Price discrimination in third class postal rates in the database marketing industry: The single-user case

NISSAN LEVIN JACOB ZAHAVI

Price Discrimination in Third Class Postal Rates in the Database Marketing Industry: The Single-User Case NISSAN LEVIN is currently teaching database marketing, operations research, and statistics at the Faculty of Management Tel Aviv University. He holds a PhD in business administration, an MSC in operations research from Tel Aviv University, and a BSc in physics and mathematics from the Hebrew University. His current research interest centers on the quantitative aspects of direct and database marketing. JACOB ZAHAVI is associate professor of management at the Faculty of Management, Tel Aviv University. He holds a PhD in Systems Engineering from the University of Pennsylvania, an MSc. in operations research from the Technion, Israel Institute of Technology, and a BA in Economics and Statistics from the Hebrew University. Previously, he has held visiting positions at the University of Pennsylvania, University of Southern California, Cornell University, and the University of California at Los Angeles. In the years 1988 – 1991 he was the Vice President for Direct Marketing Enhancement at the Franklin Mint. His current research interest is in developing decision support systems for database marketing, involving operations research models, statistical methods, artificial intelligence concepts and computer technology.

ABSTRACT This paper offers a conceptual basis for price discrimination in postal rates for large users of third class mail in the database marketing industry. We make the point that in some cases it pays to lower third class postal rates for selected mailers because it increases the revenues of the postal authority. At the core of the approach is a procedure to estimate the user’s demand function for mailing services, which is formulated as a derived demand, reflecting the characteristics and the purchasing propensity of the customers in the database. Taking the postal authority’s point of view, we seek the optimal postal rate that maximizes the postal authority’s total net profits. The problem is solved for a representative company involving realistic data, nine products, and a combined total universe of close to 10 million customers.

q 1997 John Wiley & Sons, Inc. and Direct Marketing Educational Foundation, Inc. CCC 0892-0591/97/02036-10

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1. INTRODUCTION

and even the USPS’s own Express Mail) are already competing in the first class mail market, while third class mail is in direct competition with alternative forms of advertising that have grown immensely in the last two to three decades and include radio, TV, print media, billboards, telemarketing, and others. Private agencies also are allowed to deliver print advertising messages directly to homes, provided the messages are unaddressed. Since these agencies are prevented by the USPS from using the recipient’s mailboxes, these messages are often left in plastic sacks on doorways or hanging from doorknobs. This range of alternative communications has already eroded the monopoly power of USPS, making it necessary for the postal service to take decisive measures in order to maintain its share of the huge and lucrative third class markets. These measures can range anywhere from better quality of service and faster delivery to rate reform and price discrimination schemes that would break away from the traditional uniform pricing of today. One form of the price discrimination would be quantity discount in the base postal rate, on top of the work-sharing savings, to be given to large users of third class mail in an attempt to attract more business into the market. In an interview with DM News (4) upon taking office as the Postmaster General of the United States, Marvin Runyon acknowledged, in response to a question on the rate-making process, that one of the disadvantages of the postal system is that it has only one price within a delivery class, with no options to provide discounts for large users: ‘‘I don’t know if we’ve gone to the [Postal Rate] Commission and said, ‘Listen, on an individual package, it ought to be $9.95. When someone as big as the U.S. government is going to take it, it ought to go for $3.00.’ I don’t know if we have made this approach.’’ Recognizing the major role of the DM industry as the prime user of third class mail, he went on to say: ‘‘A minimum change in rates in third class affects a large volume of our business, and I know that and the way we do business is very important. It’s one thing to make a resident, a single resident, unhappy; it’s another thing to make a third class mailer, who does $300million worth of business with us unhappy because of something we have done.’’ (4) A joint task force to the USPS’s Board of Governors and the Postal Rate Commission, set up following the 1991 postal rate increase, found ‘‘a need for

