INTRODUCTION

Some researchers in marketing contend that mail-in rebates present a great opportunity for firms to increase their profit (Mouland, 2004). Two main advantages of rebates are that few consumers redeem them and that they do not lower consumers' future price expectations of the product. Manufacturers mail-in rebates are the most common type of rebates. Lately many retailers are, however, offering store rebates.

The main focus of research has been on consumer attitudes and behaviour with respect to mail-in cash rebates and there has been little research on determining the optimal rebate strategy for sellers. Furthermore, there is less research on joint optimal pricing and rebate strategy. Like pricing, rebates are used to manipulate demand. Therefore, determining optimal rebate strategy cannot be accomplished without simultaneously determining product price.

In this paper, we develop a profit-maximisation model for jointly determining the optimal price and rebate value. Based on survey data, consumers are divided into three segments: rebate independent, fully rebate dependent, and partially rebate dependent. The data also indicate that the redemption rate is increasing and is linear or slightly convex for low rebate value and becomes concave at high rebate values. Therefore, the model is solved for linear and convex redemption rates that apply for small ticket items for which the rebate is less than $20. Results of the model indicate that three consumer attributes are critical in determining the profitability of rebates. First is the reference value, which can be thought of as the value consumers place on the time required to redeem the rebate. The larger the reference value, the larger the optimal rebate value and the higher the profit. Second is the distribution of consumers among the three segments. Profit decreases as the proportion of consumers in the partially rebate-dependent segment decreases. There is, however, a threshold value where profit increases as the proportion of consumers in the partially rebate-dependent segment decreases. This occurs because as the rebate-independent and fully rebate-dependent consumers are priced out of the market, a seller can increase both price and rebate value substantially without having to be burdened by the heavy cost of rebates for the fully rebate-dependent consumers. If the reference value for the partially rebate-dependent segment is high, rebates may prove profitable. Third is the ratio of the increase in demand due to a $1 increase in rebate value to the increase in demand due to $1 decrease in price. The closer the amount is to a $1, the more profitable the rebate programme. This ratio, termed rebate attractiveness, is in part determined by the distribution of consumers among the three segments and approaches zero as the proportion of consumers in the rebate-independent segment approaches 100 per cent.

In the next section, the literature on mail-in rebates is reviewed. In a further section, results from the data are discussed and the linear redemption model is formulated and solved and then the convex redemption model is formulated and solved followed by numerical examples and sensitivity analysis. We close with conclusions and suggestions for future research in the last section.

LITERATURE REVIEW

One marketing tool that has received little attention in the production-marketing interface literature is cash mail-in rebates. The use rebates has become very popular among manufacturers and retailers (Bulkeley, 1998; McGinn, 2003). Most rebates are offered by manufacturers, but rebates are also becoming popular among retailers. While many of the studies on delayed incentives focus on consumer perceptions and behaviour, little has been done to determine their optimal value, or to examine their impact on inventory policy and profit.

Several studies have examined consumer perception and response to delayed incentives (Soman, 1998; Folkes and Wheat, 1995; Tat et al., 1988; Jolson et al., 1987). The literature suggests a few reasons for the popularity of mail-in rebates (Jolson et al., 1987) which include negligible accumulation requirements by the consumer, ease of scheduling by manufacturers and retailers, minimal risk by consumers, and slippage. Slippage refers to the proportion of consumers who purchase the product because of the rebate but never redeem it. Preliminary evidence indicates that most consumers never redeem the rebate offers. For example, Jolson et al. (1987) reports that 70 per cent of consumers whose buying decisions are influenced by rebate offers never redeem them. Some estimates of the total proportions of consumers who redeem their rebates are in the range of 5–10 per cent (Bulkeley, 1998). Even for rebates as high as $100, some experts suggest that 50 per cent of consumers do not make an effort to collect them (McGinn, 2003). A fifth reason suggested for the popularity of mail-in rebates among manufacturers is that they allow them to offer price discounts directly to the consumers whereas with traditional price cuts, retailers may not fully pass the price reductions to the consumer (Bulkeley, 1998). A sixth reason which makes rebates preferable to using coupons or sales is that consumers' price expectations are higher for products with rebates than the same products with sales or coupons (Folkes and Wheat, 1995). In other words, when the promotions are offered using coupons or sales, empirical evidence indicate that the savings are likely to be integrated into the regular price, which means consumers will have future price expectations closer to the promoted price (Folkes and Wheat, 1995). This integration and decrease in future price expectations are much less likely to occur when the promotion is delivered using mail-in rebates.

Based on an empirical experiment, Soman (1998) tested and found support for two hypotheses. The first hypothesis is that consumer choice behaviour is more sensitive to the rebate face value at the time of purchase than it is to the level of effort required for redemption. The second hypothesis is that the redemption rate after purchase is more sensitive to the level of effort than the rebate face value. In other words, consumers underestimate the effort required for redemption at the time of purchase. Therefore, mail-in rebates provide retailers with an excellent opportunity for increasing profit, especially when the optimal rebate face value is determined jointly with price and order quantity. A retailer can increase the per unit price but at the same time can increase the mail-in rebate face value to offset the impact of the price increase on demand. If the redemption rate is low enough, then the net impact will be an increase in profit. Silk (2005) conducted a series of experiments that tracked the purchase and redemption of a real rebate offer. The experiments showed that increasing the rebate value or the length of the period for which it is valid make consumers more confident that they will redeem the rebate and increase the likelihood of purchasing the product. This increased confidence did not result in significant increase in actual redemption later on. Therefore, increasing rebate value has much less impact on redemption behaviour than purchase behaviour. Furthermore, contrary to expectations, increasing the length of valid redemption period decreases redemptions.

