-
James B. Wilcox, Roy D. Howell, Paul Kuzdrall, & Robert
Britney
Price Quantity Discounts:Some Implications for
Buyers and SellersPrice quantity discount schedules are shown to
present opportunities to buyers beyond those explicit inthe
discount schedule itself. The authors propose a taxononny of price
quantity discount schedules, andwithin that taxonomy examine the
implications of price quantity discounts for the ordering behavior
ofbuyers and the formation of alternative channels of
distribution.
BECAUSE of industry practice, for convenience,for marketing
purposes, or for all of those rea-sons, the use of price quantity
discount schedules hasbecome common in many industries. These
sched-ules, intended to act as 24-hour salespersons,
presentquantity-related prices and savings to potential buy-ers.
Though the rationale for offering such discountshas been debated
(Crowther 1964; Dolan 1978; Jeu-land and Shugan 1983; Lai and
Staelin 1984; Monroeand Delia Bitta 1978), the practice has become
ac-cepted and in many cases exf)ected in marketing.
Price quantity discounts may give buyers cost-lowering
opportunities beyond those explicit in thediscount schedule itself.
To lower cost per unit, buy-ers may order in quantities larger than
they need andenter prearranged resale (pooled buying) agreementsor
ad hoc brokerage situations. Additionally, more
James B. Wilcox and Roy D. Howell are Professors of Marketing,
TexasTech University. Paul Kuzdrall is Associate Professor of
Management,Uniyersity of Akron. Robert Britney is Professor of
Production/Opera-tions Management, Uniyersity of Western Ontario.
The authors grate-fully acknowledge the partial support of this
work under Natural Sci-ences and Engineering Research Council of
Canada Grant A-5311. Theyare also indebted to the numerous firms
whose schedules were madeavailable for the research.
formal mechanisms for the distribution of the "ex-cess"
merchandise may develop. Price quantity dis-counts recently have
been criticized as one of the ma-jor factors contributing to the
emergence of graymarkets (Donath 1985; Litley 1985), wherein
the"surplus" units reenter the market through perhapsunanticipated
and frequently unauthorized channels.Interestingly, with the
exception of some caveats byeconomists (Buchanan 1953; Oi 1970),
this possibil-ity has not been addressed adequately. The
commonpractice of offering price quantity discounts has notbeen
examined as a mechanism favoring the devel-opment of such
markets.
To address the issues raised by price quantity dis-counts, we
first briefly review the literature to developa framework. We then
describe a taxonomy of modelsthat have been found to fit actual
price quantity dis-count schedules. Next, within the context of
thesemodels, the characteristics of discounts that give riseto the
issues are examined. Finally, the seller's ra-tionale for offering
price quantity discounts is recon-sidered.
Why Price Quantity Discounts?Several reasons for the use of
price quantity discountshave been identified. Crowther (1964)
suggests that
60 / Journal of Marketing, July 1987Journal of MarketingVol. 51
(July 1987), 60-70
-
sellers save in several ways by selling fewer, largerorders to
their customers. One saving is from lowersales costs in that fewer
sales calls are made, fewerorders are processed, and so on. A
second is fromlowered costs for raw materials because quantity
dis-counts are often available to the seller. Third, the timevalue
of money is taken into account because largerrevenues are available
for reinvestment for longer pe-riods. Finally, longer production
runs without atten-dant increases in holding costs are possible
(see alsoMonahan 1984). Monroe and Delia Bitta (1978) ex-tend
Crowther's model to recognize the interactive ef-fects of these
factors, though the reasoning behind thediscounts remains
unchanged.
More recently, quantity discounts have been viewedas a tool for
achieving channel cooperation. Jeulandand Shugan (1983), for
example, see such discountsas a subtle form of profit sharing
between levels inthe channel. In their model for optimizing
channelprofits they propose negotiations between the sellerand each
individual buyer to allocate optimally thesavings referred to by
Crowther (1964). Though theybase their work on a different theory
and make dif-ferent assumptions, Zusman and Etgar (1981) providea
similar perspective.
Perhaps the most complete model has been offeredby Lai and
Staelin (1984). They argue that thoughquantity discounts are
believed to arise as a result ofpressure from large buyers,
discounts also are offeredat small quantities. They conclude that
effort on thepart of the seller to maximize profits by modifying
thebuyers' order policy is a more likely explanation forthe use of
discounts.
