A model for optimizing the refund value in rebate promotions

A Model for Optimizing the Refund Value in Rebate Promotions Abdul Ali UNIVERSITY OF MARYLAND

MARVIN A. JOLSON UNIVERSITY OF MARYLAND

Rene Y. Darmon ESSEC, CERGY-PONTOISE, FRANCE

We derive a simple model forjnding the optimal refund to be offered to

jinnl consumers by manufacturing firms during rebate promotions. The

hey features of our work are the explicit representations of the redemption

function, purchase acceleration, and rebuys by brand-switchers. Depend-

ing upon afirm’s objectives andgiven somegenerally accepted assumptions

about thefirm’s demandfunction, an optimal refund rate exists that should

be diJerent than the short-term rebate rate.

A s we scan advertisements by retailers in newspapers, magazines, catalogs, and other media, we observe on- going attempts to increase sales of designated products

by use of money-saving promotions that are distinguished from conventional price reductions. Two of the more popular methods are cents-off coupons and refund offers (rebates). This study focuses on rebate offers for consumer durable goods. For the purpose of this investigation, a rebate promotion is defined as “an offer by the manufacturer to refund a proportion of the price paid by the consumer who mails in proof-of-purchase and other forms.” At issue is how both the short-term and long-term objectives of the manufacturer contribute to the determination of the optimal refund rate.

Following numerous discussions with executives who design rebate promotions, we learned that intuition and judg- ment rather than research and methods of science guide the planning and evaluation of such programs. There are no known formulas, models, or heuristic techniques to assist management in establishing the optimal refund rate. In our model, to be presented, we recognize the existence of inter- actions among promotion-related company actions and mar- ketplace reactions such as price sensitivity and the failure of consumers to request refund to which they are entitled (slippage). In this paper, we propose a model that organizes our initial knowledge about rebate promotions within a mathematical (decision-theoretic) framework. Our representation of

Address correspondence to Abdul Ali, College of Business and Management, University of Maryland, College Park, MD 20742. The authors thank two anonymous reviewers for their helpful comments and suggestions.

Journal of Business Research 29,239-245 (1994) 0 1994 Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010

the rebate-optimization problem incorporates the redemption function, purchase acceleration, and rebuys by brand switchers. This helps us to consider the long-term effect of the rebate promotion beyond the temporal sales increase during the rebate campaign. We then use numerical analysis to explore the effects of specific factors on the optimal refund rate. Depend- ing upon a firm’s objectives and given some generally accepted assumptions about the firm’s demand function, it is shown that an optimal refund rate exists, and that this rate should be different than the short-term refund rate. Our analysis of the interrelated factors that influence the optimum refund rate may improve our understanding of the firm’s determination of the optimal rebate level and may also stimulate further work on an important topic that has been neglected by researchers to date.

Literature Review Despite the large number of vertical trade promotions involv- ing producer-retailer-consumer price-related deals, a paucity of research and literature dealing with rebate programs pre- vails. In a study exclusively focused on rebate promotions, Jolson et al. (1987) observed that infrequent rebate users far outnumbered heavy or non-users in a randomly selected sam- ple of appliance shoppers in a large eastern city. The findings indicated that in contrast to other consumers, infrequent users are more likely to establish an acceptable minimum-refund percentage, yet more likely to perceive the refund processing effort as substantial, and more likely to forego the refund request (see also Nagle, 1987).

A firm will typically undertake a rebate promotion, like any other sales promotion, only if it generates an incremental profit. The profitability of a rebate promotion depends upon whether the incremental (beyond the normal level) sales rev- enue contributed by the promotion outweighs its incremental costs (Bawa and Shoemaker, 1989). Blattberg and Neslin (1990) identify four basic mechanisms by which promotions contribute incremental sales: brand switching, repeat pur- chasing (rebuys), purchase acceleration, and category expan- sion [see Neslin and Shoemaker (1983) for a simulation model

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of these factors]. An empirical analysis of IRI scanner panel data suggests that brand switching rather than purchase acceleration accounts for most of the incremental sales due to promotion (Gupta, 1988). However, promotional offerings seem to induce purchase acceleration for consumer durable goods (Bayus, 1988).

