American Journal of Industrial and Business Management, 2013, 3, 655-673
Published Online December 2013 (http://www.scirp.org/journal/ajibm)
http://dx.doi.org/10.4236/ajibm.2013.38075
Open Access AJIBM
655
Explaining the “Buy One Get One Free” Promotion: The
Golden Ratio as a Marketing Tool
Philip Thomas1, Alec Chrystal2
1School of Engineering and Mathematical Sciences, City University London, London, UK; 2Cass Business School, City University
London, London, UK.
Received October 20th, 2013; revised November 20th, 2013; accepted November 27th, 2013
Copyright © 2013 Philip Thomas, Alec Chrystal. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In accordance of the Creative Commons Attribution License all Copyrights © 2013 are reserved for SCIRP and the owner of the
intellectual property Philip Thomas, Alec Chrystal. All Copyright © 2013 are guarded by law and by SCIRP as a guardian.
ABSTRACT
Buy-one-get-one-free (BOGOF) promotions are a common feature of retail food markets, but why are they so wide-
spread? The theory of Relative Utility Pricing (RUP) developed in this paper provides an explanation not only for su-
permarket promotional offers but also for more general pricing of packs of different sizes in supermarkets and on the
internet. A clear and simple explanation is given for the two most widely used quantity promotions: BOGOF and
3-for-the-price-of-2. The RUP model may be linked to the theory of iso-elastic utility functions, and this allows the re-
lationships amongst risk-aversion, pack-size ratio and demand elasticity to be explored. “Cautious consumers”, as de-
fined in the paper, are found to be the only sensible target for quantity promotions. It is argued that the needs of cau-
tious consumers of retail commodities will be best addressed if the vendor sets the ratio of successive pack sizes as the
square of the Golden Ratio, namely 2.62, and the price-ratio at the Golden Ratio, 1.62. Thus the Golden Ratio may be
regarded as a marketing guide for vendors considering both their best interests and those of their customers. This
proposition is supported by an analysis showing that higher profits are more likely to come from Golden Ratio sizing
than from either BOGOF or 3-for-2 when variable costs lie in most of the upper half of the range that is required for any
of these multibuy offers to generate profit. The paper’s theoretical predictions for both pack sizes and prices are sup-
ported by examples from the retail sector: grocery, paperback books and electronics.
Keywords: Relative Utility Pricing; Golden Ratio Pricing; Buy One Get One Free; Multibuy Promotions; Utility
Theory
1. Introduction
Retailers often offer products in packs of different sizes,
whether it is a food product, electronics or clothing.
Moreover, multibuy promotions such as buy-one-get-
one-free (BOGOF) are common, but why are they so
widespread?
Put formally, the research questions addressed by this
paper are as follows. Suppose that a pack larger than the
entry-level pack is offered, what price will the vendor
need to charge in order to convince the consumer that he
or she should buy it? Then if further, larger packs are put
on the market, what prices will the vendor need to set in
order to sell these? What effect will the relative sizes of
the packs have on the reaction of the consumers? Finally,
how can the retailer maximise his profit given the atti-
tudes of the consumer.
2. Background
In his 2008 article in The Daily Telegraph, the Consumer
Affairs Editor revealed that “more than 80% of all pro-
motional activity within supermarkets is a bogof [buy-
one-get-one-free] or three-for-two” [1]. So concerned is
the UK Government about the success of such promo-
tions applied to food that it is considering banning mul-
tibuy deals in an effort to tackle obesity [2].
But there has long been puzzlement about vendors’
motivation in offering quantity promotions. Back in 1984,
the Wall Street Journal carried a front-page article stating
that: “The dark secrets of packaging aren’t always easy
to unwrap. While the issue of how much product a com-

Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool
656
pany should put in a box, bottle or tube is about as basic
as any in business, it also can be fraught with complexi-
ties” [3]; see also [4].
Tabarrok [5] in his response to Pressler [6] suggests
that BOGOFs “generally increase total welfare because
the price on the last unit sold is pushed closer to marginal
cost and because of this output expands”. Certainly, it is
relatively easy to understand how vendors will profit
from the selling of larger packs of the same good when
the marginal cost associated with the extra amount of
good is small compared with the fixed costs the retail
outlet has to bear (e.g. [7]).
Nevertheless worries have been expressed for some
time in academic journals that BOGOF-type promotions
might lead people to endanger their health through over-
eating, e.g. [8]. While, according to Wallop, the distinc-
tion between perishable and non-perishable goods means
that BOGOFs in supermarkets “are almost twice as
popular in the toiletries aisles as in the meat or vegetable
section”, the same restraint on food offerings does not
carry over to fast-food chains. Here Dobson and Gerstner
[8] confirm that “supersizing” food portions and offering
the larger size at “only a few cents more” than the regular
portion can be profitable for the vendor, but claim that
this may be harmful to the health of the consumer and,
moreover, create economic inefficiencies.
However, no satisfactory economic explanation has
been available until now for the reaction of the consumer
to the multibuy promotions. It is thus necessary to pro-
vide a quantitative explanation for how the larger packs
should be sized and priced if the consumer is to be in-
duced to buy a quantity greater than that contained in the
standard size, and the logic of these promotions needs to
be understood when restrictions are contemplated by the
authorities.
The problem is a general one, applying to retail outlets
including supermarkets, restaurants, public houses, fast-
food stores and internet shopping. It also features in
market surveys where the respondent is asked to set a
price on different options.
The first part of the paper (up to and including Section
8) will be devoted to developing a model to explain
quantitatively the prices consumers will expect for dif-
ferent pack sizes. Utility theory will be used in Sections 3,
4 and 5 to develop the general Relative Utility Pricing
(RUP) model, applicable where product differentiation is
low and the goods may be regarded as commodities. The
model developed provides, inter alia, a theoretical ex-
planation for quantity discounts.
Section 6 develops the RUP model for the common
case where, when deliberating whether or not to buy
pack n, the customer chooses to compare its benefit with
the benefit he would gain by buying pack 1n

. This
leads to the Single Comparison Relative Utility Pricing
model (SCRUP). The results of this model are surprise-
ingly simple, and provide a complete explanation of why
the retailer will find it necessary to offer buy-one-get-
one-free (BOGOF) or 3-for-the-price-of-2 promotions.
Put colloquially, people will expect “to get n but pay
1n

”. The theoretical results are used in Section 7 to
explain some actual prices achieved in supermarkets and
on the internet, including in cases where the explanation
for the pack size and price structure is non-obvious.
Section 8 compares the price per unit produced by the
SCRUP model with that produced by applying an iso-
elastic utility function. It is shown that the natural targets
for multibuy promotions are cautious consumers, who
have a risk-aversion between 0.0 and 1.0. A link is de-
veloped between pack size and the average risk-aversion
amongst cautious consumers, which leads to the notion
of Golden Ratio Pricing. Here the price quotient of suc-
cessive packs ratio is the golden ratio while the pack-size
ratio is the square of that figure.
Then, in Section 9, an analysis will be made of how
the retailer may design his pack 2 offering so as to
maximise his profit while satisfying his customers’ re-
quirements on price and volume. It will be shown how
the retailer may use the knowledge of his variable costs
to maximise his profit in his choice of which multibuy
offer to make: BOGOF, or 3-for-2 or the new promotion
suggested in this paper, Golden Ratio Pricing.
3. The Utility of Packs of Different Sizes
Consider a basic good, B, that is offered for sale in a
number of differently sized packs identified by index, j,
with 0,1,2, ,, ,, ,jknN

 , where j = 0 signifies
the empty pack – no purchase is made – and the index, j,
increases with pack size. Thus indicates the
smallest, entry-level pack, while pack N is the largest
pack on sale. The indices, k and n, with
1j
kn

, have
been introduced to facilitate later comparisons between a
general pack, n, and a pack, k, that is strictly smaller. The
maximum ranges of the three indices, j, k and n are:
0
0
1
jN
kN
nN
1





(1)
The amount of good, B, in the world may be very large
indeed, but we shall be concerned normally with small
amounts, b, measured in some unit of extent, for example
mass or volume or an entirely different measure, such as
gigabytes of memory on a USB memory stick.
Let pack j contain an amount,
j
b, of B, where, as
noted above, 00b

and 1
j
j for bb 
Nj

1. We
will define the largeness, L(j), of pack j as its size relative
to the entry-level pack 1:
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Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool 657

1
for 0
j
b
Ljj N
b

(2)
Thus the largeness of pack j is the number of standard
units that pack j contains, where a standard unit com-
prises the contents of pack 1, namely . Hence
and .
1
b

00L

11L
Kahneman and Tversky [9] have argued that “people
normally perceive outcomes as gains and losses [relative
to the current asset position], rather than as final states of
wealth or welfare”, a proposition supported by Thaler
[10]. This approach of measuring the utility of the
change in the stock level rather than the utility of the
absolute value of the stock is adopted in this paper, and
leads to a consideration of the utility of the contents of
the pack. Thus the utility of the empty pack, with con-
tents 0 will be , the utility of the contents of
pack 1, namely 1, will be
0b