Tat (1994) investigated three consumer motives for rebate redemption: price consciousness, perceived time and efforts associated with rebate redemption, and perceived satisfaction from using rebates to obtain the savings. All three factors were found to be significant predictors of consumers' decisions to redeem rebates. Perceived satisfaction was the most significant, followed by price consciousness, and then perceived time and efforts. The study confirmed earlier results reported by Tat et al. (1988) on the negative relationship between the perceived effort and the difficulty of redemption and redemption rates. Tat and Schwepker (1998) empirically investigated the relationships between rebate redemption motives. The authors found that perceived price consciousness was positively related to rebate redemption; however, the relationship was not statistically significant, which implies that more price conscious consumers do not redeem rebates more frequently than consumers who are not as price conscious. Similarly, the authors found a negative, but statistically insignificant, relationship between time and effort required for redemption and rebate redemption. In other words, the time and effort needed to redeem a rebate offer does not seem to directly decrease its probability of being redeemed.

Ali et al. (1994) developed a model for determining the optimal refund rate of rebates as a proportion of the retail price. Their model considered four ways by which rebates contribute to incremental sales: brand switching, repeat purchases, purchase acceleration, and category expansion. The authors developed a myopic model that considers incremental profit due to sales during the rebate period only. The authors also developed a long-term model that also considers the repeat purchases in subsequent cycles due to brand switching and showed that an optimal long-term rebate value exists. Soman (1998) took another step toward establishing optimal rebate value by formulating a profit maximisation model in which demand and redemptions are linear functions of rebate value. The model is developed for a pre-determined price and under the assumption that total redemptions depend only on rebate face value and are independent of the quantity sold. Gilpatric (2003), using the present-biased preference model, identified market conditions under which a rebate programme is profitable. In this model, some consumers assume that their preferences will be unchanged in the future and buy the product thinking they will redeem the rebate. Later on, because their preferences change, they do not redeem it. Chen et al. (2005) introduced an explanation of rebate usage based on the idea of 'utility arbitrage'. In this model, rebates are viewed as state-dependent discounts. As redemption occurs after purchase, the rebate utility to the consumer will depend on their income utility at that time, which can be low or high. The authors compared a rebate policy with no rebate policy and presented a set of necessary conditions for an optimal rebate policy for the performance of utility arbitrage.

Mail-in cash rebates have not received as much research attention as other sales incentives such as coupons (see Anderson and Song (2004) for a brief review). Dhar et al. (1996) analysed the profit impact of package coupons on profit. They compared three types of package coupons: Peel-off coupons that are redeemed at time of purchase of the item, on-pack coupons that can be redeemed at future purchases of the item, and in-pack coupons that are similar to on-pack coupons except consumers are unaware of their presence at purchase. Other researchers dealt with the impact of other types of coupons. A distinguishing characteristic of mail-in rebates is that they require an additional effort for redemption over coupons, which may reduce their redemption rates.

Our review of the literature did not identify any models that analyse the pricing and rebate value decisions while taking into account the diverse behaviour of consumers regarding rebates. In this paper, we develop a model for determining the optimal price and rebate value of a manufacturer operating in a market with three consumer segments. Based on empirical evidence, consumers are divided into three segments with different demand sensitivity to price and rebate value and redemption behaviour.

MODEL 1: LINEAR REDEMPTION FUNCTION

Data on actual rebate redemptions are closely guarded and firms are reluctant to release it. We relied on exploratory data gathered from 204 graduate students using a questionnaire. Summary of the data is shown in Appendix A. In designing the questionnaire, we relied on literature from the present biased preferences model (O'Donoghue and Rabin, 1999, 2001). In this model, consumers are aware that they may have self-control problems in the future. In the case of rebates, this self-control problem is whether they will have the discipline to expend the time and effort required to redeem the rebate offer. Individuals are classified into one of two groups: sophisticated individuals who have full knowledge of their future selves and preferences, albeit different from their current selves and preferences, and naïve individuals who assume that their preferences will not change over time. Naïve consumers may find that after purchase of the product, their preferences have changed and they do not redeem the rebate. We therefore divide consumers into three segments with the first two making up the sophisticated consumers and the third making up the naïve consumers.

1. Rebate-independent (RI) consumers. These consumers do not redeem rebates and their demand is unaffected by rebate value. They may have been adversely affected by negative past redemption experience or realise from past redemption failures that they do not redeem rebates. The existence of this consumer segment is supported by earlier research (Oldenburg, 2005).
2. Fully rebate-dependent (FRD) consumers. These are consumers who are very sensitive to the net cost of the product and always redeem the rebate offer.
3. Partially rebate-dependent (PRD) consumers. These are consumers who are sensitive to rebates and intend to redeem them at the time of purchase, but may not act on their intentions.

The data from the questionnaire were used to provide three characteristics of the consumers: (1) The distribution among the three segments: RI, FRD, and PRD, (2) the propensity of PRD to redeem rebates, and (3) how do the PRD and FRD trade-off price decrease versus rebate increase. The data are used to identify reasonable values for the model parameters. Obviously, values of these parameters will depend on the type of product as it determines the consumer base, the ease of the redemption process, the time allowed for redemption, etc.