One additional reason for using price quantity dis-counts,
addressed primarily by economists, is pricediscrimination. Gabor
(1955) has shown that such dis-counts are actually two-part prices
composed of a fixedand variable component. Oi (1970) has
demonstratedthat, in comparison with a single-price strategy, a
two-part price is an effective means for monopolists to in-crease
profit. The two parts are a lump sum tax orfranchise fee paid for
the right to purchase the mo-nopolist's product and a per-unit fee.
The lump sumis the mechanism used to reduce consumer surplus.Oi
notes, however, that such a strategy should seldombe employed
because of the inability of the monop-olist to prevent resale. That
is, in the absence of hightransaction costs of some sort, " . . . a
single con-sumer could pay the lump sum tax and purchase
largequantities for resale to others" (1970, p. 88).
Though in some of Oi's examples the franchisefee is a one-time
payment (e.g., initiation fees for acountry club), that is not a
necessary condition. Allthat is required is a fixed and variable
component. Inthe models described hereafter, we demonstrate
thatprice quantity discounts meet this requirement.
A Taxonomy of Price QuantityDiscount Models
According to Fartuch, Kuzdrall, and Britney (1984;see also
Britney, Kuzdrall, and Fartuch 1983a,b) thereare two basic
approaches to the presentation of pricequantity discounts and
variations for each approach.The two major models are per-unit
pricing (model I)and package pricing (model II). The variations
withineach type include the presence (second degree) or ab-sence
(first degree) of quantity intervals over which acertain price per
unit applies. The models and theirvariations are shown graphically
in Figure 1.
Model IModel I price schedules are characterized by
per-unit,all-unit prices. That is, as the buyer orders
largerquantities, the price per unit charged applies to all
unitspurchased. First degree model I pricing is the limitingcase in
which a unique price is associated with eachunit. Such price
schedules may be presented as a longlist of quantities with the
price at each quantity or maybe offered simply in terms of the
fixed (F) and vari-able (V) components (e.g., $29 per day and 30
centsper mile). This model is shown in Figure 1 as havinga smooth,
curvilinear price-quantity relationship. If asimilar approach is
used but each price applies to arange or interval of quantities,
the schedule becomesa second degree model I. For example, any
quantityordered in the range of 50 to 75 units would carry thesame
price per unit. These schedules also can be de-scribed by a fixed
and variable component. In this case,price is held constant over a
range, giving rise to the"stairstep" schedules shown in Figure 1.
Note that thesteps originate from the continuous curve, either
pro-jected backward (I-A), forward (I-B), or somewherebetween
(I-A/B). Techniques for determining aschedule's fixed and variable
components (F and V)are discussed in the Appendix. Forms of model I
pric-ing are common and are used for such products assteel bars,
stud bolts, recording tape, integrated cir-cuits, photocopying,
stationery, office equipment, andexpendable computer supplies (see
Table 1).Model IModel II pricing schedules refer to package pricing
inwhich the buyer receives no credit for taking deliveryof fewer
units than the maximum quantity in the pack-age. This type of
pricing is usually the result of in-dustry practice and perhaps
physical packaging re-quirements. Like model I, model II has a
range ofvariations. Model II first degree schedules quote aunique
package price for each quantity, as indicatedby the straight-line
price-quantity relationship shownin Figure 1. Second degree
schedules involve inter-vals of package quantities to which a
single price ap-
Price Quantity Discounts / 61
-
FIGURE 1Forms of Price Quantity Discounts
TC
TCQ^
plies and hence show a stairstep price-quantity rela-tionship.
Again, the schedule is a projection from thecontinuous case. The
techniques in the Appendix canbe used to decompose the schedules
into F and Vcomponents, but the type A, B, and A/B distinctionsdo
not apply to model II second degree schedules.Though not as common
as model I, model II sched-ules are used in pricing paper,
photographic film,transistors, capacitors, and electrical
components.
In addition to models I and II, non-all-unit modelsare possible.
For example, block pricing schedules areused by electric utilities.
To get to a lower price onthe schedule, the buyer first must
acquire the lowerquantities at higher prices. Such schedules are
beyondthe scope of our discussion.
