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

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

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