The following notation is defined:

t, f, p

denote rebate-independent consumers, fully rebate-dependent consumers, and partially rebate-dependent consumers

P

price per unit, a decision variable

R

value of the rebate, which does not include its processing cost if it is redeemed, a decision variable

r

processing cost per rebate redeemed, which is incurred in addition to rebate value

h

cost per unit of the product

x_i

demand for the product by consumer segment i, (i=t, f, p)

X

total demand for the product and

profit, an effectiveness measure.

RI consumers are sensitive to only price and, therefore, their demand can be written as

FRD consumers are sensitive to both price and rebate value and, therefore, their demand can be written as

PRD consumers are also sensitive to both price and rebate value but in different degrees than FRD consumers. Therefore, their demand can be written as

Equations (1), (2) and (3) imply that demand decreases (b_t+b_f+b_p) units for each $1 increase in price and increases (c_f+c_p) units for each $1 increase in the value of the rebate. The ratio

is a measure of the effect of a $1 increase in rebate value relative to a $1 decrease in price on demand and will be referred to as 'rebate attractiveness'. An L=1 indicates that there are no RI consumers and as many consumers buy the product because of a $1 increase in rebate value as those who will buy it because of a $1 decrease in price. An L value close to zero indicates that most consumers are RI or insensitive to rebates and that increasing rebate value is much less effective in increasing demand than decreasing price by the same amount.

Total rebate redemptions depend on the total number of FRD and PRD consumers who purchase the product. All FRD consumers will redeem the rebate. For PRD consumers, the proportion of rebates redeemed depends on rebate value relative to a reference value, denoted by V, resulting in x_pR/V redemptions. The use of the linear function for the proportion of PRD who redeem the rebate results in a reasonably good fit for the self-reported data for values of up to $20 with adjusted R² value of 0.93. Therefore, the total number of rebate offers redeemed is

A reference value is used instead of unit price because as observed by Soman (1998), redemption occurs after purchase and at that time the redemption probability is no longer influenced by the price paid for the product.

To simplify the analysis, we introduce the following assumption with respect to the parameters of the demand functions:

which implies that the demand of the RI segment vanishes first, followed by the FRD segment, and finally the PRD segment. Let

All three consumer segments are served

If the demand for all three consumer segments are positive, the profit, which is revenue minus cost, is given by

Using the expressions for x_i and g from equations (1, 2, 3) and (5) in equation (10) and simplifying gives the following expression for profit:

where A=_ia_i, B=_ib_i, and C=_ic_i. Appendix B shows the conditions under which the Hessian matrix is negative semi-definite. Under those conditions, the sufficient conditions for optimality are given by

and

where

At convergence, the values of P^* and R^* should be used to compute x_p, x_f, and x_t. If a consumer segment's demand is negative, then as shown later, the optimal price and rebate value may result in no sales to that consumer segment.

Not all consumer segments are served

Let Z_fp be the profit if only the FRD and PRD consumers are served with a maximum at P_fp^* and R_fp^* (the subscript fp denotes only the FRD and PRD segments). Z_fp is obtained from equation (11) with b_t=a_t=0, which should also be used in computing A and B. Equations (12) and (13) can be used to compute P_fp^* and R_fp^*. The conditions under which the profit function without the RI segment is concave are shown in Appendix B.

Let Z_p be the profit if only PRD consumers are served with a maximum at P_p^* and R_p^* (the subscript p denotes only the PRD segment). Z_p is obtained from equation (11) with b_t=a_t=b_f=a_f=c_f=0, which should also be used in computing A, B, and C. Lemma 1 shows the optimal rebate value.

Lemma 1.

A solution is optimal if and only if

Proof.

The proof is shown in Appendix B.

Substituting for R_p^* from equation (15) into equation (12) gives

The sufficiency of equation (16) is proven in Appendix B. The global optimal solution may occur when all three consumer segments are served, only two segments are served, or only one segment is served.

Identifying the global optimal solution is based on the following properties of the profit function which derive from assumption in (6).

Property 1.

For all P and R in S₁, Z>Z_pf>Z_p.

Proof.

The proof is shown in Appendix B.

Property 2.

For all P and R in S₂ and P>h+r+R, Z<Z_fp and Z_fp>Z_p.

Proof.

The proof is shown in Appendix B.

Property 3.

For all P and R in S₃, Z_fp<Z_p and Z_p>Z.

Proof.

The proof is shown in Appendix B.

Lemmas 2–4 are used to develop an algorithm for identifying the global optimal solution.

Lemma 2

If equations (12) and (13) with all three consumer segments converge to P^* and R^* in S₂ or S₃ then it is optimal not to serve the RI consumer segment.

Proof.

The proof is shown in Appendix B.

Lemma 3.

If equations (12) and (13) with only the FRD and PRD segments converge to P_fp^* and R_fp^* in S₁ or S₃ then it is not optimal to serve only the FRD and PRD consumer segments.

Proof.

The proof is shown in Appendix B.

Lemma 4.

If equations (12) and (13) with only the PRD segment converge to P_p^* and R_p^* in S₁ or S₂ then it is not optimal to serve only the PRD consumer segment.

Proof.

The proof is shown in Appendix B.

Boundary solutions

If the optimal solution occurs at the boundary between S₁ and S₂ where P=a_t/b_t (ie the demand of the RI segment is zero), then substituting P=a_t/b_t into equation (11) and differentiating with respect to (w.r.t.) R, yields the following necessary condition, which is sufficient under the condition shown in Appendix B, for R to be optimal:

where

If the solution occurs at the boundary between S₂ and S₃ where P=(a_f+c_fR)/b_f (ie the demand of the FRD segment is zero), then substituting P=(a_f+c_fR)/b_f and a_t=b_t=0 into equation (11) and differentiating w.r.t. R, yields (17) with

as the necessary condition for optimality, which is also sufficient under the condition shown in Appendix B.