As shown in the Appendix, most quantity discountschedules can be
decomposed into fixed (F) and vari-able (V) components following
either model I or model
n pattems. As we discuss in more detail subse-quently, an
examination of a large number of pub-lished price lists shows a
surprisingly high proportionof schedules that "fit" one of the
models depicted inFigure 1 (r^ > .95). The discovery of a model
thatfits the observed data closely does not, of cotirse,guarantee
it is the only model that would fit the data.Similarly, one cannot
claim to have modeled the cog-nitive process used by the price
setter; the actual pric-ing decision could have been based on a
decision pro-cess different from the model used to fit the data.
Thisis an important point. The issues considered by thedecision
maker in establishing the discount schedule,whether cost-,
competition-, or demand-related, arelargely irrelevant to the
outcome of offering pricequantity discounts. As we demonstrate,
what reallymatters is the result (the schedule) and not the
factorsconsidered in its development.
62 / Journal of Marketing, July 1987
-
Company
TABLE 1Summarized Schedule
Product
AnalysisEstimated ($)
F VRatio
Fto VIBM Terminal (M-10)
Terminal (M-20)Magnetic cardsDiskettes (D-1)Diskettes
(D-2)Copier tonerSeries 3 tonerWatermark paper
9,096.0010,512.00
40.5054.0078.0027.0025.0024.50
1.516.001,752.00
32.2548.5087.5065.50
104.0026.00
6.0 to 1.06.0 to 1.01.3 to 1.01.1 to 1.01.0 to 1.11.0 to 2.41.0
to 4.21.0 to 1.1
Wright Worksurface panelPrintout paperBinders8V2 X 11 in
folder15 X 11 in folderBinder adapter kitRing
binderIndexesHardcover binderDiskettes (5V4 in)Diskettes (8 in)Tape
seal cartridgeTape seal beltHanging folder
21.0054.00
121.50108.0099.0072.00
237.5099.00
171.0030.0020.00
294.00350.00
72.00
77.2550.5073.0033.0042.5033.0052.5012.5046.0050.0075.0042.0055.0028.00
1.0 to 3.71.1 to 1.01.7 to 1.03.3 to 1.02.3 to 1.02.2 to 1.04.5
to 1.07.9 to 1.03.7 to 1.01.0 to 1.71.0 to 3.77.0 to 1.06.4 to
1.02.6 to 1.0
Global
DRG Envelope
DP formsElectrical ext.Extension (100')Cable (25 cond)Double
door cabinet
34.8012.0025.0011.8864.00
29.9021.9538.00
1.08189.95
1.2 to 1.01.0 to 1.81.0 to 1.5
11.0 to 1.01.0 to 3.0
Radio Shack
National Semiconductor
Daniel
Standco
Diskettes (8 in)Diskettes (5V4 in)Cassettes (C-10)Cassettes
(C-20)Flip flopRAM chipAnalog switchNumber processor12 X 2 stud
bolt12 X 15 stud bolt12 X 18 stud bolt1 % X 7 stud boltIV4 X 6 stud
bolt% X 3 stud bolt% X 9 stud bolt
40.0024.00
5.9413.20
126.96252.00
92.40139.20
15,273.1215,755.5216,242.485,395.005,245.203,384.003,741.60
49.9533.95
1.252.49
10.5721.007.7011.55
1,873.012,294.852,716.70
204.65153.3039.90
118.10
1.0 to 1.21.0 to 1.44.8 to 1.05.3 to 1.0
12.0 to 1.012.0 to 1.012.0 to 1.012.1 to 1.08.2 to 1.06.9 to
1.06.0 to 1.0
26.4 to 1.034.2 to 1.084.8 to 1.031.7 to 1.0
#7 open sideBusiness reply env.Punched card return2-fold env.#5
invitationKraft X-ray env.
88.62128.94111.30107.10171.22623.70
12.6618.4215.9015.3024.4689.10
7.0 to7.0 to7.0 to7.0 to7.0 to7.0 to
1.01.01.01.01.01.0
Canada Envelope
Blue Line Envelope
#7 open side#8 grey deco#7 open remittance#7 remittance2-fold
open side#5 invitation
69.8681.62
121.24114.6698.14
152.32
9.9811.6717.3216.3814.0221.76
7.0 to 1.07.0 to 1.07.0 to 1.07.0 to 1.07.0 to 1.07.0 to 1.0
Price Quantity Discounts / 63
-
TABLE 1 (continued)
CompanyDay-Timers
Oxford BookshopsHolmes Roberts Ltd.
