low growth and falling real wages experi-

enced by the UK economy.

The RUP model has been applied to the silicon chip

sector to explain the current prices of USB memory sticks,

how those prices have evolved and how they are likely

evolve over time. It has been shown that the silicon chip

industry’s established practice of introducing new memo-

ries with twice the capacity has had the beneficial

side-effect for the consumer that the price of the new

product that is twice as good as the old will settle down at

the old price. The mechanism will have pressured manu-

facturers to create higher capacity products due to the

previous generation’s prices quickly reaching com-

modities’ rates and thus returning low profits. Hence this

mechanism, given an economic explanation for the first

time, will have contributed to the exceptionally high rate

of technological progress in the memory industry.

Even though the data on cost and market structure are

incomplete, the RUP model has brought an understand-

ing of the key characteristics of real markets for different

packs of the same good, whether that good is a farm

product sold in a supermarket or a silicon chip offered on

the internet. The sizing and pricing of packs of the same

good have been given an economic explanation, irrespec-

tive of the good being sold. Thus the paper’s results

should be of interest to vendors, consumers and regula-

tors.

10. Acknowledgements

The authors would like to acknowledge numerous fruitful

discussions during the preparation of the paper with Mr.

Roger Jones (Hon. Research Fellow, City University

London) and with Mr. Edward Ross and Dr. Ian Wad-

dington, of Ross Technologies Ltd.

REFERENCES

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