Algorithm to identify the optimal solution

Step 1.

Using all three consumer segments in equations (12) and (13) compute P^* and R^*.

 

If P^* and R^* are in S₁ then P^* and R^* are a local maximum.

 

If P^* and R^* are in S₂ or S₃ then R^* is given by equations (17), (18), (19) and (20) and P^*=a_t/b_t.

 

Compute the maximum profit in S₁ denoted Z₁^* using equation (11).

Step 2.

Using only the FRD and PRD segments in equations (12) and (13) compute P_fp^* and R_fp^*.

 

If P_fp^* and R_fp^* are in S₂P_fp^* and R_fp^* are a local maximum. Compute Z_fp^*

 

If P_fp^* and R_fp^* are in S₃ then use equations (17) and (21, 22 and 23) to compute R_fp1^* which should be used to compute P_fp1^*=(a_f+c_fR_fp1^*)/b_f and then Z_fp1^*

Step 3.

Using only the PRD consumer segment in equations (15) and (16) compute P_p^* and R_p^*.

 

If P_p^* and R_p^* are in S₃ then P_p^* and R_p^* are a local maximum. Compute Z_p^*.

 

If P_p^* and R_p^* are in S₂ then use equations (17) and (21), (22) and (23) to compute R_p1^* which should be used to compute P_p1^*=(a_f+c_fR_p1^*)/b_f and then Z_p1^*

Step 4.

Compare all local maximum profits and select the largest.

MODEL 2: REDEMPTIONS INCREASE AT AN INCREASING RATE IN REBATE VALUE

As described by Soman (1998), one of the problems with the linear redemption function is that it does not reflect actual consumer behaviour at extreme values of the rebate where very few or very many consumers redeem them. This observation is supported by the collected data especially at large values of the rebate exceeding $25 where the redemption function clearly increases at an increasing rate. As we are restricting our attention to small ticket items, we assume proportion of rebate offers that are redeemed by the PRD is an increasing convex function in the value of the rebate and is given by:

where u is an empirically determined constant. Equation (24) implies that 100 per cent of consumers redeem the rebate (ie g=1) when its value is R=u/2. In other words, u/2 has the same meaning as the reference value in the linear redemption function. A graph of the linear redemption function versus the redemption function given by equation (24) for V=u/2=$50 is shown in Figure 1.

Figure 1.

Linear versus nonlinear redemption functions for the PRD consumer segment

Full figure and legend (12K)

Using the expressions for x_i and g from equations (1), (2) and (3) and (24) in equation (10) and simplifying gives the following expression for profit:

Let

Appendix B shows the conditions under which the Hessian matrix is negative semidefinite. Under those conditions, the sufficient conditions for optimality are given by

and

At convergence, the values of P^* and R^* should be used to compute x_p, x_f, and x_t. If a segment's demand becomes negative, then as shown later, the optimal price and rebate values may result in no sales to that consumer segment.

Not all consumer segments are served

Let Z_fp be the profit if only the FRD and PRD consumers are served (ie its demand is greater than zero) with a maximum at P_fp^* and R_fp^*. Z_fp is obtained from equation (25) with b_t=a_t=0, which should also be used in computing A and B. Equations (30) and (31) can be used to compute P_fp^* and R_fp^*. The conditions under which the profit function without the RI segment is concave are shown in Appendix B.

Let Z_p be the profit if only PRD consumers are served with a maximum at P_p^* and R_p^*. Z_p is obtained from equation (25) with b_t=a_t=b_f=a_f=c_f=0, which should also be used in computing A, B, and C. Lemma 5 shows the optimal rebate value.

Lemma 5.

A solution is optimal if and only if

Proof.

The proof is shown in Appendix B.

Substituting for R_p^* from equation (32) into equations (26) gives

The sufficiency of equation (33) for optimality is proven in Appendix B. The global optimal solution may occur when all three consumer segments are served, only two segments are served, or when only one segment is served.

Properties 1–3 and Lemmas 2–4 also hold for this redemption function. Therefore, the same algorithm can be used to find the optimal solution after replacing equations (12, 13), (15, 16), (17) (18, 19, 20), and (21, 22, 23) with equations (30, 31), (32, 33) (31), (34, 35, 36 and 37) (38, 39, 40 and 41), respectively.

Boundary solutions

Similar to the linear redemption function, if the optimal solution occurs at the boundary between S₁ and S₂, then substituting P=a_t/b_t into equation (15) and differentiating w.r.t. R, yields equation (31) with

as the necessary condition for R to be optimal, which is also sufficient under the condition shown in Appendix B. If the solution occurs at the boundary between S₂ and S₃ where P=(a_f+c_fR)/b_f, then substituting P=(a_f+c_fR)/b_f and a_t=b_t=0 into equation (25) and differentiating w.r.t. R yields equation (31) with

as the necessary condition for optimality, which is also sufficient under the condition shown in Appendix B.

