Learning and Forgetting: Optimal Product Sampling Over Time as in 10,000 supermarkets, spending over $3 million to launch its new snack (Brandweek 1995).
Learning and Forgetting: Modeling Optimal Product Sampling Over Time Amir Heiman • Bruce McWilliams • Zhihua Shen • David Zilberman Department of Agricultural Economics and Management, Hebrew University, Rehovot, Israel Department of Agricultural and Resource Economics, University of California at Berkeley, Berkeley, California Department of Agricultural and Resource Economics, University of California at Berkeley, Berkeley, California Department of Agricultural and Resource Economics, Member, Giannini Foundation, University of California at Berkeley, Berkeley, California
F
irms use samples to increase the sales of almost all consumable goods, including food, health, and cleaning products. Despite its importance, sampling remains one of the most under-researched areas. There are no theoretical quantitative models of sampling behavior other than the pioneering work of Jain et al. (1995), who modeled sampling as an important factor in the diffusion of new products. In this paper we characterize sampling as having two effects. The first is the change in the probability of a consumer purchasing a product immediately after having sampled the product. The second is an increase in the consumer’s cumulative goodwill formation, which results from sampling the product. This distinction differentiates our model from other models of goodwill, in which firm sales are only a function of the existing goodwill level. We determine the optimal dynamic sampling effort of a firm and examine the factors that affect the sampling decision. We find that although the sampling effort will decline over a product’s life cycle, it may continue in mature products. Another finding is that when we have a positive change in the factors that increase sampling productivity, steady-state goodwill stock and sales will increase, but equilibrium sampling can either increase or decrease. The change in the sampling level is indeterminate because, while increased sampling productivity means that firms have incentives to increase sampling, the increase in the equilibrium goodwill level indirectly reduces the marginal productivity of sampling, thus reducing the incentives to sample. We discuss managerial implications, and how the model can be used to address various circumstances. (Diffusion; Learning; Product Sampling; Goodwill; Forgetting; Experimenting )
1.
Introduction
Sampling is a widely used product promotion tool. In 1995, 85% of manufacturers of packaged goods engaged in product sampling as part of their promotional mix, and sampling accounted for 10% of manufacturers’ consumer promotion budgets (Shermach 1995). Pizza Hut handed out 1 million pizza samples to approximately 1% of the U.S. domestic market to Management Science © 2001 INFORMS Vol. 47, No. 4, April 2001 pp. 532–546
familiarize potential consumers with its new, reformulated pizza. In addition to the distribution of 1 million samples—or free trials, as they are often called in the industry—Pizza Hut advertised, gave out pizza coupons, and organized public events (Whalen 1997). Similarly, Frito-Lay distributed, all on the same day, more than 6 million packs of its Nacho Cheese Flavored Doritos in 100 special public events as well 0025-1909/01/4704/0532$5.00 1526-5501 electronic ISSN
HEIMAN, McWILLIAMS, SHEN, AND ZILBERMAN Learning and Forgetting: Optimal Product Sampling Over Time
as in 10,000 supermarkets, spending over $3 million to launch its new snack (Brandweek 1995). Although the above examples of product sampling are of new products, established products garnered the largest proportion of the sampling budgets in 1993, with firms increasingly interested in “tailoring sampling programs toward current customers to maintain brand equity and to enhance relationship marketing” (Direct Marketing 1994). On average, 34% of sampling funds are for new products, 38% for established products, and 28% for product extension (Donnelly 1992). This suggests that models that examine sampling behavior need to address why firms continue sampling of mature products. Sampling has not been extensively researched. It is typically considered a tool to introduce consumers to a new product (e.g., Freedman 1986, Jain et al. 1995) or to introduce a product to a new consumer group (e.g., Bettinger et al. 1979). Because sampling distributes products or miniature packages for free, the consumer’s cost of trial is reduced (Ailloni and Cheros 1984). Samples serve as a direct source of information to the consumer and have a greater effect on sales than indirect experiences such as advertising (McGuinness et al. 1992). Scott (1976) found that the effectiveness of sampling depends on the product price, while Scott and Yalch (1980) warn that if consumers interpret sampling as a signal for a poorly performing product, it might affect the consumer in a negative manner. Most of the sampling research has focused on the trial rate (the proportion of consumers who would try the product samples), finding a range of 40% to 80% (Meyer 1982, Haugh 1979a, 1979b; McGuinness et al. 1992).1 Research on adoption rates show that 20% to 60% of those who sample products end up purchasing them (Costa 1983, Meyer 1982, Schultz and Robinson 1982, Dodson et al. 1987). Rothschild and Gaidis (1981) suggest that sampling is an integral part of total marketing effort and that regular buying occurs only after several trials. 1
Bult and Wansbeek (1995) develop a theoretical framework to determine the optimal selection of target groups for sampling, resulting in the target groups differing greatly in their likelihood of receiving samples.