The incremental costs incurred in a rebate promotion m- elude advertising, distribution, and redemption processing costs in addition to the promised refund. Past research de- scribing incremental costs of consumer promotions has con- centrated on coupon redemption phenomena (see Reibstein and Traver, 1982). However, the coupon redemption rate may be different for brand switchers in comparison to loyal customers. For example, Bawa and Shoemaker (1987) suggest that because substitution and effort costs will be lower for existing customers, current users will be more likely to use coupons.

Clearly, the research emphasis has favored coupon promotions, because coupons are most frequently used for consumer packaged goods promotions (Blattberg and Neslin, 1990). However, it is likely that many of the general phenomena and conclusions associated with coupon promotions can be extrapolated to the area of rebates, the durable goods an- alog of couponing.

Model Structure In this section, we introduce a model to determine the optimal refund rate in a decision-theoretic framework. We do not consider the competitive move explicitly in order to get a tractable model to investigate the influence of various factors on individual firm’s choice of optimal refund rate. We treat the in- tensity of rivalry as fixed or exogenous in the model and assume that the firm believes its choice of the level of mail-in refund amount does not influence the refund level of its rivals. This assumption may be justified when there are many ways of achieving the same temporal sales increase through promotions (coupons, deals, quantity discounts, etc.), so that the firm’s rivals may not be offering the mail-in rebate promotion at the same time it does. These phenomena have been observed in many mail-in rebate promotional activities of low- priced consumer goods markets (e.g., Kodak films, Proctor- Silex toaster ovens and coffee makers, Hyponex soil, Fruit of

the Loom underwear). First, we consider the most general case in which no func-

tional forms are assumed for the demand and the redemption functions. Then, we will discuss the less general but more practical situation in which these functional forms are specified. These assumptions are well supported by economic the- ory and marketing practice. The basic assumptions of our model are outlined below.

vironment where the demand for its product does not fluctuate from one period to another, unless some promotional campaign is undertaken to temporarily increase sales.

Let Q,, be the quantity demanded for a product during the non-promotional period, P be the product’s retail price, and A, be the producer’s regular advertising budget. Let the demand function for the quantity, Q,, of a product during promotional period be a function g of the non-promotional period demand, Q,. the refund rate (r) as a proportion of the retail price, P, and the advertising budget, Ap, specifically devoted to sup- porting the rebate promotion. This conceptualization of the demand function is consistent with the findings that for rebates to have an impact, advertising should be used in conjunction with the offers (Tat, Cunning- ham, and Babakus 1988). Therefore, Q, = g (Q,, r, A,,) with aQJ& > 0, and aQ@A, > 0.

(A2) Redemption Rate:

Because of the slippage factor and other considerations, the redemption rate will be different for brand switchers and loyal customers (Bawa and Shoemaker, 1987). Let R, be the proportion of product units for which the refund is claimed by the loyal customers, and R, be the proportion of product units for which the refund is claimed by the brand switchers. We assume that the redemption rates R, and R, are direct functions of the rebate rate, r, and rebate-specific advertisements, Ap. Therefore, R, = h (r, A,,), with XX,/& > 0, and aR,/dA, > 0, i = L, S.

Borrowing the argument from the Weber-Fechner Law, we suggest that the redemption rate will be a function of the refund rate (r) rather than the absolute amount of the refund monies (Pr).

(A3) Purchase Acceleration:

This part of increased sales during the promotional period, as discussed in the literature review section, will be due to consumers’ tendency to accelerate their purchases that would have occurred anyway. Let A (0 s A < 1) be the proportion of increased sales during the promotional period drawn from future sales.

(A4) Rebuys:

(Al) Demand Function:

Some of the consumers’ switching from competitive brands to the promoted brand may consider rebuying the brand in the next purchase cycle. Let y (0 s y c 1) be the fraction of the purchases by newly acquired brand switchers who make a subsequent purchase in the next purchase cycle. The parameter, y, depends upon the customer satisfaction that in turn depends on

We assume that the producer operates in a stable en- the product performance and quality.

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For reasons of tractability, we assume that the parameters A and y are constant over all future time periods. Further, we assume that the inter-purchase time interval between two suc- cessive purchases is n periods. That is, the purchase cycle is n time periods from the promotional period. Also, we assume that the discount rate is S (0 c S c 1).