NUMERICAL EXAMPLES AND SENSITIVITY ANALYSIS

Numerical Example 1

Based on the collected data, we use the consumer distribution among segments shown in Table A1. We assume a total market of 50,000 consumers. In addition, we assume a loss of the consumer base of 5.5 per cent, 5.0 per cent, and 4.5 per cent for each $1 increase in price for the RI, FRD, and PRD segments, respectively. Based on the data, PRD consumers are assumed to be indifferent between $1.72 increase in rebate and $1.00 decrease in price whereas FRD consumers are assumed to be indifferent between $1.69 increase in rebate and $1.00 decrease in price. The resulting demand functions are: x_t=3,450-189.75P, x_f=20,350-1017.50P+602.78R, x_p=26,250-1181.25P+686.37R, and regression analysis of the data for rebate value up to $20 indicate that V=$24.64. The curve has a better fit at the lower values of R while the fit becomes poor as R becomes larger. At unit cost of h=$5, the product is sold to all three segments (ie the solution is in S₁) with P^*=$14.55, R^*=$2.62, and Z^*=$148,373. Also, 10.6 per cent of the PRD consumers redeem the rebate offer and 44.3 per cent of all consumers redeem the rebate offer. At unit cost of h=$10, the product is sold to all three segments (ie the solution is in S₁) with P^*=$17.09, R^*=$2.69, and Z^*=$69,936. Also, 10.9 per cent of the PRD consumers redeem the rebate offer and 42.9 per cent of all consumers redeem the rebate offer. At unit cost of h=$15, the product is sold to only the FRD and PRD segments (ie the solution is in S₂) with P^*=$21.10, R^*=$4.72, and Z^*=$23,502. Also, 19.2 per cent of the PRD consumers redeem the rebate offer and 41.3 per cent of all consumers redeem the rebate offer. At unit cost of h=$20, the product is sold to only the PRD segment (ie the solution is in S₃) with P^*=$24.08, R^*=$6.66, and Z^*=$4,776. Also, 27 per cent of the PRD consumers redeem the rebate.

Table A1 - Manufacturer's mail-in cash rebate survey responses.

Full table

Numerical Example 2

To illustrate the impact of different parameters on pricing and rebates, we consider another example of a product with demand functions x_t=40,000-3,000P, x_f=40,000-3,000P+2,500R, and x_p=100,000-6,000P+2,000R for the RI, FRD, and PRD consumer segments, respectively. The cost of the product is h=$4 per unit, r=$1 per redemption, and the proportion of consumers who redeem the rebate offer is linear in its value with a reference value of V=$50. Equation (B.9) shows that the profit function is concave for all prices below $21.00 and rebates up to 100 per cent of price. The algorithm results in an optimal price of P^*=$10.45 per unit and optimal rebate value of R^*=$2.49, which is in S₁. Equation (11) yields an optimal profit of Z^*=$365,180. The demand for the three consumer segments are x_p=42,300 units, x_f=14,883 units, and x_t=8,662 units. The rebates redeemed by the FRD and PRD consumer segments are 14,883 and 2,105, respectively.

If b_t is changed from 3,000 to 4,000, then it is optimal not to serve the RI segment P_fp^*=$11.75 and R_fp^*=$4.01, which is in S₂, and the optimal profit is Z_fp^*=$316,188. If, in addition to the change in b_t, b_f is changed from 3,000 to 5,000, then it is optimal to serve only the PRD segment in S₃ with P_p^*=$12.33 and R_p^*=$7.33 and Z_p^*=$289,560.

For the case in which the proportion of consumers who redeem the rebate offers is an increasing nonlinear convex function given by equation (24), we use u=$100, which results in 100 per cent redemptions at R=$50 (the same as the value of V in the linear redemption function). Equation (B.25) shows that the profit function is concave for all prices below $16.41 and rebates up to 60 per cent of price. The algorithm results in an optimal price of P^*=$10.65 per unit and an optimal rebate value of R^*=$3.16, which is in S₁. Equation (25) yields an optimal profit of Z^*=$369,583. The demand for the three consumer segments is x_p=42,441 units, x_f=15,964 units, and x_t=8,058 units. The rebates redeemed by the FRD and PRD consumer segments are 15,964 and 1,386, respectively. Even though both the linear and nonlinear redemption functions reach 100 per cent redemption rate at $50, the nonlinear case has an optimal price, rebate value, and profit which are $0.21, $0.71, and $4,404 larger, respectively.

If b_t is changed from 3,000 to 4,000, then the optimal solution is in S₂ where the RI consumers are not served with P_fp^*=$12.21, R_fp^*=$5.17, and Z_fp^*=$325,196. Unlike the linear redemption case, if, in addition to the change in b_t, b_f is changed from 3,000 to 5,000, then it is optimal to only serve the PRD segment with P_p^*=$13.87 and R_p^*=$11.75, which are given by equation (31) with equations (38), (39), 40 and (41), and Z_p^*=329,141. The results of the last case are counter-intuitive as the demand of the FRD consumers becomes more sensitive to price, profit increases. This is explained by the fact that if it is not profitable to serve the FRD segment (because the sum of the optimal rebate value, rebate processing cost, and unit cost exceeds the optimal price, any sales to this segment result in a loss), then an increased sensitivity to price of the FRD consumers will allow a firm to drive that segment's demand to zero at smaller values of price. Therefore, the solution space of focusing on the PRD consumers alone, that is S₃, is enlarged and the optimal profit increases.