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An industry survey (Marketing News 1995) found that, in 1994, 89% of manufacturers distributed samples in-store, 53% distributed in or on the package, 47% distributed at local events, and 28% used direct mail. All the retail chains in the survey said that they were more likely to merchandise a product supported by a sampling effort, reinforcing the use of sampling as a tool for leverage retail trade support. This helps explain the extensive use of in-store sampling and suggests that sampling increases immediate sales, one component of the model in this paper. Lammers (1991) provided samples of chocolates in a shop located in a well-known mall to determine the effects of sampling a mature product. He found that sampling increased immediate sales, with 84% of the sampled group purchasing some chocolates versus 59% of the nonsampled group, and that sampling increased purchases in all the varieties, not only in the varieties sampled. This implies a 42% increase in immediate sales. Sampling, like advertising, improves the public’s awareness and perception of a product. However, unlike advertising, sampling has a direct experiential effect that reduces the risk of product uncertainty and results in a better per-person response rate. Sampling is also more conducive to product training and demonstration (e.g., cooking or preparing a product). While sampling is expensive on a percustomer-reached basis, it can be targeted to specific groups and can be effective even with a limited promotional budget (e.g., specialty food products sampled in supermarkets). In addition, sampling can be a pleasant experience and is generally preferred by customers over advertising (see Smith and Swinyard 1983, Fitzgerald 1996, Brandweek 1997). Retailers also prefer sampling over advertising because it increases in-store purchases where the sampling is done. The direct experience, consumers’ positive attitude toward it, and the fact that it is held in stores— unlike advertising, which is usually experienced outside of the store—generate a much stronger immediate sales effect than advertisement, while both have an impact on goodwill. In advertising models with goodwill (see Feichtinger et al. 1994 and Little 1979 for surveys of advertising models), advertising affects the building of goodwill for the product, which in turn affects sales. 533
HEIMAN, McWILLIAMS, SHEN, AND ZILBERMAN Learning and Forgetting: Optimal Product Sampling Over Time
However, an important feature of sampling is that in many cases it is designed to have an immediate sales impact. This is particularly the case for products that are sampled in retail outlets and that therefore can be purchased immediately by the consumer. Because of this, we need to model both the immediate effects of sampling on sales as well as its longer term effects on building goodwill. More generally, we can speak of a class of promotional tools that differs between their immediate and longer term effects. This class of promotional activities may include not only sampling in stores but also the distribution of coupons and price reductions or other limitedtime special offers that induce immediate sales.2 Note that if sampling occurs outside the store (e.g., products received at home) or is not consumed immediately (e.g., free toothpaste samples), sampling may not induce “immediate” sales (unless coupled with a coupon or time-limited sale), and from a modeling perspective it may be sufficient to focus solely on its longer term goodwill building effects represented in goodwill advertising models. The model we develop decomposes the sampling effort into the immediate sales and longer run (goodwill building) effects, thus providing a more general framework for marketing analysis that is consistent with the needs for describing sampling efforts. Sampling affects both immediate sales and consumer perception and awareness about the product. Our paper may also be considered a more general marketing model that can be used to address advertising as well as other promotional mechanisms. The level of immediate versus long-run effects can be adjusted depending on the particular characteristics of the promotional tool, and it can support a wide family of dynamic promotional strategies depending on the tools available. In their pioneering theoretical work, Jain et al. (1995) suggested that sampling is critical in the initial stages of a product’s life because increasing the number of first adopters not only leads to a future 2
The meaning of “immediate” depends on the situation. It is the current shopping experience if products are sampled in stores and the period until a sale expires if the model is used to describe sales events.