Model Analyses In this section, we derive the optimal refund rate that follows from our model. We begin with the calculation of profit and then derive and compare the optimal rate for the short- and long-term cases.

Profit Function The profit function for the manufacturer during the promotional period can be estimated by:

% = Q,h - R,Prl + A(Q, - Q,>h - R,Prl + (1 - X>(Q, - Q,>[m - R,Pr] - A, - A, - F, - F,, (1)

where m is the unit contribution to profits, overhead and promotional expenses, and F, and F, are the distribution costs of promotion and other fixed costs, respectively. There may be a variable cost to rebates that we ignore here because most companies delegate the processing of refunds to outside agen- cies who usually charge a fixed amount for such jobs. The gross margin (excluding promotional expenses) of the product is assumed to be constant over the range of quantities sold with and without the rebate promotion.

Myopic Model In this case, the firm will be concerned with the incremental profit due to sales during the promotional period only. In the absence of any rebate promotion, the normal profit for the period would be:

=TT, = Q,.m - 4, - F, (2)

Therefore, the net incremental profit due to the rebate promotion will be given by:

7Fm=7F --p = -QJml + U(Q, - Q31m - R,W +‘(l - “x)(Q, - QJ[m - RPr] - A, - F P (3)

A necessary condition for the optimal refund rate r* which maximizes the manufacturer’s net incremental profit is given by:

&r-l& = 0. (4)

By substituting the expression of mm from the equation (3) in the first order condition (4), we get:

r’ = m(aQ,,.Mr) - {AR, + (1 - h)&JPQ, - (1 - X)(R, - RJPQ,,/(AR, + (1 - h)RJ P (aQdar> + {A(aR,/&) + (1 - A)aRJar)] PQ, + (1 - A){(aR,/ar) - (awar)]

PQn (5)

Long- term Model In this case, the firm will be concerned first with the incremental profit due to the rebate campaign after taking into account of purchase acceleration and brand switching. Since the A(Q, - Q,) sales during the promotional period have been attributed to purchase acceleration, the profit function during the post-promotional period will be:

?r pp = Q,.m - h(Q, - Q,>.m - 4, - F,, (6)

Similarly, the (1 - A)(Q, - Q,) sales during the promotional period are the results of brand switching and a proportion, y, of these brand switchers will make repeat purchases in the next purchase cycle. Thus, the profit function during the next purchase cycle will be:

rr p” = Q,.m + $1 - AXQ, - Q,>.m - 4, - F,, (7)

The combined discounted profit function for these three per-

iods will be:

rTTp + rrpd(l + S) + IT&l + s>-

In the absence of any rebate promotion, the normal discounted profit for these three periods will be:

7~,[1 + l/(1 + 6) + l/(1 + S)“] = [Q,.m - A, - F,]. [l + l/(1 + 6) + l/(1 + S)“]

Therefore, the net incremental profit due to rebate promotion will be given by:

IT = 7Fp + 7rpd(l + S) + lr,J(l + s>n - ?r,[l + l/(1 + S) + l/(1 + S)“]

= Q,[m - RPr - Am/(1 + 6) + $1 - A>m/(l + S>-] - Q,. [m - Am/(1 + S) + y( 1 - A)m/(l + S)“] -

A, - F, (8)

A necessary condition for the optimal refund rate r* which maximizes the equation (8) is given by:

am/ar = 0. (9)

By substituting the expression of IT from the equation (8) in the first order condition (9), we get:

(1 - A/(1 + S) + $1 - A)/(1 + S)n]m(aQ&%) -

r* = (AR, + (1 - AMJPQ, - (1 - AX& - RJPQ

AR, + (1 - A)RJP @Q&%9 + (A(aR,/&9 + (1 - AXdRJddlPQ, + (1 - ANdR,/ar) - (aR.Jar)]PQ,

(10)

Model Results The key result of this analysis is the nature of the first-order condition for optimality for both the short- and long-term cases. By comparing the two optimality conditions, we derive several of the results that we discuss below.