Third class mail is used primarily by the direct marketing (DM) industry in the U.S. in the promotion of goods and services through the mail. Third class mail constituted a major source of growth in income for the United States Postal Service (USPS) in the 1980s, with its share in the total revenues increasing from 16.5% in 1982 to 21.2% in 1992 (5:311). In terms of volume, third class mail made up almost 37.6% of the total volume of mailing in 1992 (5:310), a trend that is expected to grow still further in the coming years. Generally speaking, the lower the service class for mail, the lower the postal rates. Price regulations in the mail market are still in effect for first class and third class mail. These regulations are anchored on the Postal Reorganization Act (PRC) of 1970, which established the USPS in its present form as a public enterprise responsible for covering its expenses out of its own revenues. Under this act, the USPS kept its monopoly power in the first class and the addressed third class markets. One of the manifestations of this is the uniform price scheme that still prevails in these markets. This measure was justified on grounds of natural monopoly and the need to avoid the ‘‘creamskimming’’ effect that would result otherwise (10). Under the umbrella of this monopoly power, third class mailing costs have been steadily increasing in the last years, to the dismay of the DM industry. The last price increase of 14.0% took place in early 1995, bringing up the postal rate for third class bulk regular mail to close to 22 cents per piece. USPS offers a substantial discount of up to 25% for letters that have been prepared in ways that reduce the postal service’s processing costs, like presorted batches of mail, or letters that include additional information, such as bar codes, or extended ZIP codes (‘‘ZIP / 4’’). But these discounts are ‘‘work sharing’’ discounts given to users of third class mail for sharing some of the cost of processing and delivering the mail, and thus do not constitute price discounts per se. The changing times and the increasing competition in the marketplace have already resulted in an array of alternative delivery systems which have taken away some first class and third class business from the USPS. Facsimile machines, e-mail and a variety of express mail services (e.g., Federal Express, Purolator Courier, Emery, DHL, TNT, UPS,

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2. THE POSTAL RATING PROBLEM

more flexibility in pricing by the postal service, a need for greater predictability of prices, and a continuing need for full accountability in postal financial services’’ (6:4). To this end, the task force made several recommendations, which were supported by the DMA, one of which was to offer ‘‘volume discounts for large mailings.’’ (6:4) Devising a market-based pricing scheme for third class mail is indeed a very complicated venture, involving issues from different disciplines. Some of these issues have been discussed by Crew and Kleindorfer (2) and further elaborated by the same authors in a separate publication (3). These difficulties notwithstanding, one needs to start somewhere, perhaps by focusing on a simplified problem first and then proceeding into the more difficult and more generalized issues. In this paper, we take the first step in this direction by looking into the feasibility of price discrimination in the basic postal rate of third class mail by volume of mailing in the database marketing (DBM) industry. Our concern is not so much to come up with a concrete quantity discount pricing scheme as it is to make the point that reducing the postal rate for large users of third class mail is a viable alternative for all sides concerned. We concentrate on the DBM industry because here one can quantify and derive the demand curve for third class mail, which constitutes the basis for any demand-related pricing scheme. Taking up the case of solo mailings, which is an important segment of the DBM industry, we claim that under certain conditions it even pays the postal authority to reduce the postal rates for large users of third class mail because it increases its revenues. If nothing else, this should provide additional ammunition to the Direct Marketing Association in its fight against increasing postal rates in the third class market. In section 2, we formulate postal rate-setting as an optimization problem with the objective of maximizing the postal authority’s total profits. In section 3, we present the concept of the derived demand for third class mail, and in section 4 we sketch the algorithm to solve the postal rating problem. This algorithm is applied in section 5 on a realistic problem, involving a mailing audience of close to 10 million people and calculating the optimal postal rates for a variety of input data. Section 6 provides a few concluding remarks.