534
customer base but also provides a source for product promotion by word of mouth. Over time, a product will become familiar to the public, and the need for sampling will decrease. In their model, the firm invests in sampling only in the initial stage of the product’s life-cycle. Their analysis gives insight into the optimal initial level of investment in sampling and how this affects the dynamics of product diffusion. The approach in this paper examines sampling behavior as products mature. Goodwill is built through purchases and sampling. We view the dynamics of goodwill build-up as being dependent on two phenomena: learning and forgetting. Learning is the process of increasing the goodwill toward a product based on sampling and purchases (experiencing the product), while forgetting3 is the depreciation of this goodwill over time (equivalent to goodwill depreciation in Nerlove and Arrow 1962). The results of this model are that firms will initially invest more in sampling in the introduction stage and reduce sampling as product goodwill increases. Firms will continue sampling of a mature product both to increase profits from immediate sales and to counteract the effects of forgetting that reduce the goodwill level. This paper is organized as follows. We formulate the theoretical model and solve it in §2, and in §3 evaluate the dynamic optimal sampling strategy and comparative statics that arise from our model. We conclude in §4 and discuss the managerial implications of our findings.
2.
Model Formulation
Consider a nondurable product that has N potential buyers. At each point in time, the firm distributes Zt units of product samples that affect the purchasing behavior of the product. Goodwill is a cumulative 3
It is often argued that the rationale for continued promotion of mature products stems from at least three effects of a competitive nature: (1) pre-empting entry by new, direct competitors (e.g., Lane 1980); (2) inducing initial trial, category expansion, or switching from current competitors; or (3) as a strategic response to competitive sampling programs. “Forgetting” provides an alternative explanation for ongoing sampling of mature products, particularly in an environment of constant promotion from competitors.
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HEIMAN, McWILLIAMS, SHEN, AND ZILBERMAN Learning and Forgetting: Optimal Product Sampling Over Time
stock variable. Let Xt be the consumers’ average stock of goodwill in period t 0 ≤ Xt ≤ 1.4 Purchasing behavior depends on the existing goodwill toward the product as well as on immediate exposure to samples. In each period the potential consumers are divided into Z individuals who receive samples and N − Z who do not. Let hX be the probability of buying without receiving a sample, and let X be the change in immediate sales due to sampling, both of which depend on the goodwill level. The probability of purchase after receiving a sample is defined as: bX = hX1 + X. Thus, sales from sampling can be decomposed into long-run goodwill effects hX and immediate effects XhX. X = 0 implies that sampling does not affect immediate sales. Figure 1 depicts the relationship between hX and bX. We assume that both purchasing probabilities are increasing, and that concave functions of goodwill and the effect of sampling on immediate purchases decrease as goodwill increases. Therefore, 0 ≤ hX ≤ bX ≤ 1, and h X > b X > 0 b X h X < 0. The (expected) sales5 in each period, yt, are the sum of the purchases by two groups of customers: those who do not receive samples but purchase, and those who receive samples and purchase. This sum is represented as yt = hXN − Z + bXZ6
(1)
If there are no immediate sales effects, X = 0, and Equation (1) reduces to yt = hXN , which is essentially a basic goodwill model in which sales is dependent only on the existing goodwill level. By accounting for immediate sales effects as well as goodwill, this model provides a more general framework for analyzing marketing efforts. 4 In this model, the population is assumed to be relatively homogeneous. Obviously, a more complete model would account for heterogeneity and have goodwill defined for each subgroup within the population. One way to operationalize goodwill is to define it as the average likelihood of purchasing a product. 5 From here on, we will use “sales” to denote “expected sales” and assume the firm is risk neutral. 6 Here we assume that a purchaser buys only one unit. We can easily generalize to account for multiple unit sales if bX is assumed to be nonnegative and is interpreted as the average number of sales.