First, we explore the effects of specific parameters on the firm’s optimal refund rate. We observe that the optimal refund

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rate increases with the increase in rebuys and the gross margin cause it is believed that current users will be more likely to rate. The optimal refund rate decreases with the increase in use rebates (Bawa and Shoemaker, 1987). Therefore, we sug- the purchase acceleration and the redemption rate. If the re- gest that: demption rate for loyal customers relative to that for brand switchers increases, the optimal refund rate decreases. The intuition behind the results is straightforward. The manufacturer implements rebate promotions to increase the long-term demand for its brand. In determining the refund rate, the manufacturer recognizes that a proportion of the buyers who were stimulated by the refund rate will fail to apply for a refund. Although a manufacturer may initiate rebate programs for slow moving products, it will be more likely to offer a higher refund for its more profitable items. Further, we find that the optimal refund rate decreases as the inter-purchase time interval increases. In this case:

. A product with a longer life cycle will likely carry a lower optimal refund rate than a product with a shorter life cycle.

This implies that durable products (e.g., coffee-makers or toaster ovens) will have higher refund rates than non-durables (e.g., photographic film). This is because as the rebuys take longer time to materialize, they will have less impact on profitability than purchase acceleration and the resultant optimal refund rate will be lower.

Next, we compare the two optimality conditions as given in the equations (5) and (10) and show that the firm will be indifferent about the short versus long term refund rate when the purchase acceleration parameter satisfies the following condition:

A* = y/(y + (1 + 6)” - 1). (11)

The firm will underestimate the refund rate if it considers only the profitability of the promotional period when the purchase acceleration parameter, A, is smaller than the critical value given in the equation (11). Similarly, the firm will overestimate the refund rate if the purchase acceleration parameter, A, is greater than the critical value given in the equation (11). Thus we find that:

. Ignoring purchase acceleration and rebuys will more likely result in over estimation (underestimation) of the optimal refund rate if the purchase acceleration is greater (smaller) than a critical value.

An intuitive explanation of this finding is that as the purchase acceleration increases, the optimal refund rate decreases. Moreover, the decrease is steeper for the long-term case than for the short-term. The negative impact of purchase acceleration on optimal refund rate may partially explain why one observes differences in refund amounts offered by brands in different product categories (the competitive rivalry and unique characteristics of the product market may explain the rest of the differences in refund amounts). A market leader with a large base of current customers may experience a greater level of purchase acceleration than a weaker firm be-

. A market leader will have a lower optimal refund rate than a weaker brand.

This proposition is consistent with Kinberg, Rao, and Shak- un’s (1974) finding that the optimal price off and the proba- bility of the promotion diminish, as the share of the promoting brand increases.

Further, a greater proportion of the customers who switched to a promoted brand during a rebate campaign will be likely to make rebuys for a higher quality promoted brand than that for a lower quality promoted brand (Y,, > r,). The parameter, y, depends upon the customer satisfaction which in turn depends on the product performance and quality. In this case:

. A top quality brand will have a higher optimal refund rate than a low quality brand for any given level of purchase acceleration.

However, the top quality brand may not offer a higher refund rate when the increased sales during the promotional campaign comes largely from the purchase acceleration of its current customers. Narasimhan (1988) suggests that the depth of discounts will be lower for brands with more loyal customers. See also Raju, Srinivasan, and La1 (1990). It is more likely that a top quality brand will have more loyal customers and these customers will be more likely to exhibit purchase acceleration behavior than the customers of low quality brands. That is, A,, > A,. Therefore, we suggest:

. The top quality brand will offer higher optimal refund rate than the low quality brand only when

In order to further specify the value taken by r*, the specific functional forms of Q,, and R must be known. we will next discuss the less general but more practical situation in which these functional forms are specified, followed by the analysis and findings of the specified model.

Model Specification Demand Function We specify the following multiplicative type demand function for the quantity Qp. That is,

Q, = Q,[l + (1 + A,)rls, (13)

where p is a positive parameter with 0 < f3 < 1. This general form of multiplicative function is often used to model market response functions to marketing mix variables (see Lambin et

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al., 1975). This functional form recognizes that in the absence of a rebate, i.e., when r = 0, the rebate specific advertisement A, has no effect on demand and that the quantity sold increases at a decreasing rate when r and/or A, increase. How- ever, an unadvertised rebate is assumed to affect sales as we found that some customers regularly shop for rebate certificates during their visits to stores. The store managers accom- modate them by making rebate certificates available near the display counter.