0
ub
b


1
ub , and, in general, the
purchase of pack j will result in a utility,

j
ub .
Consider the case where the vendor is offering just two
packs: pack 1, with contents, 1, and a bigger pack, let
us call it pack x, with contents, 1. We will assume
that the price of the entry-level pack, pack 1, has been
accepted by purchasers as 1, implying that these con-
sumers will have a maximum acceptable price (MAP) of
at least 1. (See [11] and [12] for a discussion of the
concept of maximum acceptable price and probability
distributions.)
b
16b
p
p
Let us select one of these purchasers with a MAP of
1 or higher, who buys one standard pack. We may give
him the name, “consumer X”. Meanwhile there may be
another person, say consumer Y, who has a MAP of 1
or higher and also a need or strong desire for 1. Be-
cause no larger pack is offered, he will have no option
but to buy two of the standard packs, which will cost him
.
p
2p
p
2b
1
Suppose now that the vendor withdraws the existing
pack 1 and replaces it with a new pack 1, which contains
1. Consumer X will be put in a quandary, but con-
sumer Y, whose MAP is at least 1, will be indifferent
to this change, and will be content to continue paying
1 for the same amount of the good, B, now contained
in the new pack 1.
2b
2p
p
Let us now transfer our attention to a person, let us call
him consumer Z, with a MAP of 1 or more who has a
need or strong desire for 1. Consumer Z would need
initially to buy 4 of pack 1 at 1 each, then, after the
vendor’s doubling of the size of pack 1, would be content
to buy 2 of new pack 1 at 1 each, making his total
outlay 1. Moreover, it would make no difference to
him if new pack 1 were supplanted by a further revised
pack 1 with contents, 1, and price . His total bill
remains the same in all cases.
p
p
4b
2p
4p
4b1 4p
Now suppose that the original pack 1 containing is
reinstated, and moreover a pack 2 containing 1 is also
offered. The pack containing 1
4 is retained but it
should now be called pack 3, because there are now two
non-empty, smaller packs. The presence of the two
smaller packs has given the prospective purchaser more
choices, and this makes the pricing of the pack contain-
ing 1 more complicated than it was before. Thus we
can no longer say that the price of the pack containing
1 should be 1. Nevertheless the price, 1, retains
the significance that it is the price that customer Z would
prepared to pay for a pack of size 1 if no smaller
packs were available and the only other option was to
buy nothing – the empty pack or pack 0 in our terminal-
ogy. This limiting price for pack 3 may be denoted ,
where
1
b
2b
4p
b
4b
4b4p
4b
30
p


3
301 1
The larger pack, pack x, has not figured in the analysis
above, so that its existence or otherwise is not relevant.
This will hold true for customers, X, Y and Z, as long as
the price of pack x is not below that of the largest of the
smaller packs. (This can happen on occasion, but rarely
on a continuing basis, as it contradicts the economic
principle that more will be preferred to less.)
4pLpp
b
p
.
Generalising, we may observe that if a purchaser, say
consumer Ω, has a MAP of 1 or more for a pack con-
taining 1, has a need or strong desire for the quantity
contained in pack n, namely , and he chooses to
ignore any packs between pack 0 and pack n, then the
retailer may continue offering the smaller packs without
affecting the valuation, 0n, put on pack n by this pur-
chaser. Because he has decided to ignore smaller packs,
person Ω will be prepared to pay for pack n, given
by:
p

1
Lnb
n
p0


n
0n
pL1 p
(3)
The condition of ignoring packs smaller than pack n
will be satisfied automatically when n = 1, since, by
definition, no pack smaller than pack 1 is offered, apart
from the empty pack, pack 0. Thus putting n = 1 in Equa-
tion (3) produces the price for pack 1:


110 1pL 1
pp

 (4)
Now let us consider utility. The utility resulting from
buying pack n will be


n
bu , and the difference, 0n u

,
between the utility of having purchased pack n and that
of having bought pack 0, i.e. made no purchase, will be
given by


ub

Ln


0
b0nn uu (5)
Assume next that the purchase in question was made
on behalf of a consortium of people, each of
whom
 has a MAP of at least p;
1
 has a need or strong desire for a quantity of the good,
1
b;
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Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool
658
 is allotted 1 b from the contents,

1
Lnb, after pack
n is purchased.
If we make the assumption that the utility gain for
each person in the consortium is the same, then the gain
in the utility for each consortium member will be
0
. This will be the same as if that member
had himself gone out and bought pack 1, as we may con-
firm by putting n = 1 in Equation (5) to give the total util-
ity gain from the purchase of pack 1 as:
 
1
ub ub
 
10 10
uubub  (6)
The total utility gain for the consortium of


Ln
people may be found by adding together all the compo-
nents identified in Equation (6):
 

01010
11
1
Ln Ln
nii
uuuLn

 
10
u (7)
demonstrating that the utility gain from pack n will be
proportional to the utility gain from pack 1, with a factor
of proportionality, . For the utility gain from pack
n to be proportional to the utility gain from pack 1, then
the utility differences for the purchasers of pack n and
pack 1 will obey a very restricted version of homogeneity,
“singular homogeneity”, as explained in Appendix A.

Ln
An example of a consortium as described above would
be 4 university students sharing a house and joining to-
gether to buy a 4-pint pack of milk a day to be divided
equally between them. Another consortium could be a
mother buying the same pack of milk daily for her family.
On the other hand, an individual living alone and con-
suming milk at the same rate as one of the students might
buy a 4-pack of milk every 4 days. In this last case, the
consortium consists of the individual on day 1 and the
same individual on days 2, 3 and 4. This generalization
of the consortium so that it can include time-delayed ver-
sions of the same individual makes it reasonable to as-
sume that that every purchaser of pack n experiences the
same utility gain, 0n, whether part of a multi-person
consortium or not. This equality in the gain in utility is
taken to be independent of the price paid for the pack,
and to be valid in cases where the pack-size ratio is non-
integer.
u
Combining Equations (3) and (7), we may write the
ratio of the changes in utility in terms of the ratio of the
limiting price for pack n to the achieved price for pack 1:
00
10 1
1
nn
up nN
up

 (8)
But in general there will be packs intermediate be-
tween pack n and pack 0. In this case, when considering
the purchase of pack n, in addition to the utility differ-
ence between pack n and pack 0, the consumer may well
wish to take into account also the utility differences,
, , that will arise from comparisons between
pack n and packs in between pack n and pack 0:
nk
ukn


1
nk nk
uubub kn

 
(9)
The change in utility between buying pack n and
making no purchase may be then be expressed in terms
of intermediate utility differences:






00
0
nn nkk
nk k
uubububububub
uu
 
 
0
0
(10)
where the last step has made use of Equation (5), replac-
ing the subscripts, n, and 0 with the subscript, k, as nec-
essary. Thus the change in utility between purchasing
pack n and a smaller pack k may be written in terms of
the utility gains from buying pack n and pack k, respect-
tively:
0nk nk
uuu

  (11)
Hence
0010
00 100
1
nk nn
kk k
uu uu
uu uu
 
1

 
 (12)
Substituting from Equation (7) with the appropriate
subscripts into Equation (12) gives



0
1
nk
k
Ln
uk
uLk
0


 (13)
Since




Ln Lk, it follows that 0nk k uu will
always be positive.
4. Comparing Packs of Different Sizes
Because they are of the same good, packs may be char-
acterised by only two quantities, their size and their price.
But while a single comparison involving multiple packs
would be feasible if the packs could be characterised by
just one quantity, when each pack has even as few as two
characteristic features, it becomes difficult if not impos-
sible for the consumer to compare packs more than two
at a time, even when both those characteristics are repre-
sented numerically. This is exactly the situation ad-
dressed by mathematics when comparisons are to be
made between an array of properties, formalised mathe-
matically as a vector. Generally it is not meaningful to
say that a vector, x, is bigger than another, y, of the same
dimension, the only exception being when the vectors are
linearly related:


yx, where

is a constant. In
every other case, further mathematical operation is
needed before it is possible to make a meaningful com-
parison, such as taking the modulus: 2
i
i
x


x, and
then comparing the single number x with the single
number, y.
When the packs are characterized by size and price, it
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Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool 659
is, of course, possible to divide the price by the size to
give the price per unit. This is normally done in UK su-
permarkets by the vendor, who is obliged to display the
result to his customers. But while the prices per unit over
all n are easily ordered and hence may be compared mul-
tiply, the unit price does not give a full characterization
of the pack, for which the size and price are still needed.
For example, suppose that the price of the standard pack
containing one unit is £2 and the price of a pack contain-
ing 10 units is £19, then based on the unit price the larger
pack is clearly preferable. But the number of units may
be more than the purchaser needs or the amount of
money required may be more than the consumer wants to
lay out on this good, so that actually the consumer will
prefer pack 1. Hence the role of the price per unit for
pack n, n

, is to add to the features characterizing the
pack, meaning that the vector of characteristics has risen
to three:

n
n
Ln
p






strengthening the necessity for pair-wise comparisons.
Consumers may carry out comparisons over the whole
range of packs, or they may make an outline decision
early on in the selection process on the range of packs
that they judge may meet their desire for the good, B, and
their budgetary limitations. In each case, the pair-wise
comparison will be made by taking each pack, pack n,
within the feasible range, and comparing it with a smaller
pack, . A price, nk , will emerge from each
comparison. The consumer is presumed eventually to
eliminate from consideration all sizes greater than his
ideal size, and base the price he is prepared to pay for
this ideal size on the outcome of some or all of the com-
parisons he has made using this pack as his basis. Thus
the final price for pack n will be a function of the com-
parison-generated prices or a subset of them:
:0kk
n

p

01 ,1
,,,,,
nnnnknn
pgpp pp

 (14)
Thus the set of feasible binary comparisons for pack n
will be between pack n and the following packs, taken
one at a time:
 pack 0 (= the empty pack or no purchase).
 pack 1.
 