ProductV2 in 3-ring binderSemirigid binderCertificate
coversDecorator framesAppointment diaryDeluxe portfolioDeluxe photo
album8 x 6 custom signPhotocopyingPhotocopying
Estimated ($)F25.0016.254.204.802.292.703.20
12.00.99.45
V3.652.302.655.157.669.55
16.7517.50
.04
.15
RatioF to V
6.8 to 1.07.1 to 1.01.6 to 1.01.0 to 1.11.0 to 3.31.0 to 3.51.0
to 5.21.0 to 1.5
24.8 to 1.03.0 to 1.0
Note: The data were obtained from manufacturer/distributors'
published catalogs in the public domain. We thank those firms
thatsupplied information on request. The data were gathered and
analyzed over the period 1980 to 1982. The data are presented
toshow our research findings and not as an illustration of good or
bad pricing practices.
Price Quantity Discounts:The Buyer's Perspective
To examine the buyer's position, consider a model Isecond degree
schedule. Recall that in second degreepricing a single price per
unit applies to all quantitieswithin a range specified by the
seller. Oi (1970) re-ferred to these discounts as average price
discounts.Figure 2 represents the total cost curves for the
buyerconsidering such a schedule. For reasons
discussedsubsequently, these total costs include only the priceof
the goods purchased. Holding costs, ordering costs,transportation
costs, or other charges are not includedin this total. Figure 2
depicts a set of four curves withthree breakpoints (b,) for the
acquisition of discounts.The schedule that applies is:
TC, = P ,Qi fQ
-
FIGURE 2Windows in a Discount Schedule
MODEL1ST DEGREE 2ND DEGREE
\P--F/QtV
P=F/Q*V P-F/Q-t-V
I-A B
MODEL I
ST DEGREE 2ND DEGREE
= F+VQ
p=
P.Q* < P.^i(Q* + 1)P,Q* < Pi.iQ* + P.+i
- P..iQ* < P.+iQ*(P. - p.,,) < P...
Q* < P..,/(P, - P,.,).Thus, the new upper limit for an
interval is set at
integer (Q*). Applying this formula to the IBM sched-ule yields
a new, windowless schedule very differentfrom the original.
Quantity1-34-10
11-2324+
Unit Price ($)5795463642304056
Notice that some of the intervals disappear en-tirely. To
achieve the final schedule under these con-ditions, Q* must be
reestimated until the process con-verges. This windowless schedule
reflects the possible
endpoint of the negotiation process.The question, of course, is
why would IBM allow
a buyer to do this? In a simplistic sense, economicrationality
would demand that they do so. Such a pro-cedure would enable IBM to
acquire AT computersfor less than production cost. That is, IBM can
sell20 units at the quoted price and keep one "free,"
therebylowering their production cost. More realistically,
IBMprobably would not cooi)erate, at least to the extentsuggested
by the windowless schedule. It is more likelythat the buyer would
simply seek altemative outletsfor the extra units acquired to
achieve the lower price.At least three altematives are possible. If
other buyersare purchasing from the same schedule, an
informal"pooled buying" arrangement could be considered.When
several buyers are needed in the pool of ordersto obtain attractive
discount levels, or when order tim-ing and logistics make pairwise
arrangements ineffi-cient, an informal or ad hoc brokerage
situation couldbe arranged. The ad hoc broker may be a buyer or
anoutside party who establishes a framework for pooling
Price Quantity Discounts / 65
-
orders and handling the details of the transaction.' Instill
other cases a formalized mechanism (a gray mar-keter) may be
available for buying and reselling the"excess" units of individual
buyers. In this case, ifthe buyer is facing a schedule with
windows, it maybe possible to sell the extra units for less than
theycost IBM to produce and still come out ahead. Theresult is a
fairly attractive price in the secondary mar-ket. Because of either
negotiation with IBM or resaleto a secondary market, the windowless
schedule is morerepresentative of the true prices faced by the
buyer.
The resale market also may offer opportunities be-yond those
suggested. Given a windowless schedule,buying one more unit
increases total cost but probablyby less than the amount for which
one unit could beresold. Assume that the one extra unit needed to
qual-ify for the next quantity interval can be sold for a valueof
R. The effect will be a lowering of the upper limitto the buyer.
The determination of the new Q* is sim-ilar to creating the
windowless schedule except for therecovery of R from the
resale.