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Figure 1
Purchasing Probabilities with bX and without hX Sampling
Average Goodwill Formation Model Throughout our analysis we assume that other (nonsampling) levels of advertisement remain constant.7 At the introductory stage of the product’s life-cycle, consumers hold some initial average level of goodwill, Xt = 0. The change in the goodwill level in period t X˙ = X/t, is the outcome of the new experience effect, which we will call the learning effect, LX, minus the goodwill depreciation, or forgetting effect, F X. Formally: X˙ = LX − F X Similar to traditional diffusion models, the learning effect, LX, is modeled as an increasing function of the remaining goodwill potential 1 − X and new experience. There are three sources of direct experience: buying without sampling, buying after receiving a sample, and receiving a sample but not buying. Learning in period t is represented by Z LXt = 1 − X hX 1 − N Z Z + bX + 1 − bX (2) N N 7
Clearly, there are trade-offs and complementarities between advertising and sampling, and they differ in terms of consumer experience, cost, and efficacy. However, to keep the model manageable and focused on the dynamics of optimal sampling, we take other forms of marketing as given and model sampling alone.
535
HEIMAN, McWILLIAMS, SHEN, AND ZILBERMAN Learning and Forgetting: Optimal Product Sampling Over Time
where
= goodwill updating factor after buying without a sample, = goodwill updating factor after receiving a sample followed by buying, = goodwill updating factor after receiving a sample followed by not buying, and Zt = probability of an average consumer N receiving a sample.8
Learning is affected by the sum of three factors: hX1 − Z/N , the expected goodwill change of consumers (from here on “consumers” will refer to the average consumer) who purchased the product without receiving samples; bXZ/N , the expected goodwill change of consumers who purchased the product after receiving samples; and 1 − bXZ/N , the expected goodwill change of consumers who did not purchase the product after receiving samples. The parameters and are discussed in detail later in this section. Equation (2) implies that LX is concave in X. We model forgetting as a linear function of the expected level of goodwill F Xt = Xt where > 0 is the depreciation coefficient. Combining the learning and the forgetting effects yields the dynamics of goodwill: Z X˙ = 1 − X hX 1 − N Z Z + bX + 1 − bX − X (3) N N The traditional diffusion literature assumes that = and = 0. However, may be greater than if sampling can provide the consumer with valuable information about how to use the product. For example, a
study done by Agrexco on avocado purchases in Italy found that sampling of Carmel brand avocados led to repeat purchases. Many consumers had little knowledge about avocados and ate them as though they were eating an apple—without adding salt, pepper, or lemon—and did not wait until they were ripe. Sampling provided the consumers with this vital information and increased the sales of Carmel avocados dramatically. Carmel managers claim they receive a 20% price premium over competitors for their avocados and that sampling is one of their most important activities in creating sales and brand strength (Agrexco). On the other hand, in rare cases, can be less than . < 0 represents the case where samplers who did not purchase had a negative experience of the product. This can arise if consumers who purchase after receiving samples at the buying place (in-store demonstrations) are disappointed that they cannot repeat the sample results at home. Finally, we expect , the after-sampling-but-not-buying goodwill coefficient, to be positive. However, it can be negative if the sampling procedure communicated the wrong message to the consumers or was poorly done. Profit Maximization Behavior. Let be the gross profit from selling one unit of the product (price minus unit cost of production), constant over time, and exogenous to the firm. This specification assumes price-taking behavior, although the firm can monopolistically influence sales through its sampling effort. Nevertheless, the relative simplicity of this model will help us to focus on the role of learning and goodwill accumulation and allow us to examine the optimal dynamic evolution of product sampling. Then total gross profit is Z X = hX× N − Zt + bXZt. Let CZ be the cost of sampling in period t, with properties such that CZ =
8
Equation (2) models the average population and not the three segments separately (the segment not receiving samples, the segment receiving and buying, and the segment receiving but not buying). Treating each segment separately would complicate the model considerably and would not allow us to find analytical solutions since we would have to solve three sets of equations over time instead of one. In this paper we focus on aggregate changes in sampling, goodwill, and sales. Addressing consumer heterogeneity is a major issue for further research.