 pack k.
 
 pack n – 1.
At each comparison between pack n and pack k, the
consumer is presumed to ask himself the question:
“Suppose I am on the point of agreeing to buy pack k,
what will it take to make me want to buy the greater
quantity contained in pack n?”
He may then reason that the best answer to this ques-
tion will be found by consulting a mentor with greater
relevant experience, namely someone who has actually
bought pack k. This strategy may be characterised as
“phone a friend”. The mentor has something the con-
sumer does not: he knows what it feels like actually to
own pack k. He has knowledge borne of experience, the
knowledge of possession.
Let us now consider the perspective of the mentor. As
a purchaser of pack k, the mentor may be presumed to
have most of his need satisfied by pack k, which contains


Lk units. Buying pack n in addition will give him




LkLn units. But will be significantly
greater than his original target, , and so those

Ln

Lk


Ln units may be presumed to meet his wants in full.
Hence


Lk units out of the will now be
surplus to requirement, and bring the mentor no utility.
As a result, after purchasing packs k and n, the person
who originally bought pack k will take the utility gain
from pack k and n together as equal simply to the utility
gain from pack n. So in considering whether to buy pack
n, given that he possesses pack k, the mentor will be pre-
pared to pay for only the extra utility that pack n adds
over and above that provided by pack k, that is to say,
nk

L

Ln k
u

. We may presume that he would advise the pro-
spective purchaser accordingly.
We may further suppose that the prospective purchaser
of pack n making the comparison with pack k will accept
the advice of his k-n mentor and take the utility gain from
pack n as relative to the gain brought by pack k, namely,
nk
u

. By Equation (11), nk u

is the margin between
the utility gain from of pack n and the utility gained from
pack k, and this is smaller than the utility gain, 0n
u

.
Thus, when the needs or desires of a prospective pur-
chaser of a commodity are close to being satisfied by a
pack smaller than pack n, that purchaser will see the ac-
quisition of pack n bringing him less than its full utility.
It will be apparent to the reader at this point that the
prospective purchaser does not actually need to “phone a
friend” at all. He can work it out for himself simply by
imagining how he would feel if he had, in fact, bought
pack k.
Such an approach is likely to work well with com-
modities, which are by their nature simple. The demands
on the consumer’s imagination might become more tax-
ing with more complex and differentiated goods, where
real, third-party advice might be needed. But such dif-
ferentiated goods tend to be sold as one-offs rather than
in packs, taking away the need for a k-n comparison. The
vendor may, of course, attempt to provide differentiation
for goods that many would regard as commodities, and a
discussion of such situations is given as part of the field
studies included in [13].
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Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool
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5. Prices Arising from a Binary Comparison
between Pack n and Pack 1
If the consumer narrows his final pair-wise comparison
down to either buying pack n or making no purchase
(buying pack 0), the price he will be prepared to pay will
be 0n, as given by Equation (3). This represents the
case of a consumer with a MAP of at least who con-
siders that he has such a strong need for standard
units that he is not prepared to take any lesser quantity
and decides to ignore all packs of intermediate sizes. It is
this that causes him to value those units on the
full utility gain rather than their utility relative to some
intermediate purchase.
p
1
p

Ln



Ln
It may be, however, that the consumer will grant him-
self the flexibility to make a comparison between pack n
and the first pack. Let the price that he is prepared to pay
for pack n when his comparison is between pack n and
pack 1 be 1n. We now assume that the price per unit
gain in utility stays constant for all transactions he is
considering – a standard economic assumption, e.g. [14].
Hence
p
11
11
n
n
pp
uu


0
(15)
Meanwhile the ratio of utility differences, 110n uu,
may be found by putting k = 1 in Equation (13):

1
10
1
n
uLn
u

 (16)
Since , the price that the consumer will be
prepared to pay for pack n, if his comparison narrows
down finally to the purchase of pack n or pack 1, may be
found by combining Equations (15) and (16) and also
using from Equation (4):

11L
10 1
pp


1
11
10
max 1.0,1
n
n
u
pp Ln
u

 


1
p (17)
A maximum function has been added in Equation (17),
embodying the general economic principle that, all other
things being equal, less will not be preferred to more, and
so the larger pack will not attract a lower price. (Without
it, a largeness of , for example, would imply
that the price of pack n would be lower than that of the
smaller pack 1. While such a pricing tactic is not un-
known as a marketing promotion, its use is rarely main-
tained in the long term, as noted previously.)

1.5Ln
Noting that , it will be seen by comparing
Equations (3) and (17) that the price derived by compare-
ing pack n with pack 1 cannot be greater than the price
achieved by comparing pack n with pack 0. Since some
consumers of pack n are almost certain to make the for-
mer comparison, the retailer will need to accommodate
the likely lower valuation to some degree, and will set a
price for pack n that is lower than would be implied by
proportionality to size. In general, the achieved price,
n, for any pack, n: n > 1, is almost certain to be less
than :

1Ln
p
0n
p
0nn
pp

(18)
This provides the theoretical basis for quantity dis-
counts.
6. Prices Arising from a General Binary
Comparison between Pack n and Pack k;
the Single Comparison Relative U tilit y
Pricing Model (SCRUP)
When the purchase of pack n is being considered relative
to the purchase of pack k rather than pack 1 the assump-
tion of constant price per unit utility gained produces
Equation (19), which is analogous to Equation (15):
0
nk k
nk k
pp
uu


(19)
where k is the achieved price for pack k. As noted
previously in Section 3, the same gain in utility, 0k
pu

, is
assumed for all purchasers of pack k, as found from
Equation (5) after substituting k for n.
Combining Equations (13) and (19) gives the price for
pack n emerging from the comparison of pack n and pack
k as:


max 1.0,11
nk k
Ln
pp
Lk

k





 (20)
Once again, the maximum function embodies the gen-
eral economic principle that, all other things being equal,
less will not be preferred to more, and so the greater
quantity will not attract a lower price. Equations (20) and
(3) together give nk for all , and, of course,
Equation (20) takes the form of Equation (17) when k =
1.
p0k
The customer considering buying pack n, of largeness,


p
Ln, will have the following possible prices available
to him, based on his comparisons with other, smaller
packs, ranging from the empty pack, pack 0, to the pack
immediately below pack n in size, namely pack n – 1:
01 ,
,,,,,
nnnknn
pp pp

 1 (21)
The good in pack n is effectively competing with itself
contained in smaller packs. The maximum acceptable
price (MAP) for individual, i, interested in purchasing
pack n, , will therefore be a function of the prices
shown in list (21). It is suggested that the dependency
can be represented by the weighting function:

i
n





 

00 111,1
1
0
ii iii
nnnknknn
ni
knk
k
pwpwp wpwp
wp
n




 


(22)
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Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool 661
where , are a series of weighting
factors particular to the individual, which will be subject
to the conditions:

i
k
w0,1,,1kn

pp


1
0
01
1
i
k
ni
k
k
w
w




 (23)
The aggregation of these MAPs will produce a prob-
ability density for MAP for the target population, which
will be constrained to the interval, , as
shown in Appendix B.
 
11
,
n
pLnp

In considering whether or not to purchase pack n, of a
commodity, the greatest weight will usually be given to
the comparison between pack n and the pack just below
it in size, pack n – 1. In the limiting case where this
comparison is completely dominant, .

11
i
n
w
All other comparisons are now either not made or are
considered irrelevant. The achieved price for pack n will
be ,1nnn when all the prospective purchasers
adopt this policy. We may describe this as the Single
Comparison Relative Utility Pricing (SCRUP) model,
which will be shown to have the power to explain a
number of everyday pricing strategies the economics of
which have been understood only poorly until now.
pp


7. Using the SCRUP Model to Explain the
BOGOF and 3-for-the-Price-of-2
Promotions
To illustrate the results and the power of the SCRUP
model, we shall consider first the case of the pricing of
pack 2, when there is only one pricing comparison that
can be made, that is to say between pack 2 and pack 1.
Here n = 2 and k = 1. Let us consider first the case where
pack 2 holds twice the contents of pack 1 so that
. If buyers choose to compare the two packs,
we have, after substituting the appropriate values into
Equation (20):

22L

211 1
max1.0, 21p (24)
This means that if the purchasers of a commodity have
needs or desires that are close to being satisfied by pack
1 and so are making a serious comparison between pack
1 and a pack 2 that is twice the size, then they will be
prepared to pay only the same for the larger pack as they
were prepared to pay for the smaller pack. Prima facie
surprising, this result nevertheless provides a complete
explanation for why the vendor needs to offer the appar-
ently generous BOGOF promotion.
By a combination of technological convenience and
custom and practice, universal serial bus (USB) memory
sticks are offered with storage capacities ascending in
multiples of 2. The lack of novelty associated with the
lower-capacity sticks means that they are regarded purely
as commodity item. Hence, by Equation (24), we can
expect the price of the two lowest-capacity sticks to be
equal or nearly so. Hence in December 2009, the en-
try-level 1 Gb USB memory stick was on sale on the
internet at £6.96, while a 2 Gb stick from the same
manufacturer was offered at a comparable price—in fact
slightly lower at £6.44 [15]. This pricing was not a
one-off, as demonstrated by the pricing, early in August
2013, of the entry-level pack, now containing 4 Gb of
memory at £5.79, while the next pack up from the same
manufacturer, containing 8 Gb, was on sale for almost
the same price, £6.03. The pricing of such memory sticks
is considered in more detail in [13].
The SCRUP model can also explain immediately a
perplexing instance of everyday pricing. It used to be
normal practice in the UK to sell eggs by the dozen, but
supermarkets have taken to making half a dozen the size
of pack 1 for “value” eggs. Surprisingly, UK supermar-
kets no longer sell “value” eggs in packs of a dozen [16].
But if they did, Equation (24) shows that they would
find themselves compelled to sell both 6 egg and 12 egg
packs at the same price, a situation with which they
would probably not be happy on a continuing basis. In-
stead major UK supermarkets have taken to selling their
“value” eggs with a pack 1 set at 6 eggs and a pack 2
containing either 15 eggs or 18 eggs.
Considering the case of packs of “value” eggs of 6 and
18, substituting, n = 1, k = 1 and into Equation
(20) produces:

32 L


1121 213,0.1max ppp 

(25)
The achieved price predicted by the SCRUP model for
pack 2 will be equal to the result of Equation (25), viz.
1212 2ppp


. Equation (25) provides the explanation
for the other common sales promotion: 3-for-the-price-
of-2.
Validation for this prediction of the SCRUP model
comes from the prices charged by one of the UK’s larg-
est supermarket chains, namely Sainsbury. In that su-
permarket, a pack of 6 “Basics Barn Eggs” (pack 1) was
on sale for £0.91 in December 2009, while pack 2, which
contained 18 eggs, retailed for £1.85 [16]. This is within
2% of the theoretical figure from Equation (25) of
82.1£91.0£2


. Further discussion of prices achieved
will be given in Section 8.3 and, in greater detail, in [13].
An example of a quantity offer that the SCRUP model
makes no claim to explain is the promotion, “Buy one get
one half price”. How is it that the retailer is able to com-
mand 1½ times the price for two items, rather than just
the same price for two items as for one item? In fact, this
promotion is commonly used in selling paperback books,
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Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool
662
for example by the large U.K. retailer, W. H. Smith. W.
H. Smith applies this offer both to fiction and to paper-
backs on management, economics, business and self-help,
or to any mixture of the categories. But given that the
customer is unlikely to buy two of the same book, the
offer may be seen to concern two complex and different-
tiated goods, which do not fall within the capability of
the SCRUP model, as explained at the end of Section 4.
The interesting feature here is that the retailer has de-
cided to increase his sales of paperback books by re-
garding these items as half-way to being commodities,
and this allows him to use a variant of the promotional
techniques that apply to retail commodities.
8. Comparing the Price per Unit from the
SCRUP Model with the Price per Unit
from the Iso-Elastic Utility Function
In this Section we will restrict our attention to the com-
mon case where the packs increase in size by a constant
pack size ratio,

.
8.1. Price per Unit from the SCRUP Model
In this analysis we will drop the maximum function from
Equation (20), which may therefore be written in the
simpler form:
  

1
k
nk
p
pLnLk k
Lk
  (26)
Equations (20) and (26) will give identical results
when
 
12Ln Ln  , and dropping the restric-
tion of the maximum function will allow the exploration
of inelastic demand.
The achieved price predicted by the SCRUP model
will be ,1nn , allowing us to write ,1nnn . Substi-
tuting n – 1 for k and dividing both sides of Equation (26)
by the number of units in pack n, , gives:
ppp



Ln


 
1
1
12
1
nn
Ln
pp
n
LnLn Ln



 



 (27)
The price per unit for pack n, n

, is:

n
n
p
Ln
  (28)
and so Equation (27) may be rewritten:


1
111
11
n
n
Ln n
Ln




 2 (29)
Prices per unit relative to the price per unit for pack 1,
1n

 may be found successively from:
2
1
2
33 2
112
1
1


 
 







(30)
and so on, so that the general expression is
1
1
11
n
nn



 



 (31)
These per unit price ratios will be valid at the discrete
points corresponding to the various pack sizes.
8.2. The Price per Unit from the Iso-Elastic
Utility Function
If we assume that the amount of good, B, that may be
contained in a pack is a continuous variable, b, the gain
in utility may be modelled by the continuous utility func-
tion,


ub. The price per unit or price density will now
also be continuous in b:


b 
b. Hence the cost of buying
an extra, small amount,

, starting from an initial
amount, b, will be


bb

 . In the case where the quan-
tity is equal to the amount contained in the starting pack,
pack 1, then 1 bb

, and the cost of a further small
amount, b

, will be approximately

1
bb

 .
Meanwhile, the extra utility associated with having
acquired an extra, small amount, b

, of B in addition to
amount, b, will be




ububb ub  

(32)
Correspondingly, the extra utility associated with a
small amount, b

, of good, B, added to , the contents
of pack 1, will be
1
b




11
ububb ub  

1
(33)
The standard relationship between price and marginal
utility used in Equations (15) and (19) may be extended
to the case of a continuous quantity, b, so that the ratio of
Equations (33) and (32) gives:





11
bb ub
bb ub


 (34)
or








1
11
as 0
bubububb
bbbub
 



 
 (35)
An iso-elastic utility function, , giving the utility
of a continuous quantity, b, has the property that the elas-
ticity of marginal utility is constant, specifically inde-
pendent of b. Pratt [17] and Arrow [18] showed that the
negative of the elasticity of marginal utility was equal to
the coefficient of relative risk aversion,

ub

, which we
shall denote “risk-aversion” for short. Thus
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Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool 663


 

ddd dd
d
ub bubbub
b
bbu


 b
(36)
For an iso-elastic utility function to be smooth and
regular in risk-aversion, it must take the form [19]:

11 for 1
1
ln for 1
b
ub
b











(37)
This has the derivative with respect to b:

1for allub b


 (38)
Combining Equations (35) and (38) gives:


1
1
bb
bb




 (39)
If we convert b into standard units, with 1 standard
unit corresponding to the contents of pack 1:
1
1b

,
then:


1
1
b
b



 (40)
The elasticity of demand with respect to price or “de-
mand elasticity”,

, is the normalised change in quan-
tity sold with price, given by
d
d
bb



 (41)
Applying Equation (41) to Equation (40) shows that
the demand elasticity,

, is the negative inverse of the
risk-aversion:
d
d
bb

1


 (42)
When risk-aversion, 1


1
, the demand elasticity will
take the neutral value,



, neither elastic nor inelas-
tic. Furthermore, inspection of Equation (40) shows that
the locus of
 
1  b  vs. b will be a rectangular hy-
perbola when 1

.
Since all packs will be at least as big as the standard,
entry-level pack, we need consider only that part of the
curve where . For this sector of the graph, the
price-ratio curve will lie above the rectangular hyperbola
when the risk-aversion,
1b

, conforms to 1

, which
means that the demand elasticity, 1

 , and the de-
mand is elastic. The price-ratio curve will lie below the
hyperbola when the risk-aversion obeys 1

, signify-
ing a demand elasticity, 1

 , implying inelastic de-
mand.
The contents of pack n will be . Hence, noting
also that b is now denominated in standard units, so that
, we may set b to:
1
1
nb


11b
1n
b


 (43)
The price per unit required to clear 1n b


 units of
the continuous good will be given by substituting Equa-
tion (43) into Equation (40)




1
1
1
1
n
n 




 (44)
8.3. Equating Prices from the SCRUP and Utility
Models
Equating the prices required to clear 1n
b


 units of
the continuous good with the price the SCRUP model
suggests for a pack containing 1n

 units means that we
may equate the right-hand sides of Equations (31) and
(44):
1 1
11
n n



 
 

 
(45)
which gives the relationship between risk-aversion,

,
and pack size ratio,

, as:

ln 1
1ln




 (46)
Equation (46) shows immediately that a continual dou-
bling of the pack size, viz. 2

, is consistent with a
risk-aversion, 1


, which will give rise to a logarithm-
mic utility function, as recommended by H. M. Treasury
for the evaluation of public-service schemes [20]. As
demonstrated above, this produces the neutral demand
elasticity, 1.