We view the DM business as a ‘‘game’’ with three ‘‘players’’: the USPS, which provides the service, the direct marketer who consumes the service, and the consumer who receives the mail. If the price is right, everybody benefits from larger volume of mail; the USPS because it increases its revenues, the direct marketer because it boosts his sales, and the consumers (provided they receive only the mail they want) because they get more information to make better and more intelligent purchasing decisions. The last-mentioned certainly applies in the DBM industry, where targeting of the audience takes place prior to the promotion campaign to ensure that only the most-likely-to-purchase customers receive the mail. Clearly, the lower the postal rate facing the direct marketer, the more mailings he or she is willing to make, up to the point where the expected net profit from an additional piece of mail is equal to the promotional cost. In the case that the demand for mail is elastic, reducing the postal rates also increases the USPS’s revenues. Last but not least, even though customers do not directly bear the cost of mailing, it is also in their best interest to have the postal rates go down so they can get more mailings and increase their awareness of the purchasing opportunities available to them. The question, then, is how much the postal rate can go down and still make everybody happy? Facing only one user, however big, it is assumed that the USPS will be willing to deliver any quantity of mail at the predetermined price. This is equivalent to assuming that the supply curve for mailing services is infinitely elastic. But, as with any other product or service, we assume that the demand curve of the direct marketer for mailing services varies with the postal rate that he or she faces: the lower the postal rate, the larger the quantity of mail demanded; i.e., the demand curve decreases from left to right (see Figure 1 for a downward-sloped demand curve). Viewed from the perspective of the postal authority, we seek the postage rate that maximizes its total net profits. This problem can be formulated as an optimization problem, as follows: Max G(p) Å (p 0 c)N(p) where:

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following sections for the case of solo mailing in which a customer is offered only one product at a time, and for which the customer’s response is a simple yes (purchase) or no (no purchase).

3. THE DERIVED DEMAND FOR MAIL Basically, the demand function facing any direct marketer in the DBM industry is a derived demand, reflecting the customers’ purchase probabilities for the product offering. Clearly, the larger the purchase probability the ‘‘better’’ the customer, and the more likely he or she is to make it into the mailing. From the mailer’s point of view it is worth mailing to a customer as long as the expected profits from purchasing the product exceed the cost invested in generating the order, i.e., the cost of the promotion, which includes the brochure cost and the postage cost. This criterion can be translated into a cutoff response rate such that if the customer’s purchase probability exceeds the cutoff point, it is worth mailing to him or her, otherwise it is not. Since the cutoff point is a function of the postage cost, one can vary the postal rate over the relevant range to obtain the set of promotable customers at each rate, and derive the demand function for third class mail, N(p). The detailed algorithm is described in the Appendix. The core of the process to derive the demand function consists of estimating the customers’ purchase probabilities for each product offering. In this work, we estimate the purchase probability by means of a discrete-choice logistic regression model. The logistic regression model is a non-linear model which in the binary 0/1 (purchase/no purchase) case has the form (1):

FIGURE 1 Representative demand function

p Å the postal rate ($/mailing price) c Å the postal authority’s variable cost of delivering the mailing piece to its destination ($/mailing piece). N(p) Å the demand function of the direct marketer, as a function of p, and G(p) Å the resulting USPS net profit

pÅ

k

1 / e0(b / ∑ b x ) 0

(2)

i i

iÅ1

where: xi, i Å 1, 2, . . . , k Å the customer’s ith attribute bi, i Å 0, 1, 2, . . . , k Å the corresponding coefficient estimates

To solve the optimization problem (1), one needs to estimate the demand function N(p), which is a function of the customers’ response pattern, on the one hand, and the direct marketer’s decision process, on the other. We take up these issues in the

The model often is calibrated on the basis of past observations (a live market test or a previous mailing activity), estimating the coefficients bi using the

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method of maximum likelihood. The model then is applied against the customer database to calculate a purchase probability for each customer in the list, a process often referred to as ‘‘scoring.’’ The logistic regression model and its underlying theory, ramifications and estimation have been wildly discussed in the literature and will not be reviewed here. Some works are quoted in the reference list (1,7,8).