536
C >0 Z
and
CZZ =
2C > 09 Z 2
The firm is interested in maximizing the cumulative benefits from sampling over the life of the product, 9
Convex sampling costs CZ arise if firms target the cheapest locations or time slots first, increasing their costs as they expand sampling efforts. Sampling is defined as an effective unit of sample, i.e., samples that are distributed and used by the consumers.
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HEIMAN, McWILLIAMS, SHEN, AND ZILBERMAN Learning and Forgetting: Optimal Product Sampling Over Time
which, for simplicity, is assumed to be infinite:10 Max e−rt Z X − CZt dt (4) 0 Z s.t. X˙ = LX Z − F X = 1 − X hX 1 − N Z Z + bX + 1 − bX − X N N
where %E is the elasticity of E with respect to X (i.e., %E = E/XX/E. From the concavity of hX and bX, it can be shown that %E < 1. Therefore, LX < 0 in Equation (9) when X > 1/2, reflecting the fact that at high levels of goodwill, incremental learning declines as goodwill increases.
The current Hamiltonian, H, for the above maximization problem is
3.
H = Z X − CZt + #LX Z − X
(5)
where # is the shadow price of goodwill. The firstorder necessary conditions are H (6) = Z − CZ + #LZ = 0 Z H ˙ (7) = X + #LX − FX = r# − # X H = X˙ = LX Z − X (8) # Equation (6) states that at the optimum, the firm will sample until the marginal immediate gross profit from an additional unit of sampling Z = bX − hX, plus the future benefits from learning, represented by the marginal effect of sampling on learning LZ = 1 − X−hX + bX + 1 − bX1/N multiplied by its shadow price #, equals the marginal cost of sampling CZ . This will be true at every point in time. In the analysis that follows in the rest of this paper, we assume that ≥ ≥ 0 and ≥ 0, which is sufficient for sampling to be effective in increasing learning LZ > 0. Equation (7) describes the movement of the shadow price of goodwill over time, where X = b X − h X > 0 FX = ; and LX = −E + 1 − XEX , where Z Z Z E = hX 1 − + bX + 1 − bX N N N To sign LX we rewrite it as 1 L EX X LX = L − = % − E 1−X X E 1−X 10
(9)
Our problem approximates that of a long-term planner. If the planning horizon is short enough, the firm will not sample unless the increased profits from immediate sales bX−hX justify this.
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Dynamic Analysis
The dynamics of the shadow price of sampling and sales for a profit maximizing firm are summarized in the following three propositions. Proposition 1. The dynamics of # follows: X #˙ = r − L %E − + − X # 1−X #
(10)
This result follows from integrating Equations (7) and (9). The dynamics of #˙ are not straightforward because they are affected by factors that operate in opposite directions. The interest rate r and depre˙ which suggest ciation have positive effects on #, that capital costs and depreciation delay investment in sampling. The marginal income from sales associated with improved goodwill X > 0 also increases investment in goodwill building. Because %E < 1, when %E < X/1 − X, which occurs for X larger than 0.5, improved goodwill increases the marginal experience after buying or after receiving a sample. In those situations, the positive effect of X on experience will increase goodwill build-up and, in turn, this will have ˙ a negative effect on #. Proposition 2. The dynamics of sampling follows: ZX ˙ LZX ˙ C − Z r+ X+ X Z˙ = Z CZZ C Z − Z LZ X LZ − − LX − FX (11) C Z − Z Equation (11) is derived by totally differentiating Equation (6) and replacing #˙ with Equation (7) and # with CZ − Z /LZ (Equation (6)). We know that CZ − Z /LZ is positive and that CZZ > 0. Therefore, ZX ˙ LZX ˙ X+ X sign Z˙ = sign r + CZ − Z LZ X LZ − − LX − FX CZ − Z 537
HEIMAN, McWILLIAMS, SHEN, AND ZILBERMAN Learning and Forgetting: Optimal Product Sampling Over Time
The factors that determine the change in sampling over time and can be decomposed into five effects. First, the interest rate, r, is always positive, encouraging the decision maker to delay investments in sampling. The higher the interest rate, the higher is the incentive to delay the investments, and with very high interest rates, sampling activity over time will be close to zero. We assume that the interest rate is sufficiently low that sampling will occur. The second term, ZX /CZ − Z X˙ < 0, represents the marginal productivity of sampling when goodwill is increasing. The numerator, which represents the impact of an increase in goodwill on the marginal short-term profitability from sampling, is negative i.e., ZX = b X − h X < 0. The denominator is positive from the first-order condition. The X˙ will initially be positive because firms will initially sample to build goodwill. Sampling will decline over time as marginal productivity of sampling declines, until an optimal goodwill level is achieved. The third term, LZX /LZ X˙ < 0, represents the impact of a change of goodwill on marginal learning from sampling. As noted previously, LZX is assumed to be negative because at high levels of X sampling will be less productive. Therefore, when X is increasing X˙ > 0, firms will reduce future sampling because the marginal learning from sampling is declining. The fourth term, X LZ /CZ − Z , is the crossinteraction between marginal learning from sampling LZ and marginal short-term profits from building goodwill X , which are nonnegative for all X. This represents the impact of increased sampling on the marginal profitability of goodwill. Thus, we have a force that encourages high levels of sampling when the product is introduced and reduced sampling as X increases. The fifth and last term, LX − FX , represents the marginal net learning effect of goodwill. When X is low, LX − FX is positive but decreases as X increases, similar to the previous three terms. At high levels of X, this term can be negative if marginal forgetting outweighs marginal learning. The above discussion suggests that, when X is small, all the forces except the interest rate will encourage a higher initial level of sampling that 538
decreases over time. As we will show later in our stability analysis, even when the goodwill level is high, if the rate of forgetting is high the firm may find it optimal to continue sampling to maintain a high goodwill level. Proposition 3. The dynamics of sales follows: ˙ X˙ Z˙ yt = Sh %h + Sb %b + SZ y X Z
(12)
where %h = hX/X · X/hX and %b = bX/X· X/bX represent the elasticities of a change in goodwill on the probability of purchases with and without and sampling, respectively, N − ZhX/y = Sh is the share of sales without samples, ZbX/y = Sb is the share of sales after samples, and bX − hX/yZ = SZ to be the increase in share of sales due to sampling. Equation 12 was derived by taking the derivative of firm sales in Equation 1 with respect to time. ˙ ˙ Sb %b X/X + SZ Z/Z represents the indirect and direct effects of sampling. For customers without samples, the increase in sales is generated only by increasing goodwill build-up as customers purchase the product. In their case, changes in sales over time are ˙ ˙ represented by yt/y = Sh %h X/X, which is similar to traditional diffusion models in that goodwill is built up by sales rather than by sampling. Steady-State Analysis. This subsection derives the optimal dynamic sampling strategy. It is of interest to find the steady-state sampling behavior and, in particular, to determine whether the firm will stop sampling at some period or if it will continue to sample at some level to maintain its former achievements. In addition, it is useful to understand the dynamic path that leads to this steady state. Figure 2 presents a phase diagram analysis of goodwill level X and sampling Z. Isocline X˙ = 0 is positively sloped (proof given in Appendix A). Because a higher X increases the forgetting rate, Z must increase to raise total learning so that X ∗ does not decrease. We do not observe positive Z values with X˙ = 0 when X Only when X is greater than is less than a critical X. X will the forgetting effect fully reverse the learning effect, and attaining X˙ = 0 will require ongoing positive sampling effort. We prove in Appendix A that the Management Science/Vol. 47, No. 4, April 2001
HEIMAN, McWILLIAMS, SHEN, AND ZILBERMAN Learning and Forgetting: Optimal Product Sampling Over Time
Figure 2
Stability Analysis and the Saddle Path
the firm will optimally choose an initially higher level of sampling effort. As goodwill increases, the firm gradually decreases the level of sampling effort over time and keeps a positive level of sampling at the steady state. If goodwill is excessively high to begin with, the firm will initially reduce sampling, essentially “mining” goodwill, and then increase sampling as X approaches X ∗ from above. Sampling follows a smooth transition to steady state that arises as firms take advantage of the inherent capacity of the diffusion model to increase goodwill (represented by the learning parameters , and ). Steady-state sampling depends on two factors. From Equation (6), Z + #LZ = CZ