T C Q.
P,Q*T C Q + I R
- P , . . Q * < P , . , - RQ * ( P , - P , , , ) < P , , ,
- R
Q* < [(P,.,)/(P. -= new limit.
- R/(P. -
If the one additional unit can be resold for R =the lowest price
in the schedule, R = P,+ i for the lastinterval and
Q* < [(P..,)/(P, - P,..)] - R/(P, - P..,) = 0.That is, the
schedule must collapse entirely. It is alsoworth noting that a
windowless schedule is not re-quired for this to work; the presence
of windows sim-ply accelerates the process by providing lower
aver-age costs.
Essentially, the economists' p)erspective is that aslong as
there are other buyers, resale is possible un-less prohibitively
high transaction costs are present.Such costs could include order
processing, inventorycarrying, transportation (Levy, Cron, and
Novack1985), and all marketing costs incurred by the originalbuyer
in order to deal with the secondary market.
Transaction Costs and Barriersto Resale
Though specific data are not available, several factsabout
transaction costs can be deduced. First, in Oi's1970 presentation,
barriers included the need for thebuyer to be physically present to
receive the good orservice (e.g., amusement park fee or country
club dues).In addition, custom tailoring of goods for the
recipientwould create an adequate barrier.
Much less attention has been directed to the fi-nancial side of
transaction cost barriers. It seems likelythat such costs would
create barriers to resale only ifthey were larger than the total
"rebate" available fromthe discount schedule. If the rebate were
greater thanthe additional cost of reselling the surplus units,
thebuyer would come out ahead (lower total cost for thegoods
required) by seeking cooperative and/or sec-ondary markets. Because
rebates are a function of thefixed component in the schedule, the
issue dependson the magnitude of F and the additional costs
in-curred by the buyer. The proprietary nature of costand pricing
data makes direct comparison of F andtransaction costs difficult.
The two can be consideredseparately, however.
The Fixed ComponentAbsolute judgments cannot be made, but the
likeli-hood of profitable resale clearly increases as F in-creases.
Table 1 lists values of F determined from nu-merous published
discount schedules by techniquesdescribed in the Appendix. Notice
that the values ofF range from a low of $.45 to more than $16,200
forstud bolts. The F to V ratio in Table 1 serves as anindex of the
magnitude of the potential rebate.' Thehigher the ratio, the
greater will be the rebate (in dol-lars and as a percentage of
price) for any given quan-tity. As the rebate constitutes a larger
percentage ofprice, the likelihood of profitable resale
increases.
Assessing how a particular schedule is determinedis beyond the
scope of our article. However, severalpatterns emerge. Consider the
case of the IBM model20 display terminal. The relatively high fixed
com-ponent of $10,5(X) might suggest that IBM does notwant to
handle small orders (Lambert, Bennion, andTaylor 1983). Notice that
the ratios of fixed to vari-able components are all equal (7:1)
regardless of theitem for DRG Envelope Company, despite fixed
com-ponents that range from $88 to more than $600 perorder. Even
more remarkable is that this seven-to-oneratio is the same for the
two other envelope compa-
'This ad hoc situation may be formalized over time as an
additionallevel in the channel of distribution closer to the
manufacturer; that is,a distributor or wholesaler large enough to
purchase in quantities thatqualify for lower prices may emerge. In
such cases, the pdce quantitydiscount acts as a trade or functional
discount.
'Because P = F/Q + V, as Q increases, P approaches V.
Therefore,/ can be used as a single-valued surrogate for P.
66 / Journal of Marketing, July 1987
-
nies. Notice also the difference in pricing approacheslisted for
the two photocopying services at the bottomof Table 1. Oxford
prices with a high fixed but lowvariable component whereas Holmes
Roberts does justthe opposite. Finally, compare ihe fixed
componentsassigned by Radio Shack to orders of its C-10 and C-20
cassettes, products that differ only in quantity oftape. Though the
variable components are expected toincrease, the fixed components
are in more than a two-to-one ratio.
The preceding issues are somewhat peripheral toour topic, but do
suggest that insights can be gainedby decomposing a schedule into
its fixed and variablecomponents. As the schedules in Table 1 are
not arandom sample, broad generalizations about the mag-nitude of F
and the F to V ratio are not possible. How-ever, several of these
values appear to offer the op-portunity for substantial
rebates.