ER1 ú ER2 ú rrr ú ERJ then, as described in the Appendix, it is worth promoting to customers in the first j groups if there exists: ERj/1 ° [pj / b(Nj )](1 / M) ° ERj

(5)

Using equation (5) followed by (3), one can find the mailing audience worth promoting to, Nj, for each product. Plugging in the resulting value of Nj and the corresponding value pj in (1) yields:

4. SOLVING THE POSTAL RATING PROBLEM

Gj Å (pj 0 c)Nj

To facilitate the solution process, we discretize the problem by partitioning the customers in the universe into J more-or-less homogeneous groups in terms of purchasing pattern (this partitioning can be the same for all products), and denote:

(6)

Gj is the net profit of the postal authority for the current product if it charges the postage rate pj. Summing up the Gj values for all products involved yields the total expected profits at the postal rate pj. One can then find the optimal postage rate by an enumeration method, checking all values of pj which exceed c. (Note that each value of pj results, by equation (4), in a different mailing audience, and thus in a different profit.) The process is demonstrated below for a practical problem.

ERj Å the expected profit from each customer in group j, pj Å the purchase probability of each customer in group j, nj Å the number of customers in group j, and Nj Å the cumulative number of customers in the first j groups, i.e., j

N j Å ∑ ni

(3)

5. AN APPLICATION TO A REALISTIC SYSTEM

iÅ1

For convenience, we assume that the possible values of p, over which we seek the optimal solution, are given by the discrete values pj, each corresponding to an audience level Nj. These values are calculated for each product separately based on its economic parameters, using the following equation (for further details see the Appendix): pj Å

ERj 0 b(Nj), 1/M

j Å 1, 2, . . . , J

The postal rating problem described above was solved for a realistic system involving nine different products and a total combined universe of close to 10 million people. The products, all solo mailing pieces, were selected to provide a representative sample of the firm’s products, to render a typical demand function for third class mail for the firm involved. The application process consisted of several steps:

(4)

r Calibrating a probability model for each of the nine products, based on the results of a live market test which was conducted for each product separately. r Calculating the purchase probabilities (‘‘scoring’’) for each customer in the corresponding universe, for each product. r Deriving the demand function for each product, as a function of the mailing cost, based on

where: b(Nj) Å the brochure cost as a function of the number of mailings Nj and M Å the required marginal profit margin for the direct marketer Now, arranging the customer groups in decreasing order of ERj,

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TABLE 1 Representative Demand and Revenue Table Data Smoothed by Regression

Original Data

Postal Rate (cents)

Audience (%)

Revenue (M$)

Audience (%)

Revenue (M$)