slope of isocline Z˙ = 0 is negative around the steady state. When X is large enough and X˙ < 0, the isocline of Z˙ = 0 might change signs from negative to positive. The steady state is the intersection point of the two isoclines (point A in Figure 2). The diagram reveals that point A is unstable,11 and a saddle path with negative slope runs through point A. The saddle path is downward sloping when X is small and around the steady state and may become positive as X approaches 1. (See Appendix B for proofs.) The shadow path provides both the necessary and sufficient conditions under which the optimal sampling strategy is realized. Given any level of goodwill (X), it is optimal for the firm to chose a sampling level Z along the saddle path that leads to the steady state. The phase diagram analysis implies that when a product is newly introduced (low goodwill level) 11
It is quite common for problems with dynamic optimization to have saddle points with unstable solutions that can only be reached through a unique shadow path (e.g., Ramsey’s 1928 growth model). The significance of the saddle point is that it ensures a unique optimal level of sampling for any given level of goodwill. Given an explicit functional from, the firm can identify the optimal sampling path. Because the system is unstable, monitoring goodwill will help the firm identify when it has diverged from this path (or steady state) and allow it to adjust its activities to get back to the path or steady state. Thus the firm may incur monitoring costs that are not modeled in this paper but may be considered part of an overall fixed monitoring budget for the firm’s marketing efforts.
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Therefore, ongoing sampling will occur only if the immediate profits from sampling (Z ) plus the future benefits from learning (#LZ ) justify the costs of sampling. If profits from immediate sales are small (Z ≈ 0), sampling may be justified if it leads to learning (LZ > 0). This will be true only if there is forgetting. On the other hand, if there is no forgetting, goodwill will reach its upper potential in the steady state and LZ = 0. In this case the firm will sample only if immediate sales impacts justify this (bX > hX. Sampling costs clearly affect the sampling decision. If they are high enough, they can eliminate sampling altogether. Comparative Statics. It is interesting to examine the factors that influence the steady state and to determine what their managerial implications are. The results of a comparative static analysis of the key variables goodwill (X ∗ ), sales (Y ∗ ), and sampling efforts (Z ∗ ) when the parameters change are shown in Table 1. We discuss the derivations and interpret the comparative statics for on Z ∗ in detail below, with the proofs given in Appendix C. We define g as the change in sampling and f as goodwill building over time, which are given in the right-hand side of Equations (11) and (3), respectively: ZX ˙ C − Z r+ X gX Z = Z˙ = Z CZZ CZ − Z LZX ˙ X LZ + − LX − FX X− LZ CZ − Z 539
HEIMAN, McWILLIAMS, SHEN, AND ZILBERMAN Learning and Forgetting: Optimal Product Sampling Over Time Table 1 Parameter
Comparative Statics of Changes in Steady State Goodwill X ∗ , Sales Y ∗ , and Sampling Z ∗ When Parameters Increase Description
X∗
Y∗
Goodwill updating effect when buying after sampling Goodwill updating effect when buying without sampling
+
+
+/−a
+
Z∗ +− if
g f < > gX fX −
Goodwill updating effect when not buying after sampling
+
+
+− if
g f < > gX fX
b
Immediate purchasing effect due to sampling
+
+
+− if
g f < > gX fX
Goodwill depreciation rate
−
−
+− if
r
Discount rate
−
−
g f > f /fZ . Recall that bX = hX1 + ≥ 0. For the comparative statics only, we treat as a parameter rather than a function of goodwill. b
Z ˙ f X Z = X = 1 − X hX 1 − N Z Z +bX + 1 − bX − X N N When the goodwill-building effect of buying after sampling () increases, X ∗ at the new steady state will be larger because marginal learning more than offsets marginal forgetting, causing X ∗ to increase. A higher increases firms sales Y ∗ at the new steady state. The intuition behind this is the Envelope Theorem in optimal control, in which the net indirect effect of a higher on sales via the changes in Z ∗ and X ∗ is zero. Because a higher directly increases sales, the firm ends up with higher sales and profits at the new steady state. However the effect of a higher on sampling level (Z ∗ ) is indeterminate because increasing has two opposing effects on Z ∗ . The first effect is an increase in the marginal productivity of sampling, which increases future goodwill and therefore sales, thus giving incentives to increase sampling effort. The second effect of is that while an increase in goodwill (X) is achieved, the marginal productivity of sampling declines at higher levels of X, thus reducing incentives to engage in sampling. dZ ∗ /d is positive if and only if g /gX < f /fX ; that is, the impact of a higher 540