Transaction CostsRecall that transaction costs comprise all
marketingand distribution costs incurred by the original buyerin
reselling excess quantities purchased to take ad-vantage of lower
prices. These costs can differ con-siderably depending on the type
of resale arrange-ment. For the pooled buying and ad hoc
brokeragesituations, the only costs may be a few phone callsand
some additional transportation. These costs, how-ever, may be
substantial (Levy, Cron, and Novack1985).
The gray market situation differs from the otherarrangements in
several ways. Additional marketingfunctions must be performed
because ultimate buyershave not been identified in this case. Risk
is involved,as are promotional and order-taking costs. However,the
gray marketer has the advantage of having manymarketing functions
performed by the original (in-tended) channel and need not
duplicate these efforts.This fact, coupled with technological
advances (WATSlines, direct marketing via mass media and
catalogs,etc.), may reduce transaction costs to the point wherethey
are no longer a barrier to resale (Howell et al.1986).
Implications for the SellerThe factors and processes we describe
would tend toincrease the proportion of purchases made at the
higherquantity intervals of a price quantity discount sched-ule.
That is, given (1) informed and aggressive buy-ers, (2) the
negotiation opportunities provided by win-dows in the schedule, (3)
the possible development ofpooled buying, (4) the possible
development of ad hocbrokerage, (5) the possible development of
higher levelchannel intermediaries, and (6) possible
development
of altemative gray market channels,' sellers who offera price
quantity discount schedule should be preparedto sell a large
proportion of their output at the lowestprice on the schedule.
The seller should be indifferent to where on theschedule the
orders fit (assuming channel structure andcontrol is not an issue)
if the fixed component (F) istruly refiective of a fixed cost per
order. That is, theseller should be indifferent if dealer
commitment toand support of the product in a traditional channel
arenot diminished by distribution through altemativechannels or if
the presence of an additional interme-diary is not objectionable.
However, if F reflectssomething other than true fixed cost per
order, in-creasing order sizes may result in lower than
expectedprofit for the seller when the total quantity sold doesnot
change.
In certain situations the distribution of orders maybe
concentrated at lower quantities. Customized prod-ucts with little
or no utility for other than the originalbuyer are an obvious
example. Likewise, products withlow brand recognition/preference or
for which an es-tablished dealer network is not available to
providenecessary support activities are not candidates for
graymarkets (Howell et al. 1986). In still other cases, theproduct
may not be important enough to warrant theadditional effort
required for negotiation or resale(Shapiro 1979).
Other ConsiderationsWe do not explore the impact of price
quantity dis-counts in the case of elastic demand in the
end-usermarket or a segment thereof. If, through the mecha-nisms we
discuss, intermediary buyers are able to pur-chase at lower price
intervals on the discount scheduleand thus sell the product at
lower prices, the totalquantity sold by the manufacturer may
increase in-stead of staying constant with fewer orders. It is
par-ticularly interesting to speculate on the use of pricequantity
discounts to encourage sales through a graymarket. A manufacturer
may be able to engage in pricediscrimination, selling to a more
elastic segment (re-quiring fewer dealer support services) at a
lower pricewhile maintaining an established, full-service
dealernetwork. Any "leakage" to the lower priced marketof buyers
who would have paid the higher price (seeGerstner and Holthausen
1986) normally would gen-erate demands for protection and
complaints from thedealer network. However, it is the dealers
themselveswho are supplying the lower price channel with
mer-chandise.
'Price quantity discounts are neither a necessary nor sufficient
causeof gray markets. Many factors (e .g. , arbitrage, currency
fluctuations)may contribute to their formation.
Price Quantity Discounts / 67
-
ConclusionsWe attempt to provide a taxonomy of price
quantitydiscounts and a set of methods for decomposing theminto
fixed and variable components. Using this infor-mation, we examine
the implications for price quan-tity discount use. The issues
addressed are not ex-haustive of those that could be considered,
butpreliminary findings suggest the impact of price quan-tity
discounts is more complex, more subtle, and morepervasive than work
to date has suggested. We do notimply that price quantity discounts
are either good orbad, but rather that many factors must be
consideredin assessing the advisability of their use.