Redemption Function In order to get tractable results, we assume that R, is equal to R, (say, R, = R, = R). This assumption may lead to under or over estimation of the optimal refund rate. We now specify the following functional form for the redemption rate:

R = a(1 + A&{1 + (1 + A$), (14)

where (Y is a positive parameter. This function suggests that as r gets close to zero, R also tends to approach zero. As (1 + A& increases, R increases at a decreasing rate, and as (1 + A,)r becomes very large, R approaches 01 as a limit. In fact, for large (1 + A& values, common sense suggests that cx should be close to one, because almost all purchasers should request the promised refund.

Optimal Refund Rate Myopic Condition: Using the above functional forms and replacing Q,, R, JQ@r, and 8U& by their values in (5), after algebraic simplification we get:

cxP(1 + A,)(1 + p)r*2 - [Pm(l + AJ - 2 aP] X r* - pm = 0

Solving for r*, we get,

(15)

r* = [pm(l + AJ - 2cxP + [@m(l + AJ - 2c~p)’ + 4r$mP(l + A,)(1 + ~>]“2]/{2aP(1 + A$

X (1 + P)l (16)

Long-term Condition: Using the above functional forms and replacing Q,, R, aQ#.Yr, and aR/& by their values in (lo), solving for r*, we get,

r* = [pm(l + A,){(1 + S>n - A(1 + S>,-l + $1 - A)] - ZolP(1 + 8)” + [{pm(l + A,)((1 + 8)” - A(1 + s>n 1 + $1 - A)) - ZcYp(1 + 6>“)2 + 4apmP(l + A,)(1 + p)(l + S)n{(l + 8)” - A(1 + S>n ’ i- ~(1 - A)}]“2]/{201P( 1 + AJ

X (I + P)(l + 6)“) (17)

It can be shown that there exists a value of r* (0 < r* < 1) which always satisfies equations (16) and (17), and if p lies between 0 and 1, we can show from the equation (17) that r* asymptotically approaches to m(( 1 + S)n - A(1 + S)- + $1 - A)]/(aP( 1 + 6)“). This result implies that

. there will be a maximum ceiling up to which the optimal refund rate can be increased.

Model Findings In this section, we first derive the comparative statics to show the sensitivity of the refund rate to various parameters, p, (Y, A, y, and 6. Figure 1 shows the optimal refund rate, r*, as a function of p, OL, A, y. The main findings from Figure 1 confirm our earlier discussion of the effects of specific parameters on the firm’s optimal refund rate (see Model Results section). Further, we suggest that the optimal refund rate r*:

. increases at a decreasing rate as demand increases, i.e., as the market becomes more responsive to this type of promotion. However, there is a maximum ceiling up to which the optimal refund rate can be increased,

. decreases as the market propensity to claim refunds increases, i.e., as the redemption rate increases.

We also used numerical analysis to obtain selected results of our model for the selection of the optimal refund rate. These results are obtained in an assumed market environment. We have attempted to make realistic assumptions about the model parameter values in setting up the market conditions for our analyses. For example, a mail-in rebate promotion is more prevalent with products that retail for 50 dollars or less; con- sequently we vary the price of the product in our analyses from 10 to 50 dollars. In order that our numerical results are applicable to a broad spectrum of market situations, we consider different scenarios which vary in the assumed levels of demand, purchase acceleration, rebuys, and redemption rate. The parameter p associated with the demand function is allowed to take five values, viz, 0.1, 0.3, 0.5, 0.7, and 0.9. This is consistent with our discussion in the model specification section that the parameter I3 should lie between 0 and 1. Sim- ilarly, the parameters A, y, and (Y both are allowed to take five values, viz, 0.1, 0.3, 0.5, 0.7, and 0.9. Based on our discussions with executives who design rebate promotions, we vary the gross margin rate, m/P, in our analyses from 0.1 to 0.4 and rebate specific advertisement, A,, from 0 to 50,000 dollars. Finally, we set the discount parameter, 6, at 0.05, 0.10, and 0.15 and consider the purchase cycle, n, to be 1, 2, or 3 unit period(s) from the promotional period.