 Figure 1 shows the plot of risk-
aversion versus pack-size ratio.
Meanwhile a risk-aversion, 0

, may be seen to re-
quire an infinite pack size ratio:

. This may be
deduced from Figure 1 or directly from Equation (46)
and verified intuitively by the following argument. The
impossibility of constructing an infinitely large pack
corresponding to


 means that pack 1 will be the
0
1
2
3
12345
Pack-size rati o, 
Risk-aversion,

Figure 1. Risk-aversion versus pack-size ratio.
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Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool
664
only pack. Hence all quantities will be supplied as multi-
plies of pack 1, implying that the per-unit price will stay
constant. By Equation (35), an unvarying per-unit price
means that the slope of the utility function will be con-
stant so that the utility function must be linear. By in-
spection, the iso-elastic utility function of Equation (37)
becomes linear in b only for 0


.
Equation (46) will not predict a risk-aversion with a
negative value, 0

. This means that the analysis in
this Section will include only those who see either con-
stant or diminishing returns from extra amounts of the
good, B, and will preclude those who see increasing re-
turns from extra amounts of the good, B. This does not
constitute a significant practical limitation, since it is
assumed generally in economics that the individual will
perceive diminishing marginal utility from larger
amounts.
Amongst those with positive risk-aversions, who
therefore will see strictly diminishing returns from extra
amounts of the good, B, let us define as “cautious” those
with risk-aversions in the range, 01

. Cautious
individuals will always perceive a benefit from more of
the good, even when the quantity is very high, since their
utility, , will increase without limit as b increases.
By contrast, let us call those with a risk-aversion strictly
greater than unity,

ub
1

, “ultra-cautious”. The utility,
, of ultra-cautious individuals will approach an up-
per, asymptotic limit as b. Moreover, it is clear
from Figure 1 that a risk-aversion of greater than unity
will call for a pack-size ratio of less than 2. Therefore the
limit contained in Equation (17) will always need to be
invoked, and even then such ultra-cautious individuals
will want to pay the same price for all packs, irrespective
of size.

ub 
While the vendor may face a market composed of cau-
tious and ultra-cautious individuals, only the cautious
consumers will always want more and only the cautious,
not the ultra-cautious, may be prepared to pay more for a
larger pack. Therefore it is only the cautious consumers
whom it is sensible for the vendors to target with quan-
tity promotions.
In the absence of more precise information, it is rea-
sonable to assume a uniform probability distribution for
risk-aversion amongst the cautious consumers, over the
open interval, 01


0.5
. The average value of this dis-
tribution is


0.5
, and, indeed, the same mean will
result from any symmetrical distribution over the same
interval. Thus, in the absence of evidence to the contrary,
the figure,

, may be regarded as the risk-aversion
for the average person likely to be interested in a sales
promotion offering greater quantities. This value of
risk-aversion requires from Equation (46) that
21
   (47)
where

is the golden ratio (see e.g. [21]). (The result is
easily confirmed from Equation (46) by using Equation
(47) to replace


ln 1 

with ln

and substituting
2
ln2 llnn

   . Hence

1ln   2ln 0.5  .)
The properties of

mean that when the pack size ratio
is set to its square, 2

, then the normalised difference in
size between two successive packs will be equal to the
golden ratio:





1
1
Ln Ln
Ln



 (48)
We may combine Equations (27) and (47) to show that
the ratio of the price of pack n to the price of pack n – 1
will also be equal to the golden ratio:
1
n
n
p
p


 (49)
The value of

, found from solving the quadratic
equation of Equation (47), is approximately 1.62, so that,
from the same equation, .
22.62
 
On this basis, if the vendor is to increase pack sizes by
a constant ratio,

, within a range of quantities known
to be desired by cohorts of his customers, the vendor can
expect to be match his market most accurately if he sets
the size ratio to the square of the Golden Ratio (Equation
(47)) and his price ratios at the Golden Ratio (Equation
(49)). This result will clearly apply to the case where just
two packs are offered, when the size ratio of pack 2 to
pack 1 should be the square of the golden ratio and the
price ratio should equal the golden ratio. The matching of
the pack size and its prices to the values desired by the
average customer liable to be tempted by a multibuy of-
fer means that there is a strong case for regarding Golden
Ratio sizing and pricing as delivering what the customer
wants most.
Figure 2 gives the normalised price per standard unit
for packs of different sizes when the comparison is al-
ways made with the pack immediately below (the
SCRUP model), and when the pack-size ratio,

, stays
constant. The Figure illustrates the cases where this ratio
takes the values, 2, 2.5, 2

and 3, and shows how the
locus derived from iso-elastic utility functions with cor-
responding risk-aversions 1.0, 0.56, 0.5 and 0.37 respec-
tively will intersect the discrete data points. Also plotted
is the locus corresponding to a pack size ratio of 50: the
risk-aversion, at 0.005, is now close to zero, implying an
absolutely very large demand elasticity of −200, and a
price density vs. quantity curve that is nearly horizontal.
Figure 2 demonstrates that when the pack-size ratio is
2, the SCRUP Model will predict a price equal to that
produced by a logarithmic utility function. This corre-
sponds to the case where a pack containing double the
quantity will attract the same price as the first pack, as
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Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool 665
may be seen from Equation (24). The demand elasticity
is neutral under this condition. The retailer may move the
market into a region of elastic demand by using a higher
pack-size ratio, and can expect to achieve better per unit
prices as a result. However, as illustrated by Figure 2
and as a moment’s thought will confirm, these higher per
unit prices will come at the expense of poorer coverage
of the range of potential purchases. While higher size
ratios (e.g. a ratio of 3) should give higher per unit prices,
the square of the golden ratio (~2.62) should offer an
optimal compromise between higher per unit price and
coverage of the range of quantities required by consum-
ers.
More detail on the prices achieved by major UK su-
permarkets will be given in [13]. Here we will content
ourselves with a brief discussion of one result given there,
concerning “value” eggs where the size of pack 1 was 6
eggs and the size of pack 2 was either 18 eggs (Sainsbury)
or 15 eggs (Asda and Tesco) and where pack 1 was re-
tailing at the same price, £0.91, in all three supermarkets.
The price per egg achieved by Sainsbury for its 18-egg
pack 2 was £1.85/18 = £0.103, which is within 2% of the
theoretical figure of

101.0£1891.0£1618

.
As noted in the previous paragraph, pack 2 for value
eggs in Asda and Tesco contained 15 eggs, giving a
pack-size ratio of 2.5, which is significantly closer to the
ideal ratio derived above of . By selling pack
2 at £1.50, these stores achieved a price per egg of
£0.100 with their pack 2, which is 9% higher than the
predicted price of
22.62
 

091.00£615 £1591. . The ideal
number of eggs in pack 2 based on this theory would be
or 16 after rounding. A pack 2 containing
16 eggs should command a price of
26 15.72
 

52.1£91.0£1616  , giving a price per egg of
£0.095. The premium price per egg achieved by Tesco
and Asda with their pack 2 may result from the pack size
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0123456789101112
Number of units in pack
Normalised price per unit
Pack size ratio = 2.0
Pack size ratio = 2.5
Pa ck s ize ratio = 2.62
Pack size ratio = 3.0
Risk-aversion = 1. 0
Risk avers ion = 0. 56
Risk-aversion = 0.5
Risk-aversion = 0. 37
Risk avers ion = 0. 005
Figure 2. Effect of pack size ratio on the price per unit.
satisfying better people’s sense of the most appropriate
next size up. On the other hand, it may well be that they
priced their pack 2 based on the price per egg that their
competitor, Sainsbury, was achieving, and might have
done better through higher sales by pricing their pack 2
at


37.1£91.0£1615



. The theory just developed
suggests that they might have done better still by putting
16 eggs into their pack 2, which they would then have
sold at £1.52, since this would have matched the re-
quirements of the average cautious customer best.
Interestingly, by August 2013, Tesco was pricing its
value eggs at £0.87 for a half dozen and £1.35 for 15 [13].
Using the SCRUP model, the predicted price for 15 eggs
would be


31.1£87.01615



. The actual price had
thus drawn significantly closer to the predicted value—a
3% difference as opposed to the earlier 9% difference.
9. The Retailer’s Problem: Choosing the
Largeness of Pack 2 to Maximise Profit:
BOGOF vs. 3-for-2 vs. Golden Ratio
Pricing
Fundamentally, although the customer will want quantity
discounts, the retailer will be prepared to offer them if his
variable costs are low enough for each sale to make a
non-negative contribution to covering his overall costs.
The purpose of this Section is to examine the circum-
stances when the vendor can maximise his profit by set-
ting the largeness of pack 2 as , as in a BO-
GOF, when it is best to set it as , as in a 3-for-2
offer, and when the most profitable strategy might be to
set the largeness of pack 2 at the square of the Golden
Ratio,

22L

23L


2
22L  .62. This Section is therefore con-
cerned with the overall design of pack 2, in terms of both
size and price.
9.1. The Optimising Equation
The exercise of choosing the largeness of pack 2 to
maximise the vendor’s profits may be treated in a similar
way to that presented in [11] by using a probability dis-
tribution to model the maximum acceptable price (MAP)
amongst those making up the target market. The further
complication in this case is that the MAP for pack 2 var-
ies with its largeness. For the cases that we want to con-
sider, where the pack-size ratio is at least two: 2

,
the relationship is given by putting n = 2 into Equation
(27) and rearranging:

 
211
211
1
L
ppp
L

 2

 


 (50)
where, since


11L

, the largeness of pack 2 is identi-
cal with the pack-size ratio,

2L

. For convenience,
let us define the price of pack 1 as our base money unit
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Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool
666
so that 1 and the price of pack 2, 2, is expressed
as a multiple of the price of pack 1. Using this system of
units, we may then simplify Equation (50) into the form:
1pp
21p2

   (51)
The variable cost of pack 1, v, may be expressed in
terms of the same base unit, the price of pack 1. Assum-
ing moreover that the income from selling pack 1 will
cover at least its variable cost, will lie in the defined
range:
c
v
c
01
v
c.0 (52)
Since pack 2 contains a factor,

, times the contents
of pack 1, then the variable cost associated with pack 2
will be v
c

, or, using Equation (51), .