APPENDIXTypes of Discount Schedules
Model I. Unit PricesMost simply, these schedules present
increasing quan-tities and decreasing unit prices. They are common
inmany industries. Calling F a fixed component (in-cluding profit
and fixed costs) and V a variable com-ponent (including variable
costs and profit) we cangenerate the model I schedule in its pure
form from
P = F/Q + V (1)where Q is a quantity. Observe that for each
quantity,a unique unit price is generated.
Example of Model I Schedule GenerationSuppose processing a
customer's order costs $10.00,the seller wants to make $5.(X) on
every order (re-gardless of quantity), the product costs $2.00 per
unit($1.00 direct labor, $1.00 direct material), and a 30%markup on
cost is desired. First, F is determined tobe $15.00 by adding the
quantity-independent (yetorder-dependent) factors. The variable
component Vis determined by the cost and profit objectives. In
thiscase it is $2.00 plus the profit of $.60 per unit or $2.60each.
Now the schedule can be generated. Substitut-ing the values of F
and V in equation 1 gives a pricefor any desired quantity.
P = 15/Q + $2.60 (2)The following price quantity discount
schedule is ob-tained.
Quantity Unit Price ($)1234567
17.6010.107.606.355.605.104.74
Determining the Pricing Parameters F and VGiven the
ScheduleThree ways of working backward (i.e., given theschedule,
finding F and V) may be useful.
Method 1. The first method is most straightfor-ward. It is
simply asking the developer how theschedule was generated.
Method 2. Pick a price from the schedule, or havethe supplier
quote a price, at a very large quantity.Equate that price to V and
substitute any other sched-ule price and quantity into equation 1
to obtain F. Ifthe price at quantity 1000 for our data has been
quotedat $2.62, it is a good estimate of V. The first term
ofequation 1 is a diminishing function of quantity. Sub-stituting
the first price and quantity data into equation1 using the
estimates gives
17.60 = F/1 + 2.62F = 14.985
which is close to the "true" F of $15.00. One mustremember the
procedure is one of estimation and oftengives good, not exact,
results.
Method 3. For persons who have a computer orhand-held calculator
with regression capabilities, theprice equation can be regressed
with a simple trans-formation of variables. If the factor 1/Q in
its firstterm is called X, it becomes
P = F(l/Q) + V= FX + V.
In this form, the price-quantity relationship is graph-ically a
straight line. Parameters can be estimated di-rectly from the graph
or the regression coefficients.In either case, care must be used as
the estimates mustbe "retransformed" to the original equation.
The quality of the estimate can be an issue in thiscase as it is
in the preceding example. The coefficientof determination should be
high (in excess of .95). Ifnot, some "kinks" in the line may be
present. Theycan represent changing schedule parameters that
mayapply to a relevant quantity range. Beyond that rangethe
production process may change (usually a substi-tution of capital
for labor typified by increasing F'sand decreasing Vs) , indicating
economies of scale.Thus this type of schedule shows the changing
coststructure of the producing firm (as it should) and
givesadditional insight as part of the decomposition anal-ysis.
This point leads to another benefit of scheduleanalysis.
Discontinuities of this nature must be con-sistent and cost
justified. A quick examination of theprice-quantity graph will show
any deviations that couldportend trouble under cost justification.
If, for ex-ample, a "favorable zone" for one class of customersis
found, it may be construed as discriminatory.
68 / Journal of Marketing, July 1987
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Finally, if the schedule is well-behaved, the price-quantity
relationship can be estimated by using linearalgebra and solving
two equations for two unknowns.Simply take any two prices and
associated quantitiesfrom the schedule and substitute into the
previous for-mula. In the following example we use prices
asso-ciated with quantities 2 and 5.
Subtracting:
10.10 = F/2 + V5.60 = F/5 + V
4.50 = F ( l / 2 - 1/5)F = 15.
The preceding examples are generated for illustra-tive purposes.
Often prices must be "rounded" to thenearest cent, which introduces
some inaccuracies inthe estimation procedure. Again, we emphasize
thatthe procedure is one of estimation and generally pro-duces good
results that may not be exact.
Quantity IntervalsThe pricing formula will produce very lengthy
sched-ules if a market with a broad spectrum of
quantityrequirements is served. Such schedules can be accom-modated
by collapsing them into brackets or quantityintervals, a common
practice for the model I sched-ule. They do raise some issues,
however. Specifi-cally, what should be used for Q in the
formula?