Numerical sensitivity analysis

Numerical analysis indicate that there are three key consumer/market attributes that determine the effectiveness of a rebate programme. First is the reference value, V. Figures 2 and 3 show the optimal profit, rebate value, and price and for values of V up to $100. Figures 2, 3, 4 and 5 are constructed based on the distribution of consumers shown in Table A1 with unit cost of h=$10. As the figures shows, optimal profit, price, and rebate value increase with V. The rate of increase in the optimal profit, price, and rebate value changes at a value of V between $20 and $30 and becomes larger. This is due to the optimal solution moving from region S₁, where all three consumer segments are served to region S₂, where only the FRD and PRD segments are served. As the RI segment is only sensitive to price, when it is priced out, the optimal price increases and so does the optimal rebate value. As price can be increased without losing as many consumers, the rate of increase in the optimal profit with V becomes even larger.

Figure 2.

Optimal profit as a function of reference value

Full figure and legend (13K)

Figure 3.

Optimal price and rebate value as a function of reference value

Full figure and legend (13K)

Figure 4.

Optimal profit as a function of rebate attractiveness, V=$100

Full figure and legend (44K)

Figure 5.

Optimal price and rebate value as a function of rebate attractiveness

Full figure and legend (16K)

The second key consumer/market attribute is the rebate attractiveness defined in equation (4). As Figure 4 shows, as rebate attractiveness increases, the optimal profit increases at an increasing rate. Figure 5 shows the optimal price and rebate value, which are both increasing in rebate attractiveness. As the figure shows, there is a jump in both optimal rebate value and price at L of about 0.52 as the RI consumer segment is priced out of the market. An important conclusion from Figure 4 is that having a large RI segment can have a negative impact on profit. The reason is that even if consumers in the PRD and FRD segments are indifferent between $1 increase in rebate value and $1 decrease in price but the RI segment is large, the rebate attractiveness will be small. This is an important motive for firms to simplify the redemption process and ensure prompt refunds upon redemption to avoid causing consumers becoming RI due to negative redemption experience.

The third key consumer/market attribute is the distribution of consumers among the three consumer segments. To analyse the impact of different consumer demographics that may exist for different products on rebate profitability, we again use a market of 500,000 consumers and a product with a unit cost of h=$10.00. For each consumer segment, 5 per cent of the demand is lost for each $1 increase in price. PRD consumers are assumed to be indifferent between $1.72 increase in rebate value and $1.00 decrease in price, whereas FRD consumers are assumed to be indifferent between $1.25 increase in rebate value and $1.00 decrease in price. Figures 6 and 7 show the optimal profit, optimal rebate value, and optimal price and for V values of $75 and $100. Figure 6 shows that as the relative size of the RI and FRD increase, profit tends to decrease but not over all ranges. For V=$75 there is a region around (RI=3 per cent, FRD=38 per cent, PRD=59 per cent) where the profit shows an increase. A more significant increase occurs for V=$100 around (RI=5 per cent, FRD=40 per cent, PRD=55 per cent). Both of these increases in profit occur when the optimal solution moves from the boundary of regions S₂ and S₃ into region S₃. Such a phenomena is counter-intuitive as it implies that as the proportion of consumers who are RI and FRD increase, there is a threshold where the optimal profit increases. The explanation is that at a certain threshold in terms of the distribution of consumers among the three segments, it becomes optimal to price the FRD out of the market. As Figure 7 shows, at this threshold distribution of consumers among segments, optimal prices show a sudden increase to price the FRD out of the market. As FRD are 100 per cent redeemers, having them priced out of the market causes the optimal rebate value to also increase. In some cases, such as for V=$100, optimal rebate value becomes even larger than the optimal price.

Figure 6.

Optimal profit as a function of consumer distribution among segments

Full figure and legend (21K)

Figure 7.

Optimal price and rebate value as a function of consumer distribution among segments

Full figure and legend (23K)

Figures 6 and 7 have an important implication in terms of the time dimension. As consumers gain better understanding of their behaviour with respect to rebates over time, they may move from the PRD segment to the RI or the FRD segments. A consumer who finds that she/he did not redeem any rebates, in spite of intending to at the time of purchase, may decide that they will no longer take rebates into account in making the purchase decision and becomes part of the RI segment. Similarly, a consumer who finds that she/he have successfully redeemed all rebate offers over time may place more emphasis on rebates in making a purchase decision and becomes part of the FRD segment. As Figure 6 shows, this change in the relative size of the consumer segments in favor of the RI and FRD segments will cause rebate effectiveness to decrease over time, which may be a contributing factor to some retailers eliminating cash mail-in-rebates altogether (Albright, 2005).

CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH

In this paper, we have developed a profit-maximisation model for a firm using mail-in cash rebates. The model takes into account the presence of three heterogeneous consumer groups: a RI segment whose demand is unaffected by the rebate and whose consumers do not redeem rebates, an FRD segment whose consumers always redeem rebates, and a PRD segment whose consumers intend to redeem rebate offers at the time of purchase but may not do so later. The proposed model is solved for a demand that is linearly increasing in rebate value and decreasing in price and for redemption functions that are increasing (linear or convex) in rebate value. These redemption functions are valid for rebate values up to about $20 beyond which the redemption function becomes concave.

Our analysis indicates that there are three consumer/market characteristics that determine the profitability of a rebate programme: The first is the reference value of the consumers in the PRD segment. The reference value can be thought of as the value at which consumers believe that the rebate is fully sufficiently large to compensate for the time and effort they will spend on redeeming it. The larger the reference price, the larger the increase in profit a rebate programme brings about. This suggests that rebate programmes may be more profitable for products targeted toward consumers with relatively large disposable income since these consumers may place a larger value on their time. The second characteristic is the rebate attractiveness which is given by the ratio of the increase in demand due to a $1.00 increase in rebate value to the increase in demand due to a $1.00 decrease in price. The larger the rebate attractiveness, the larger the increase in profit. The third characteristic, which partially determines the second characteristic, is the distribution of consumers among the three segments. The larger the proportion of consumers in the PRD consumers, the larger the increase in profit from using rebates. This observation, however, is not true in certain regions of the solution space where the optimal solution changes in terms of the composition of the served market.