1
2v
cp

dv v

Let the probability density for the MAP for pack 2,
2, be . Using the concept of the “uniconsumer”
introduced without loss of generality in [11] to charac-
terise a person prepared to buy one but only one item if
the price is right, the fraction of the uniconsumers pre-
pared to pay 2 or more for a single pack 2 will be
, given by
p

Sp

2
hp


2
Sp
p

p
2
 
22
20
d1
m
pp
p
hv vh
 (53)
where 2m is the most that anyone in the target market
is prepared to pay for pack 2. Those prepared to pay
more would ideally want a larger pack 2 for their money,
but it is assumed that they would be content with the
smaller pack 2, provided it cost less, if that was all that
was offered.
Let there be N uniconsumers in the target group. The
vendor’s total profit, , from selling packs 2 will be his
total income for pack 2 less both the total variable costs
and the fixed costs associated with pack 2,

F
C

1C
:
 
222 2v
ppNSpcp F NS (54)
The retailer will seek to maximise this profit, which,
for a constant size of target population, N, is equivalent
to maximising the average profit per consumer,

:
  
2
2
0
11d
p
F
vv C
pcc h
NN
  


vv



 (55)
where use has been made of Equation (53) in the second
step.
The maximum value of profit,

, may be found by
differentiating Equation (55) with respect to pack 2 price,
2, and then setting p2
dd 0p


. This gives the opti-
mal price as the solution, , of
2
p
  


22
d 1
vv
vvpchp

 



2
0
11
p
ch
0
v
c (56)
which will be seen to depend on the variable cost associ-
ated with pack 1, v, and the probability distribution, c


2
hp , for pack 2 price, .
2
A limit in the range of risk-aversions is implicit in the
assignment of the maximum price that anyone would pay
for a pack 2, 2m, in Equation (53), and it is appropriate
now to consider the basis for this limit. The maximum
price, 2m, that anyone is prepared to pay for pack 2,
will impact on the maximum largeness,
p
p
p


max , via
Equation (51): 2
2L
1
mm
p


 where m

is the maxi-
mum pack-size ratio and equal to the maximum largeness
for pack 2:


max . The maximum pack -size
ratio, m
2L
m
 

, will imply a lower limit on the risk-aversion
via equation (46):

ln 1
1ln
m
mm



 (57)
When he is considering assigning a largeness for pack
2 of between 2 and 3, we can expect that the vendor will
have an idea of the maximum size for pack 2 wanted by
even the consumers possessed of the largest appetite for
that good. The lowest maximum largeness we can take to
be 3 (otherwise why would he be considering this as a
possible size for pack 2?), and twice that figure, namely
6, would seem to constitute a reasonable estimate of the
highest maximum largeness in the absence of more pre-
cise information. Putting 3
m

 into Equation (57)
gives a minimum risk-aversion of 0.369
m

, while
inserting 6
m


gives m0.101


0.101
. In the latter case,
about 90% of the possible range of risk-aversion for cau-
tious consumers is covered: 1.0
m

.
9.2. Distributions for Risk-Aversion
The distribution of risk-aversion has been modelled in
the first instance as a uniform distribution over the re-
stricted range: 1.0
m



, so that the probability den-
sity,


g

for risk-aversion,

, is:

11.0
1m
m
g 


 (58)
Meanwhile the probability density for the maximum
acceptable price (MAP) among the target group for pack
2, , is given by
2
p
  
222
2
d1
dm
hpgpp p
p


 (59)
where the relationship between pack 2 price, 2, and
risk-aversion,
p

, is found by combining Equation (46)
and (51) to give:




2
2
2
ln1 ln
ln 1
pp
pp 


2
(60)
Differentiating Equation (60) with respect to
2
p
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Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool 667
gives 2 ddp

:

 

22 22
2
222 2
ln1ln1
d
d1ln 1
pp pp
ppp p

 

(61)
The resultant probability distribution,


2
hp
6,0.10
, is
shown in Figure 3, for the case when 1
m

  .
It is clear that the probability density for MAP falls away
quickly as MAP increases.
The probability density for risk-aversion,

, has also
been modelled using the generalised, Double Power den-
sity [12], which provides an analytically tractable model
for a variety of smooth distributions with a wide range of
modes. Let
1.0
mm

  (62)
so that the minimum value of

is zero, while the
maximum value, m

, of

is
1
mm

  (63)
The Double Power probability density for

is then
defined on

0, m


by:

0
cd m
fab

    (64)
where, from [12]



1
11
c
m
cd
adc



 (65)



1
11
d
m
cd
bdc



 (66)
where c and d are free parameters to be specified by the
user.
The corresponding probability density,

g

, for

is then
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
012345
MAP for pack 2, p
2
( standard units)
Prob ab ility density, h(p
2
)
Figure 3. Probability distribution for maximum acceptable
price for a uniform distribution of risk-aversion with
m

0.101 .
 


d
d
cd
m
gfa b





m
 (67)
since dd 1



from differentiating Equation (62).
Putting 1c

and 2d

produces a symmetrical
distribution on m1.0



, see Figure 4, where the
pack-size ratio has been set at 6
m

 so that
0.101
m


. Applying Equations (59) and (61) gives the
probability density for MAP for pack 2 given in Figure 5,
showing how the probability density falls away for high
values of MAP in a way similar to when the distribution
is uniform over the same interval.
Skewed distributions for risk-aversion were produced
by setting c = 1.0 and varying the parameter, d, in the
Double Power Equations (65) and (66). Setting 0.4d

produced a distribution skewed towards less caution,
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
00.20.4 0.60.81
Risk aversion, 
Probability density, g(

)
Figure 4. Double power probability distribution of risk-
aversion wh en , c = 1, d = 2.
m
 0.101
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0123456
MAP f or pack 2, p
2
Probability density, h(p
2
)
Figure 5. Probability distribution for maximum acceptable
price for a symmetric double power distribution of risk-
aversion wit h , c = 1, d = 2.
m
 0.101
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Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool
668
with a mode of mode 0.3

, as shown in Figure 6. A
distribution skewed towards greater caution was found
by putting , with a resultant mode of mode 8d0.77


.
See Figure 7. The probability density falls away for high
values of MAP in both these cases also.
9.3. The Optimal Largeness of Pack 2
To facilitate the discussion of the results, we shall intro-
duce the concept of the generosity of the offer, as meas-
ured by the price per unit of its contents. The price per
unit for pack 2, 2

, is given by:


 
2
2
21 1
1
22
L
p
LL L


 
2
(68)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
00.2 0.4 0.6 0.81
Risk aversion, 
Pro bability d e nsity, g(

)
Figure 6. Double power probability distribution of risk-
aversion skewed towards less caution; , c = 1, d
= 0.4.
m
 0.101
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
00.2 0.4 0.6 0.81
Risk aversion,

Probability density, g(

)
Figure 7. Double power probability distribution of risk-
aversion skewed towards greater caution; m

0.101 , c = 1,
d = 8.
where the price of pack 1 provides the base unit for price
once again. Hence
  
2
2
dd21 20LL  , imply-
ing that increasing the largeness of pack 2 will raise the
per-unit price, thus decreasing the generosity of the offer.
In these terms, BOGOF is the most generous offer of
those under consideration and 3-for-2 is the least.
We may also note that the vendor will wish to cover
at least the variable costs in any long-term offer. The
money brought in per sale of pack 2 will be


221pL

, while the variable cost associated with
pack 2 will be


2
v
cL . Hence covering the variable cost
of pack 2 requires that


212 0
v
LcL or

1
21v Lc  (69)
Turning now to the problem of maximising profits, a
solution may be found for the optimising Equation (56)
in terms of v. The equation may be solved iteratively
by first assigning a value to the optimal pack price, 2,
and then finding the variable cost, v, associated with
pack 1 that reduces the left-hand side of the equation to
zero. (The reverse process of fixing v and finding the
corresponding value of 2 is equally valid, but tends to
be less well conditioned.) The optimal largeness may
then be found from the golden ratio, Equation (48), not-
ing that
cp
c
c
p


2L


.
Figure 8 shows the optimal largeness, , of pack
2 versus the variable cost, v, associated with pack 1 for
four distributions for risk-aversion. It plots additionally
the largeness of pack 2 needed to cover the variable cost
(Equation (69)), which provides a base line. Clearly
when

2L
c
0.5
v
c

, selling 2 for the price of 1 (BOGOF)
will cost 1 standard unit and bring in 1 standard unit, so
that the variable cost of pack 2 is only just covered.
When 0.66
v
c7

, selling 3 for the price of 2 (BOGOF)
will cost 2 standard units and bring in 2 standard units,
just covering the variable cost of pack 2 again. Similar
calculations will apply for all offers in between.
Let us consider the profit-maximising offer based on
the assumption that the population follows a symmetrical
Double Power probability density for risk-aversion with
a lower level of 0.369
m


by the assumption that


max
2L3

. From Figure 8, the largeness offered when
0.5
v
c

is just less than 2.3, which is less generous than
the largeness of 2.0 which would just cover the variable
costs. A largeness of 2.3 can sustain a profit because
there are sufficient consumers in the target population
prepared to pay 1.3 standard units or more for an appro-
priately sized pack 2. When , the optimal
largeness has gone up to 2.65, which may be compared
with the largeness of 2.5 needed to cover variable costs.
A largeness of 2.65 can be sustained because there are
still enough consumers “left in” prepared to pay 1.65
standard units for the appropriately sized pack 2. The
0.6
v
c
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Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool 669
margin between the optimal largeness and the variable-
cost-covering largeness has now reduced, however. This
margin disappears altogether when the variable cost rises
to , when the optimal largeness of 3 will
bring in only enough money to cover the variable costs.
Figure 8 demonstrates how the optimal largeness asso-
ciated with the Uniform distribution converges to the
same point as that associated with the Double Power
distribution. This convergence will be independent of
the precise form of the distribution of risk-aversion
above the lower limit of
0.667
v
c
0.369
m

 that is consequential
on

max
largeness to a less generous value above 3, there will be
no consumer prepared to pay the correspondingly higher
price, which would be more than 2 base units.
2L