The optimal refund rate for the short-term as given in the equation (16) is a function of five variables, p, CY, m, P, and A,. By varying the values of these variables as mentioned above, we obtained 3,000 different market scenarios. Simi- larly, we obtained 675,000 different cases for the long term optimal refund rate as given in the equation (17). Based on our numerical analyses, we obtained 2,850 different cases out of 3,000 where the optimal refund rate for the short-term exists. Similarly, in 650,635 cases out of 675,000 cases the optimal refund rate for the long-term exists. If we look at the frequency distribution of the optimal refund rate for these market scenarios, we observe that both the distributions for

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0 1 B-

0 1 a_

Demand !Sedtiiity Parameter Redemption Pammeter

t

r’ max

Refund Rate

-------__

t I-

Refund Rate

i r

Refur Rate

I 0 5- Y-

Acceleration Parameter Rebuy Parameter

Figure 1. Optimal refund rate, r*, as a function of (Y, f3. A, and y.

1

short- and long-term refund rate are right-skewed. The mode, median, and mean of the distribution for the short-term optimal refund rate are 0.05,O. 14, and 0.20 respectively. Similar terms for the long-term optimal refund rate are 0.02, 0.09, and 0.15 respectively. It seems that the long term optimal refund rate is lower than the short-term rate. In other words:

. Ignoring purchase acceleration and rebuys will more likely result in overestimation of the optimal refund rate.

We also explore the effects of specific factors on the optimal rebate rate via a set of ordinary least square (OLS) regression models for the short- and long-term models. The main findings from the regression analysis confirm our earlier discussion of the effects of specific parameters on the firm’s optimal refund rate.

Conclusions Hopefully, the optimization model introduced in this paper will facilitate management’s understanding of the interaction of marketing mix and demand related variables as they affect the amount of the advertised refund available to purchasers of the rebate product. Far from being a substitute for executive opinion, this model should be viewed as a tool to supplement those judgments and enhance managerial thinking.

In our modeling of the mail-in rebate optimization problem, we have included the need to represent the manufacturer’s long-term interest in the promotional campaign and have shown that the long-term optimum refund rate exists. To the extent that there will be some purchase acceleration and the likelihood that some of the customers who switched to the

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promoted brand may be sufficiently satisfied to make a repeat purchase in the next purchase cycle, a brand manager may under or overestimate the refund rate of its rebate promotion if these effects are ignored.

We have assumed that the rebate campaign is in effect for one period. Further, for reasons of tractability, we assume that the accelerated sales come from the next period, and rebuys comes from the next purchase cycle that is n time periods from the promotional period. This is logical if we assume that most durable goods manufacturers will experience a maximum of one rebuy by a given consumer during the strategic planning horizon they usually set. In reality, accelerated sales and rebuys would be drawn from more than one period. This assumption will then definitely understate the value of these factors on the optimal refund rate, and this underscores our earlier point that managers should be concerned about long- term effect in determining the optimal refund rate. Further, the length of the rebate period, which is considered fixed here, itself will be a decision variable for managers. For example, if the length of rebate period is also a decision variable, Q, and R would change depending on this length. These issues pro- vide some interesting dynamics which we leave for future research.

We determine the optimal refund rate in a decision theoretic framework. This paper deals with the influence of various factors on individual firm’s choice of optimal refund rate, it does not deal with issues dealing with market behavior-in- terdependence among rivals’ promotional plans. A game theoretic framework needs to be employed for such analyses. Also, “rebate clutter” due to competitive activities may reduce the effectiveness of a rebate campaign (Marketing Communi- cations, March, 1989, pp. 42-43).

Another important issue that remains unanswered in this paper is the empirical validation of the results. We used numerical analyses to explore the effects of specific -factors on the optimum refund rate. However, the model parameters need to be estimated. A firm could get good parameter esti- mates by using data from past rebate experiences or by run- ning a rebate promotion in two or more market tests, over a short period of time (only the time necessary to obtain reliable results), and to monitor the results of the promotional campaign. If reliable promotions are used frequently, the para-

meters can be updated regularly, taking into account the results of the latest campaign to decide upon the next. Clearly, there is much potential for further work in this area to improve our understanding of the criteria and relevant variables that should be considered in determining the optimal refund rate.

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A model for optimizing the refund value in rebate promotions