There are several important research questions that remain. A very important question in light of the increased negative perception of rebates among consumers (Oldenburg, 2005; Chuang, 2003, Spencer, 2002) is the effect of simplifying the rebate redemption process and requirements on redemption rates. The negative perception of rebates has even caused some retailers such as Best Buy to phase-out mail-in rebates altogether (Smith, 2005) and causing many consumers to become RI. If simplifying the redemption process has only a small effect on the redemption rates, then manufacturers and retailers may be able to avoid the negative perceptions that is overtaking the market place and may be causing many consumers to become RI while maintaining profitable rebate programmes. Another area or research involves determining reference values of customer groups and its relationship to disposable income. Obviously, the reference value of consumers may be positively correlated with their disposable income. While some products are targeted to a broad consumer market, some products are targeted to only higher disposable income consumers. Different pricing and rebate strategy may greatly enhance the profitability of these products.

References

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Appendices

Appendix A

A pilot study was conducted in order to develop a profile of consumer behaviour relative to redeeming manufacturers' mail-in cash rebates. A brief four-part questionnaire was designed and distributed to graduate students over a three-week period during the spring semester of 2005. Of the 204 questionnaires collected, consumers fell into the three categories as shown in Table A1.

Respondents indicating they always or sometimes redeem mail-in cash rebates were asked the following question: 'If you were willing to purchase a product priced at $40 that has a $10 mail-in cash rebate, how much would the rebate have to be if the price were raised to $50?' The average responses were $17.21, $16.88, and $17.07 for the PRD, FRD, and the PRD and FRD combined, respectively.

PRD consumers were asked to indicate their future probability of redemption based on nine dollar amounts ranging from $1.00 to $100. After collecting 147 questionnaires, it became clear that the rebate increments were too large to estimate redemption rates for rebates that fall below $20. Therefore, in the next 57 questionnaires, the increment in mail-in cash rebate values were reduced to seven, ranging from $1.00 to $20. Figures A1 and A2 represent the data from the two parts of the survey. Both graphs indicate that for rebate value up to $20, the rebate redemption function is well approximated by a linear function. Actually, regression analysis for the second group of respondents using a linear regression model results in an adjusted R² of 93 per cent and a reference value of $24.64. While this value may be a good estimate for rebate values up to about $20, beyond this value, Figure A1 shows that the redemption function becomes concave. Furthermore, Figure A2 shows that at smaller rebate values, of up about $10, there may be slight convexity in the rebate redemption function. It is worthwhile to stress that many firms who have used rebates in the past have data on the amounts of each rebate they offered and the percentage of rebates redeemed, which may enable them to better estimate the redemption function and not rely on self-reported data.

Figure A1.

Self-reported redemption probability as a function of rebate values up to $100

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Figure A2.

Self-reported redemption probability as a function of rebate values up to $20

Full figure and legend (10K)

Appendix B

Derivatives and Hessian matrix of profit for linear redemption function

Demand for all three segments

The first partial derivatives of Z with respect to P and R are

and

The solution to setting the partial derivatives in equations (B.1) and (B.2) to zero are given by equations (8) and (9). The Hessian matrix of Z is

The determinant of the first principal minor of H is

The determinant of the second principal minor of H is

As D₁<0, Z, is concave if D₂>0. Let

Substituting from equation (B.6) into equation (B.5) and simplifying gives

equation (B.7) has a single root at

D₂ switches signs at P_s. Let be a small positive number. For P=P_s-, D₂=4Bb_p/V>0. Therefore, Z is concave for P<P_s. A plot of P_s for 01 shows the values of P for which Z is concave for all positive rebate values up to price.

Demand from only the FRD and PRD segments

Following the same analysis for the three-segment case, Z_fp is concave for P<P_s, where P_s is given by equation (B.8) with a_t=b_t=0. A plot of P_s for 01 shows the values of P for which Z_fp is concave for all rebate values up to price.

Demand from only the PRD segment

Substituting from equation (15) into the profit function and taking the first and second derivative w.r.t. P gives d²Z_p/dP²=-2b_p<0. Therefore, Z_p is concave and equation (16) is a sufficient condition for optimality.

Proof of Lemma 1.

1. By contradiction, suppose R₁=(c_pV-b_pr)/2b_p+ is optimal. Then decreasing the rebate by and decreasing the price by c_p/b_p will leave demand unchanged. The change in revenue is:

The change in total rebates paid is

As the revenue decreases and rebate cost also decreases, the net change in total profit is

Therefore, R₁ cannot be optimal.

2. By contradiction, suppose R₂=(c_pV-b_pr)/(2b_p)- is optimal. Then increasing the rebate by and increasing the price by c_p/b_p will leave demand unchanged. The change in revenue is

The change in total rebates paid is

The net change in total profit is

Therefore, R₂ cannot be optimal.

Proof of Property 1.

Subtracting Z_fp from Z gives

Subtracting Z_p from Z_fp gives

From equations (B.15) and (B.16), Z>Z_fp and Z_fp>Z_p.

Proof of Property 2.