30.0

11.5

340.6

11.2

332.3

29.5

11.8

344.4

12.4

360.3

29.0

12.2

348.2

12.6

361.8

28.5

12.5

352.0

13.0

366.0

28.0

12.9

355.8

13.3

366.9

27.5

13.2

359.5

13.6

368.3

27.0

13.6

363.2

14.0

373.4

26.5

14.0

366.9

14.3

373.5

26.0

14.4

370.5

14.6

376.2

25.5

14.9

374.0

14.9

375.8

25.0

15.3

377.4

15.2

375.6

24.5

15.7

380.8

15.6

377.5

24.0

16.2

383.9

15.9

377.0

23.5

16.7

387.0

16.9

391.8

23.0

17.2

389.9

17.3

391.9

22.5

17.7

392.6

17.7

392.8

22.0

18.2

395.1

18.1

393.7

21.5

18.7

397.4

18.6

394.4

21.0

19.3

399.5

19.0

393.6

20.5

19.8

401.4

19.5

395.0

20.0

20.4

403.0

20.0

395.6

19.5

21.0

404.4

20.6

397.2

19.0

21.6

405.5

21.1

395.9

18.5

22.2

406.3

21.7

396.7

18.0

22.9

406.7

22.1

393.4

17.5

23.5

406.8

23.0

398.2

17.0

24.2

406.6

23.5

395.1

16.5

24.9

406.0

24.2

394.4

16.0

25.6

405.0

24.8

392.1

15.5

26.4

403.6

25.4

388.5

15.0

27.1

401.8

26.1

387.3

14.5

27.9

399.5

26.8

383.5

14.0

28.7

396.8

27.5

380.3

13.5

29.5

393.5

28.9

385.2

13.0

30.4

389.8

30.2

387.5

12.5

31.2

385.5

31.1

383.4

12.0

32.1

380.7

31.9

378.6

11.5

33.0

375.3

33.1

376.2

11.0

34.0

369.3

34.0

369.3

10.5

35.0

362.6

35.0

363.1 Continued

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TABLE 1 Continued Data Smoothed by Regression

Original Data

Postal Rate (cents)

Audience (%)

Revenue (M$)

Audience (%)

Revenue (M$)

10.0

36.0

355.3

36.1

356.8

9.5

37.0

347.3

37.7

353.7

9.0

38.1

338.6

38.7

344.3

8.5

39.2

329.1

39.8

334.4

8.0

40.4

318.8

40.7

321.7

7.5

41.6

307.8

42.6

315.3

7.0

42.8

295.8

43.6

301.1

6.5

44.1

283.0

45.0

288.9

6.0

45.4

269.3

46.0

272.7

5.5

46.9

254.5

47.5

257.8

5.0

48.3

238.7

49.0

242.0

4.5

49.9

221.8

51.3

227.8

4.0

51.6

203.7

52.5

207.5

3.5

53.3

184.3

53.8

186.1

3.0

55.2

163.7

55.3

164.0

the procedure described in section 3 and the Appendix. r Applying the enumeration algorithm sketched in section 4 to find the optimal mailing cost.

a regression model. Several nonlinear curves of the postal cost p were used to fit the data. The one that seems to provide a consistent fit, for almost all cases involved, with an adjusted R 2 value in excess of 99%, is the function:

In the absence of information about the variable cost of processing the mail, we maximize the total USPS revenues, instead of profits. This is equivalent to assuming c Å 0 in the objective function (1). This objective is compatible with the revenue requirement criteria often used in setting up the postal rates (6). To explore the sensitivity of the postal costs to variations in the input data, we ran the optimization model for 145 combinations of brochure costs and minimal acceptable rate of return. The brochure costs consisted of a fixed cost component and a variable cost component, which reflect real costs. The rates of return were selected in accordance with acceptable mark up factors used in the DBM industry. To avoid the instability of the optimal solution that often results in the discrete case, the optimization problem also was solved for the continuous equivalent of demand. The latter was obtained by smoothing out the discrete demand function using

q

N(p) Å g0 / g1 p / g2p / g3p2 where the intercept g0 and the coefficients g1, g2 and g3 are estimated by the regression model. The ‘‘observations’’ for the regression model are the demand for mail at increments of 0.1 cents in the postal rate, over the interval of 1–50 cents, 500 points in all. As each input data yields a different combined demand function for mailing, we present here the demand function only for a representative case of M Å 50% and a brochure cost which consists of a fixed cost component of $20,000 and a variable cost component of 7 cents per brochure. Table 1 presents the demand for mail at selected postal rate values, expressed in terms of the percentage of the universe worth mailing to at each postal rate for both the original (discrete) and as the continuous equivalent of it as well as the resulting postal authority total

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optimal postal rate is lower than the uniform rate currently charged for third class mail of about 22 cents per piece. Definitely, this is true for rate of returns in excess of 20 percent, which are more prevalent in the DBM industry. We also note, that because of the fact that the revenue function is pretty flat in the vicinity of the optimal solution, even a deviation of 2 to 3 cents from the optimal price, in either direction, does not translate into big revenue loss to the postal authority, giving the USPS more latitude in determining the postal rate. Thus, while confined to a somewhat simplified and limited problem, the analysis above suggests that at least in some cases there is indeed ample room for lowering the charges for third class mail, even from the perspective of the postal authority, for the benefit of both the direct marketers and the customers.