3. While the vendor might want to set his
The vendor’s plight will be eased if there are more
people with lower risk-aversions in the target population.
Thus when

max and so 26L0.101
m

, the ven-
dor will be able to set his largeness at less generous,
higher values throughout the range. Thus he will be able
to assign pack 2 a largeness of 3 for a variable cost,
v, for the symmetrical Double Power density
and for a variable cost, , for a Uniform distri-
bution over the same range.
0.59

c
0.50
v
c
Based on Figure 8, we may state generally that, if the
vendor expects to be able to sell a 3-for-2 offer, so that
max , and requires that at least his variable costs
should be covered, then the variable cost, v, associated
with pack 1 must be at or below 66.7% of its selling price.
More generally, a multibuy offer that is at least as gen-
erous as 3-for-2 cannot be profitable unless:
2L3
67
c
00.6
v
c (70)
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
0 0.10.40.50.60.7
Variable dard units)
Optimal largeness of pack 2, L(2)
20.30.
cost, c
v
(stan
DP, L(2)max = 3
DP, L(2)max = 6
Uniform, L(2)max = 3
Uniform, L(2)max = 6
Just covers cv
Golden Rat io s qua red
Figure 8. Optimal largeness of pack 2 vs. variable cost for
Uniform and Double Power (DP) distributions for risk-
aversion, with


L
max
23 and


L
max
26. Also shown
is the locus of the lowest largeness needed to cover the
variable cost of pack 2.
Figure 9 compares the effects of skewing the symmet-
rical Double Power distribution for risk-aversion towards
lower caution and towards greater caution. The minimum
risk-aversion is 0.101
m


in all three cases. The effect
of a more cautious population of consumers is to lower
the optimal largeness at any given v, while the optimal
largeness of pack 2 increases when the population is less
cautious.
c
Pinpointing the optimal largeness for pack 2 requires a
knowledge of both v and the distribution of risk-aver-
sion in the target population. While v may be deter-
mined accurately, the vendor is unlikely to know the ex-
act form of the latter. It is argued in [19] that risk-aver-
sion stays constant during any decision (whether to buy
or not, in this instance), but nevertheless risk-aversion
will vary with the importance of the decision. See also
[22]. It is also likely to vary from person to person, de-
pendent on both temperament and personal wealth. In the
absence of more precise information, it is necessary to
employ a range of credible distributions, leading to a
plausible range of optimal largeness for a given variable
cost, .
c
c
c
v
The interval for v for which it is sensible to consider
multibuy offers in the range, is given by
condition (70). If the retailer is to restrict for simplicity
his selection of the size and hence price of pack 2 to a
ternary choice between BOGOF, Golden Ratio Pricing
and 3-for-2, then, based on Figures 8 and 9, the follow-
ing approximate v ranges are appropriate for the three
possible values of largeness for pack 2:
c

22L3
c



2
0.00.4,22.0
0.4 0.6,22.62
0.60.667,23
v
v
v
cL
cL
cL

 

 
(71)
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
00.10.2 0.3 0.4 0.5 0.6 0.7
Variable cost, c
v
(standard units)
Optimal largeness of pack 2, L(2)
Less c autious
Symmetric risk-aversion
More caut ious
Just covers variable cost
Golden Ratio squared
Figure 9. Optimal largeness of pack 2 vs. variable cost for
Double Power d istrib utions for risk -aversion, with


L

max
26
with c = 1 and d = 0.4 (less cautious), d = 2 (symmetric), and
d = 8 (more cautious).
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The curves suggest that the BOGOF provides the
closest approach to optimality over roughly the lower
three fifths of the range of variable cost where any of the
three multibuy offers is able to generate a profit. The
Golden Ratio will then be the best of the three offers in
roughly the next three tenths of the profitable range of
v, with 3-for-2 becoming the optimal selection in
roughly the highest decile. The boundaries suggested
here are all approximate, as is clear from Figures 8 and 9,
and should be regarded as indicative only.
c
Clearly if the entry pack contains a single, discrete
item (a white shirt, for example), then fractional quanti-
ties, as implied by the Golden Ratio offer, will be impos-
sible. Hence the vendor will need to adopt either a BO-
GOF or a 3-for-2 as an approximation.
The analysis suggests that the generous BOGOF can
be justified on pure economic grounds only when the
variable cost is below about 40% of the selling price of
pack 1. On the other hand, BOGOF will be attractive to
the most cautious of the target market of cautious con-
sumers, namely those with a risk-aversion, 1

, and an
offer that attracts the most cautious of the cautious con-
sumers will also be attractive to rest. The BOGOF is thus
ideal for eye-catching promotions that will appeal to all
consumers who can be tempted by a multibuy offer.
However, it should be offered on a temporary basis only
unless the variable costs are low.
10. Conclusions
At a time when legal restrictions on quantity promotions
are being given serious consideration in the UK, it is vital
to have a proper understanding of multibuy promotions
such as BOGOF. The theory of Relative Utility Pricing
(RUP) developed in this paper takes account of the reac-
tion of the customers to provide a quantitative economic
explanation not only for supermarket promotional offers,
but also for more general pricing of packs of different
sizes in supermarkets and on the internet.
Grounded in standard economic thinking, the RUP
model allows the derivation of a general formula that
explains clearly and simply for the first time the two
most widely used quantity promotions: BOGOF and
3-for-the-price-of-2. It also provides a general explana-
tion for why the price of a pack 2 that contains twice the
amount of a commodity as the entry level pack, pack 1,
may need to be priced exactly the same to sell.
The paper has linked the RUP model to the theory of
iso-elastic utility functions, allowing the relationship
between risk-aversion and pack-size ratio to be stated.
The same theoretical development also allows a simple
relationship to be set down between risk-aversion and the
demand elasticity for the product in question.
Successive doubling of pack sizes has been shown to
correspond to a logarithmic utility function, where the
risk-aversion is unity. This is associated with a neutral
demand elasticity, viz. 1


. Finite pack-size ratios
greater than 2.0 are suitable for individuals with a
risk-aversion value that lies above zero but no higher
than unity – cautious consumers. They are also associ-
ated with a lower (more negative) values of demand elas-
ticity: 1


, which implies elastic demand.
Cautious consumers are shown to be the only sensible
target for quantity promotions. There is no asymptotic
limit to their gain in utility from additional quantities of a
good, and they may be prepared to pay more for larger
packs. Satisfying the desires of the average cautious
consumer will result in a ratio of successive pack sizes
equal to the square of the golden ratio, namely 2.62,
while the price-ratio will be the golden ratio, 1.62. It is
arguable that this pair of ratios is necessary in order to
best satisfy the needs of consumers. Thus the golden ra-
tio may be regarded as a marketing tool that vendors
should consider using in the interests of their customers.
Golden Ratio Pricing can also be in the best interests
of the vendor, as shown by an analysis of profit maximi-
sation in the presence of the consumer attitudes reported
in the paper. Promotions ranging from BOGOF through
Golden Ratio Pricing to 3-for-the-price-of-2 can be prof-
itable only if the variable cost of pack 1 is less than two
thirds of the price of that pack. But higher profits are
likely to come from Golden Ratio Pricing than from ei-
ther BOGOF or 3-for-2 when variable costs as a fraction
of pack 1 price lie in most of the upper half (~0.4 to ~0.6)
of the range required to generate profit from any of these
multibuy offers.
The RUP model has been applied to the prices of gro-
ceries, where it has been able to illuminate the pricing
structure for “value” eggs in supermarkets. A significant
degree of validation of the model comes from the close-
ness of its predictions both for pack size and price.
The application of the RUP model to the electronics
sector has explained the non-obvious prices of USB
memory sticks of increasing capacity. Linked to the bi-
nary system at the heart of every computer, the ingrained
practice of the silicon chip industry to double the capac-
ity of its products each year is enshrined in the influential
Moore’s Law [23]. This has the beneficial side-effect for
the consumer that the price of the new product that is
twice as fast or has twice the storage capacity as the old
will settle down at the old price. Undoubtedly this has
produced great benefits to the consumer. Although prima
facie the practice may not be ideal for the vendor, it will
be a driver for the exceptionally high rate of technologi-
cal progress observed in the memory industry. Manufac-
turers are pressured to create higher capacity products
due to the previous generation’s prices quickly descend-
ing to commodity rates and thus returning low profits.
Open Access AJIBM

Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool
Open Access AJIBM
671
Hence, the RUP model has explained the structure of
quantity promotions, in terms of both the sizes of packs
and the prices. Different packs should be able to com-
mand. This new theory should be of interest and value to
vendors, consumers and regulators.
11. Acknowledgements
The authors would like to acknowledge numerous fruitful
discussions during the preparation of the paper with Mr.
Roger Jones, Honorary Fellow, City University London,
and with Mr. Edward Ross and Dr Ian Waddington, both
of Ross Technologies Ltd.
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[1] H. Wallop, “Buy-One-Get-One-Free Offers Are One of
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waste-Why-supermarkets-will-never-say-bogof-to-buy-on
e-get-one-free.html
[2] T. Evans, “The End of Buy One Get One Free? Govern-
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83/The-end-buy-one-free-Government-considers-plans-sc
rap-multibuy-deals-tackle-obesity.html
[3] J. Koten, “Why Do Hot Dogs Come in Packs of 10 and
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[5] A. Tabarrok, “Buy One Get One Free,” 2004.
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[6] M. W. Pressler, “Something Racy in the Male. Spilling
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[7] T. Harford, “The Undercover Economist,” Abacus, Lon-
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Why Supersize Unhealthy Food,” Marketing Science, Vol.
29, No. 4, 2010, pp. 770-778.
http://dx.doi.org/10.1287/mksc.1100.0558
[9] D. Kahneman and A. Tversky, “Prospect Theory: An
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No. 2, 1979, pp. 263-291.
http://dx.doi.org/10.2307/1914185
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published 2008 in Marketing Science, Vol. 27, No. 1,
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[11] P. Thomas and A. Chrystal, “Generalized Demand Densi-
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pp. 279-294. http://dx.doi.org/10.4236/ajibm.2013.33034
[12] P. Thomas and A. Chrystal, “Retail Price Optimization
from Sparse Demand Data,” American Journal of Indus-
trial and Business Management, Vol. 3, No. 3, 2013, pp.
295-306. http://dx.doi.org/10.4236/ajibm.2013.33035
[13] P. Thomas and A. Chrystal, “Using Relative Utility Pric-
ing to Explain Prices in Supermarkets and on the Inter-
net,” American Journal of Industrial and Business Man-
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[14] R. G. Lipsey and K. A. Chrystal, “An Introduction to
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[15] Amazon.co.uk, 2009.
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products/ref=tag_tdp_sv_istp#page=2:sort=relevant
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http://en.wikipedia.org/wiki/Moore%27s_law

Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool
672
Appendix A. Singular Homogeneity
Consider the following transformation between two func-
tions:
 
21
for all numbers,
H
mxmHxm (A.1)
The function,


1
H
x

2
, may be seen to be transformed
into the function,

H
mx

, by multiplying both its input,
x, and its output,

1
H
x, by the constant, m. Now sup-
pose that is a homogeneous function of degree
one. In this case, by the property of first-degree homoge-
neity:

1.H
 
11
for all numbers,
H
mxmH xm (A.2)
Linear operators possess this property of homogeneity
of degree one. Comparing Equations (A.1) and (A.2), it is
clear that if is a homogeneous function of degree
one, then the second operator will be the same as the first:
.

1.H

..H

H
21
Now consider Equation (7), which may be written in
terms of a continuous variable for extent, x, as:
  


010
0
n
uLnxLnux xb
1
 (A.3)
By Equation (A.3), the function, , may be
transformed into the function, , by multiplying
both the input, x, and the output, , by the con-
stant, . Clearly Equation (A.3) bears a similarity to
Equation (A.1), but it differs in the fact that Equation
(A.3) applies only for the single value of m:

10 .u

.

10
ux
0n
u


Ln


mLn.
Hence we may describe Equation (A.3) as having the
property of “singular homogeneity”.
Under singular homogeneity, and

0. n
u


10 .u
will be different, nonlinear functions, except for the lim-
iting case when is linear in its argument. Such a
situation will occur only when risk-aversion is zero, when
the risk-neutral utility function emerges. (This contention
becomes evident after substituting

10 .u
0

 into equations
(A.5) and (A.6) below. At this point, the function,
, has become homogeneous of degree 1, because
the utility function is linear in its argument.)

10 .u
An alternative but equivalent formulation of Equation
(A.3) arises after replacing the extent variable, x, by a
new variable of extent, b, where so that

bLnx

x
bLn. Then equation (A.3) becomes:
  
010
0
nb
ub LnubLnb
Ln




 1
(A.4)
As an example, let us assume a pack-size ratio, 3


,
as used by Sainsbury for its value eggs. From Equation
(46), the matching risk-aversion is 0.369

. Using the
utility function given in Equation (37) to represent the
utility that the person will gain from using the contents of
pack 1 containing an amount, , of good, B, we achieve,
after noting that :
1
b

11L

11
0.631
10 1
11 1.5850
111
bb
ubb bb





 
 
(A.5)
Meanwhile the utility of that the person or consortium
of people will gain from using the contents of pack n may
be found by combining Equations (A.4) and (A.5) to give
  




11
0
0.369 0.631
1
1
11
1.585 0
nbb
u bLnLn
Ln
Lnbb Lnb






 





(A.6)
Figure 10 shows the utility gain from pack 1, 10 u

,
and that from pack 2, 20 u

, plotted against the amount
of the good, B, measured in standard units of the contents
of back 1, so that 11b

and 2. These graphs
might represent the case of one person buying pack 1
containing 6 eggs and a family unit of three buying pack
2 containing 18 eggs. The final utility gain for the indi-
vidual, at b = 1, will be 1.585, while the final utility gain
for the family of three will be 3 × 1.585 = 4.755, at b = 3
standard units. The right-hand arrows on the graph show
how this result may be derived by multiplying by the
factor,
3b


23L

, both the x and the y co-ordinates asso-
ciated with the complete consumption of pack 1.
Because Equation (A.3) has been formulated in con-
tinuous terms, we may apply the same transformation
within the contents of pack 1. Hence we may take the
case where the individual has used 1/6 of his pack 1, one
egg in this case, and multiply each of the co-ordinates of
10
u

, namely (0.167, 0.51) by a factor of three to give
the corresponding co-ordinates (0.5, 1.53) of 20 u

.
These show that the utility gained by the family of three
in consuming half a standard unit, that is to say 3 eggs, is
three times greater than that achieved by the single per-
son eating one egg. The smaller arrows on the left-hand
side of the graph indicate the graphical process.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.51 1.5 2 2.53 3.5
Pack contents in standard units
Utility gain, u
10
, u
20


Figure 10. Graphical analysis of singular homoge ne ity.
Open Access AJIBM

Explaining the “Buy One Get One Free” Promotion: The Golden Ratio as a Marketing Tool
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673

21
Appendix B. The Range of Maximum
Acceptable Price (MAP) for Pack n But, by Equation (3),
020n
, so that we may sim-
plify the right-hand side of condition (B.7) to:
pp


20 0200
max, n pp pp
B.1 The Highest MAP for Pack n n
(B.8)
Comparing Equations (3) and (17), it is clear that the
price, 1n, resulting from a comparison of pack n and
pack 1 will be no greater than the price arising by com-
paring pack n with pack 0:
pSubstituting from condition (B.8) into condition (B.6),
the price arising from the comparison of pack n with pack
2 will obey:
20
for 2
nn
pp n

 (B.9)
10
for 1
nn
pp n (B.1) If we put n = 3 into Equation (22), the MAP for pack 3
emerges as It should be emphasised here that the comparison price,
nk , arising from comparing pack n with smaller packs
will depend on the value of pack largeness and the
achieved prices of for the lower packs, but not on the
opinion of the individual, allowance for which comes
from the weightings, .
p

i
k
w
 
30 301312 32
iii i
pwpwpwp (B.10)
Since by condition (B.1), 3130 , and by condition
(B.7),
pp
32 30
pp

, the highest value of any individual’s
MAP, , will occur when ,

3
i
p0 w110w

and
20w

, so that
Putting n = 2 into Equation (22) gives the individual’s
MAP for pack 2 as

330
for all
i
ppi
30
(B.11)
 
20201
iii
pwpwp (B.2) Moreover, since , it follows that the achieved
price for pack 3, , will obey

33
i
pp
3
p
By condition (B.1), 2120 , and so, from equation
(B.2), the highest value of the MAP, , for any
individual will occur when that person sets
pp

2
i
pp

0
i
w
20
1

and
. Hence we may write:

10
i
w
3
pp

(B.12)
where, from equation (3),


30 1
3pLp
i
p
(B.13)

220
for all
i
ppi
0
(B.3)
It is clear that this process may be continued indefi-
nitely, so that for any pack, , the upper limit of
MAP will be defined by
1n
The achieved price, 2, must be less than or equal to
the highest MAP in the customer cohort, and so it follows
that
p

0for all
i
nn
pp (B.14)
22
pp (B.4)
where, from Equation (3), In other words the upper limit, , for MAP for
pack n is given by:
maxn
p

20 1
2pLp (B.5)

max 01nn
ppLn (B.15)
We may now consider the highest MAP for pack 3.
Putting k = 2 into Equation (20) and then using condition
(B.4) gives the price of pack n coming from a comparison
with pack 2 as:
B.2 The Lowest MAP for Pack n



 
 
 

22
1
11
20 020
max 1.0,1
2
max 1.0,12
2
max2 ,2
max ,2
n
n
Ln
pp
L
Ln Lp
L
LpLnL p
pp pn












A price for pack n, n, equal to the price of the next
lowest pack, 1n
p
p

, is possible, as demonstrated by the
buy-one-get-one-free promotion and as explained above.
However, bearing in mind the economic principle that
more will be preferred to less, we may presume that the
vendor will not allow the price, n, of pack n to fall
below the price,
p


(B.6) , of pack n – 1.
1n
p

The lower price that the vendor would countenance,
1nn
pp


, will have the effect that any potential cus-
tomer for pack n who has a MAP lower than 1n p

will
exclude himself from the cohort of customers of interest
to the retailer. Thus the lowest MAP for pack n will be
simply the price, 1n p

, of the pack next down in size:
where Equation (B.5) has been used twice in the devel-
opment. Clearly, for any strictly positive values,
:
20 0
,n pp
min 1n
pp
n

(B.16)

20 02020 0
max ,max ,
nn
pp ppp (B.7)