From equation (B.15), Z-Z_fp<0 for all P and R in S₂. Therefore, Z<Z_fp. From equation (B.16), Z_fp-Z_p>0 for all P and R in S₂ and P>h+r+R. Therefore, Z_fp>Z_p.

Proof of Property 3.

From equation (B.16), Z_fp-Z_p<0 for all P and R in S₃ and P>h+r+R. Therefore, Z_fp<Z_p. Subtracting Z_p from Z gives

From equation (B.17), Z<Z_p.

Proof of Lemma 2.

By Property 1, if P^* and R^* are in S₂, then Z(P^*, R^*)<Z_fp(P^*, R^*). Therefore, there is a P₂ and R₂ in S₂ such that Z(P^*, R^*)<Z_fp(P^*, R^*)Z_fp(P₂, R₂) and P^* and R^* cannot be optimal. By Property 2, if P^* and R^* are in S₃, then Z(P^*, R^*)<Z_p(P^*, R^*). Therefore, there is a P₃ and R₃ in S₃ such that Z(P^*, R^*)<Z_p(P^*, R^*)Z_p(P₃, R₃) and P^* and R^* cannot be optimal.

Proof of Lemma 3.

By Property 3, if P_fp^* and R_fp^* are in S₃, then Z_fp(P_fp^*, R_fp^*)<Z_p(P_fp^*, R_fp^*). Therefore, there is a P₃ and R₃ in S₃ such that Z_fp(P_fp^*, R_fp^*)<Z_p(P_fp^*, R_fp^*)Z_p(P₃, R₃) and P_fp^* and R_fp^* cannot be optimal. By Property 1, if P_fp^* and R_fp^* are in S₁, then Z(P_fp^*, R_fp^*)>Z_fp(P_fp^*, R_fp^*). Therefore, there is a P₁ and R₁ in S₁ such that Z(P₁, R₁)>Z(P_fp^*, R_fp^*)>Z_fp(P_fp^*, R_fp^*) and P_fp^* and R_fp^* cannot be optimal.

Proof of Lemma 4.

By Property 1, if P_p^* and R_p^* are in S₁, then Z_p(P_p^*, R_p^*)<Z(P_p^*, R_p^*). Therefore, there is a P₁ and R₁ in S₁ such that Z_p(P_p^*, R_p^*)<Z(P_p^*, R_p^*)Z(P₁, R₁) and P_p^* and R_p^* cannot be optimal. By Property 1, if P_p^* and R_p^* are in S₂, then Z_p(P_p^*, R_p^*)<Z_fp(P_p^*, R_p^*). Therefore, there is a P₂ and R₂ in S₂ such that Z_fp(P₂, R₂)Z_fp(P_p^*, R_p^*)>Z_p(P_p^*, R_p^*) and P_p^* and R_p^* cannot be optimal.

Derivatives and Hessian matrix of profit for nonlinear redemption function

The first partial derivatives of Z with respect to P and R are

and

The solution to setting the partial derivatives in equations (B.18) and (B.19) to zero are given by equations (30) and (31). The elements of the Hessian matrix of Z are

and

The determinant of the second principal minor of H is

As q₁₁<0, Z is concave if D₂>0. As (u-R)⁴>0, it is sufficient that the term inside the brackets be positive. Let

Substituting from (B.24) into (B.23) and simplifying gives

equation (B.25) has a single root at

D₂ switches signs at P_c. Let be a small positive number. For P=P_c-, D₂=-4b_pBu²(r+u)(1-). Therefore, as long as <1 (ie rebate values less than price), D₂<0 for P<P_c and Z is concave. A plot of P_c for 0<1 shows the values of P for which Z is concave for all rebate values up to price.

Demand for the FRD and PRD only

Following the same analysis for the three-segment case, Z_fp is concave for P<P_c, where P_c is given by equation (B.26) with a_t=0 and b_t=0. A plot of P_c for 01 shows the values of P for which Z_fp is concave for all rebate values up to price.

Demand for the PRD only

Substituting from equation (32) into the profit function and taking the first and second derivative w.r.t. P gives d²Z_p/dP²=-2b_p<0. Therefore, Z_p is concave and equation (33) is a sufficient condition for optimality.

Proof of Lemma 5.

The proof of Lemma 5 can be established in the same way as Lemma 1 using and

Solution at boundary — Linear redemption

At P=a_t/b_t, d²Z/dR²<0 for R>[a_tb_p-b_t(a_p+c_fV+c_pr)]/(3b_tc_p) and Z is concave. At P=(a_f+c_fR)/b_f, d²Z/dR²<0 for R<(b_pa_f-a_pb_f)/[3(b_cc_p-b_pc_f)]+(c_fV–b_fr)/(3b_f) and Z is concave.

Solution at boundary — Nonlinear redemption

At P=a_t/b_t,

where

For k₀<0 (which holds for realistic problems) and R>0, d²Z/dR² switches sign at

If k₁<0 then d²Z/dR²<0 for all R>R_c and Z is concave. If k₁>0, then d²Z/dR²<0 for all R<R_c and Z is concave.

At P=(a_f+c_fR)/b_f, d²Z/dR² is given by (B.28) where

If k₀>0 and k₁<0, k₀<0 and k₁>0, or k₀<0 and k₁<0, then d²Z/dR² switches sign at R_c given by equation (B.31) and for all R<R_c, d²Z/dR²<0 and Z is concave. If k₀>0 and k₁>0 then d²Z/dR² switches sign at R_c given by equation (B.31) and for all R>R_c, d²Z/dR²<0 and Z is concave.