The decision maker has two extreme options: (1)the Q associated
with the lowest quantity in an inter-val can be used and the
associated price applied tohigher quantities in the interval or (2)
the largestquantity in the interval can be used and the price
ex-tended to the lower quantities in the interval. For dis-cussion,
these variations on unit price schedule gen-eration are termed
model I-B and Model I-A,respectively.
Model I-B schedule generation. Using the same Fand V, we modify
the formula to refiect I-B strategy.Assume intervals of a 10-unit
width are desired. Call-ing QL the quantity associated with the
lower boundof the interval, we obtain the new schedule.
Quantity1-10
11-2021-3031-40
Unit Price ($)17.603.353,103.08
This schedule was obtained by substituting values of1, 11, 21,
and 31 for QL into
P = + V.
for price determination, extending the price to lowerquantities
within the interval. Using the same F andV, we obtain the following
model I-A schedule.
Quantity1-10
11-2021-3031-40
Unit Price ($)4,103.353.102.98
Calling the upper quantity in its associated intervalQ', we
modify the pricing formula to reflect I-A pric-ing as follows.
P = F/Q' + VThe preceding schedule uses the values of 10, 20,
30,and 40 for Q' to generate interval prices.
There is a difference between the two approaches,given the same
F and V. Clearly, interval width andmodel selection have a major
role in the appearanceof the schedule and, more importantly, how it
relatesto the market demand and the firm's profitability
ob-jectives. As quantities become very large, models I-Band I-A
converge.
Decomposition is more difficult for I-A or I-Bschedules than for
a "pure" model I schedule. Onemust make an assumption as to whether
the scheduleis I-A, I-B, or somewhere between (I-A/B). We sug-gest
multiple runs using the regression technique bemade and the run
with the best coefficient of deter-mination be used.
Model II. Package PricingIn model II pricing, schedule prices
increase propor-tionately with quantity, in contrast to model I
pricebehavior. The underlying price-quantity relation is
P - F + VQ.In this instance, F is translated directly into the
sched-ule price. In models I-A and I-B, the importance ofF to price
is affected by quantity. In model II pricing,F is always recovered
in the price, thus making it use-ful in forcing the buyer to
discrete quantity points ifintervals are offered. This pricing
technique may havearisen from an unwillingness to "break down"
quan-tities shipped according to industry trade practices.
For the same F and V given in previous examples,the following
schedule can be generated.
Quantity Package Price ($)1234
17.6020.2022.8025.40
Model I-A schedule generation. In the other ex-treme, model I-A
uses the upper bound of an interval
One can apply the previously noted methods (linearalgebra, ask
the supplier, regression, graphic analy-sis) in decomposing the
schedule with the following
Price Quantity Discounts / 69
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exceptions: (1) transformation of quantities is unne-cesssary as
the price-quantity relationship in theschedule form is a straight
line and (2) the selectionof an exti-eme point (in this case
quantity 1 ) is not veryaccurate.
Model II pricing can be applied to intervals by tak-ing the
highest quantity in the interval and substitutingit into the
pricing equation. Using the lower quantitymakes no sense as it
exposes the supplier to losses.Thus, calling the upjjer quantity Q'
and establishinga schedule with an interval of 10-unit width, we
ob-tain the following figures.
Quantity1-10
11-2021-3031-40
Package Price ($)41.0067.0093.00
119.00
Buying 20 units once rather than 10 units at two sep-arate times
results in savings (only a single F is paid).Buyers may find this
an attractive discount schedule.
Summary Model I andModel II Schedules
The schedules presented are related linearly to quan-tity either
in the given form for model II or whentransformed (X = 1/Q) in the
case of model I. Eachtype can be presented by a continuous function
ap-plying to all quantities, which is called first degreeprice
differentiation. When intervals are presented,discrete prices arise
and the decision maker has somelatitude in the selection of the
quantity at which to pegthe price. This option applies in model I
but not inmodel II pricing. In each case, the schedule is
discreteand appears as stairsteps. This second degree
pricedifferentiation makes schedule decomposition moredifficult in
the case of model I.
In all cases, several techniques are available to es-timate F
and V, depending on the type of schedulepresented. The analysis and
resultant estimates of Fand V have several uses, including
negotiation, find-ing economies of scale, lot sizing, determination
ofproduction process, and detection of illegal schedules.
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70 / Journal of Marketing, July 1987