6. CONCLUSIONS In this paper, we take the first step in addressing the price discrimination problem for third class mail in the database marketing industry. Limiting the discussion to a single user, we show that there are situations in which both the USPS as well as the mailer can benefit from lower postal rates—the USPS because it increases its revenues, the direct marketer because it boosts its sales and profits. At the core of the approach is a procedure to estimate the user’s demand function for mailing services which was formulated as a derived demand, reflecting the characteristics and the purchase probability of the customers in the database. Observed from the USPS’s point of view, the optimal postal rate was solved to maximize the postal authority’s total net revenues. The third class rating problem is indeed a very complicated one, involving many issues that were not touched upon in this paper. The purpose of the paper is much more modest—to make the point that the postal rate for third class mail may be too high, even from the perspective of the USPS. This we hope will provide the direct marketing industry with additional ammunition in its fight against the high postal costs. Especially complicated is the issue of expanding the single-user analysis of this paper to the entire DBM industry. Clearly, because of the relatively

FIGURE 2 Representative revenue curve

revenue. Figure 1 exhibits the resulting ‘‘smooth’’ demand function emerging from the regression model: Figure 2 exhibits the corresponding revenue function. As can be seen from Figure 1, the combined demand function is a well-behaved, convex decreasing from left-to-right function. Figure 2 indicates that the revenue function is a unimodal function having a fairly wide peak in the range of 15.0–20.0 cents per mailing piece. The corresponding solution for the discrete demand, observed from Table 1, falls in the same range, even though it exhibits some fluctuations in the vicinity of the optimal solution. Table 2 presents the optimal postal rate (cents per brochure) for several combinations of brochure costs and acceptable rate of return. By and large, the postal costs seem to be declining with an increase in the rate of return for the same brochure cost components. A quick glance through Table 2 indicates that there are many input combinations for which the

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TABLE 2 Optimal Postal Rates for Several Input Data Combinations Brochure Costs M (%) Fixed (M$)

Variable (Cents)

20

40

50

60

80

10

7

24.1

21.5

20.6

19.8

18.5

10

varying

24.8

21.9

21.0

20.2

19.0

15

10

24.2

21.8

20.8

19.4

17.3

15

varying

23.6

20.9

20.1

19.0

16.6

20

7

22.1

19.2

17.6

16.1

14.6

20

varying

22.5

19.7

17.9

16.5

15.2

Note: Varying brochure cost implies a different brochure cost per product. The variable costs used for the nine products are 10, 15, 5, 6, 5, 10, 6, 7, and 8 cents, respectively.

large number of users involved, there is no way to negotiate a separate deal with each individual user; also, the derived demand function for each individual user may not always be known. Instead, a pricing scheme is needed that will determine the postal rate as a function of the volume of mailing involved, e.g., defining several volume ‘‘brackets’’ for third class mail, with a separate rate for each bracket. In all likelihood, the modeling efforts in this case may involve issues from game theory, economics, statistics, optimization, and others. Notwithstanding the difficulties involved, the potential benefits of a price differentiation scheme in the postal rates, and the huge magnitude of the DBM industry, renders this work extremely useful, holding much promise for all sides concerned.

R Å the net profits per order, after providing for covering the manufacturing cost of the product, the fulfillment cost, the financial cost, bad debt and returns and all other cost elements which can be directly related to the product involved ($/order). b[N(p)] Å The brochure cost as a function of the number of brochures N(p), ($/brochure). Typically, the brochure costs consist of a fixed cost component, FIX, and a variable cost component, VAR, i.e.,

b [N (p)] Å

APPENDIX: THE ALGORITHM TO FIND THE DERIVED DEMAND

FIX / VAR, N (p)

which goes down with the number of brochures.

Denoting by: M Å The minimum required rate of return to allow for covering the indirect costs which cannot be attributed to the product involved and leave some profits for the direct marketer.

ER Å the direct marketer’s expected net profits from a customer, excluding the postal and brochure cost. In the case of solo mailing, we have: ER Å prR

(A.1)

Now, the total cost ‘‘seen’’ by the direct marketer per mailing piece consists of the brochure cost b[N(p)], and the postal cost p, i.e., p / b[N(p)]. Thus, by definition, the rate of return per mailing (customer) is given by:

where p Å the customer purchase probability

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rate corresponding to each audience level. The algorithm proceeds as follows: Let pn denote the purchase probability of a customer located at the nth position in the customers’ list. Then, the postal rate at which it is worth mailing all customers above n is the one for which pn is equal to the cutoff point. This rate is given by solving (A.2) as an equality.

ER ER 0 {p / b[N(p)]} Å 01 p / b[N(p)] p / b[N(p)] and the customer is worth promoting to if his/her rate of return exceeds the minimum accepted rate of return, i.e., ER 01¢M p / b[N(p)]

pn Å y(p)/R Å {p / b[N(p)]}(1 / M)/R,

or

yielding ER ú y(p)

R p Å pnr 0 b[N(p)] 1/M

where

Now, one can derive the demand function for mailing, by varying the mailing audience N by small increments, say DN Å 100, going from the top of the list and down, and solving for the postal rate for each mailing audience, using (A.3), to yield the demand function N (p).

y (p) D {p / b[N(p)]}(1 / M) Using (A.1), and expressed in terms of purchasing probabilities, the promotability criterion is given by: p ¢ y(p)/R

(A.2)

REFERENCES

The quantity y(p)/R in (A.2) defines the cutoff point such that if the customer’s purchase probability exceeds it, he/she is mailed, otherwise he/she is not. Since the brochure cost varies by the number of brochures for any given value of p, the cutoff point also varies, attaining a different value for each mailing size. A special case is where the brochure cost is constant, in which case the cutoff point is also constant for any mailing quantity (though it varies with the postal cost). Thus, to find the demand function for mail, one needs to find the set of promotable customers at each postal rate p. This can be done by arranging the customers in decreasing order of their purchase probabilities for the product, from the top probability and down. Then, people at the top of the list, who fall above the cutoff point for the product, as determined by (A.2), are worth mailing to. But since the cutoff point depends on the mailing audience (because it is a function of the brochure cost which varies by the number of brochures), it is easier to go in a ‘‘reverse’’ mode and find the postal

1. Ben Akiva, M., and Lerman, S. R. (1985), Discrete Choice Analysis, Cambridge, MA: MIT Press. 2. Crew, M. A., and Kleindorfer, P. R., eds. (1991), Competition and Innovation in Postal Services, Boston, MA Kluwer Academic Publishers. 3. Crew, M. A., and Kleindorfer, P. R. (1992), The Economics of Postal Service, Boston, MA Kluwer Academic Publishers. 4. DM News, December 21, 1992, ‘‘What’s in Store for the US Postal Service.’’ 5. Direct Marketing Association (1992), Statistical Fact Book, New York: DMA. 6. Direct Marketing Association (1992), DMA Report on Postal Issues, New York DMA. 7. Green, W. H. (1990), Econometric Analysis, New York: MacMillan Publishing Company. 8. Judge, G. G., Griffiths, W. E., Carter Hill, R., Lutkepohl, H., and Lee, T. (1980), The Theory and Practice of Econometrics, New York: John Wiley and Sons. 9. Shepard, D. (1994), The New Direct Marketing, 2nd ed., Homewood, IL: Dow Jones-Irwin. 10. Sherman, R. (1991), ‘‘Competition in Postal Service,’’ in Competition and Innovation in Postal Services, ed. Michael A. Crew and Paul R. Kleindorfer, Boston, MA Kluwer Academic Publishers.

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