on sampling productivity (relative to X) is bigger than its impact on goodwill building (see Table 1).12 The effect of on sampling is determined by the attributes of sampling. Therefore, if has a greater short-run effect through the increase in marginal productivity (increasing goodwill in the near future) than it does on long-run learning and goodwill, the equilibrium sampling level will increase. On the other hand, if is more likely to increase long-run learning, the new equilibrium sampling level will be lower, and the firm reduces total sampling effort and costs while achieving a higher level of sales and goodwill. The comparative statics of on X ∗ and Z ∗ is demonstrated in Figure 3 where the initial steady state is at point A. With an increase in , the isocline 12
The comparative statics of on Z is: f /fZ gX /gZ −g /gZ fX /fZ dZ ∗ + = fX /fZ − gX /gZ fX /fZ − gX /gZ d
where the first term represents ’s positive direct effect on the marginal productivity of sampling −g /gZ represents the gross positive change in sampling Z due to a higher while holding X constant, and fX /fZ fX /fZ − gX /gZ is a positive adjustment factor (less than 1) that accounts for the slope of the X˙ = 0 and Z˙ = 0 isoclines. The second term represents the negative indirect effect of on sampling productivity as increases X through learning and can be similarly decomposed.
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HEIMAN, McWILLIAMS, SHEN, AND ZILBERMAN Learning and Forgetting: Optimal Product Sampling Over Time
Figure 3
As Increases, X˙ =0 Shifts to the Right and Z =0 Shifts Up, ∗ Ending Up with Higher XZ and Either Higher or Lower Z2∗
Z˙ = 0 shifts up as the marginal productivity of sampling increases. This is the first effect and is equivalent to ending up at point B with a higher Z ∗ and X ∗ . The second effect shifts the X˙ = 0 isocline down, ending at point C with a lower Z ∗ and higher X ∗ . The net impact on Z ∗ is the sum of these two effects, ending up at point D (AD = AB + AC), where Z ∗ is generally indeterminate while X ∗ is unquestionably higher. To give further meaning to these results, the adjustment factors (shown below) for the two effects can be shown to reflect slope effects for the two isoclines. This leads us to the following Corollary (a proof can be requested from the authors). Corollary 1. When the slope of the X˙ = 0 isocline is relatively flat (steep), dZ ∗ /d is more likely to be negative (positive). When the slope of the Z˙ = 0 is relatively flat (steep), dZ ∗ /d is more likely to be positive (negative). This corollary allows us to identify factors that affect the slope of isoclines and, therefore, to help determine the effect of a change of on optimal sampling. In particular, we find that when the depreciation rate is high, the X˙ = 0 isocline will be steeper while the Z˙ = 0 isocline is flatter. Therefore, when the depreciation rate is high, equilibrium sampling will tend to increase with an increase in . Intuitively, Management Science/Vol. 47, No. 4, April 2001
with a high depreciation rate we can expect the direct learning effect to be smaller so that the increases in equilibrium X will not be great, while an increase in will allow short-term benefits to increase as the direct effect on sampling productivity increases. For example, products that do not have unique characteristics or a strong brand name are likely to have high forgetting rates associated with them. This suggests that if were increased for “me too” products, sampling will also increase in the equilibrium steady state. By contrast, for products that have distinctive features or a strong brand name where forgetting is low, an increase in will lead to decreases in equilibrium sampling as firms realize that a more effective sampling method means they can reduce sampling expenditures while maintaining, and even increasing, goodwill. Also, when immediate sales impacts are very high, it can be shown that a change in will increase equilibrium sampling. This arises because acts as an accelerator for , so that an increase in has high short-term effects, which increase equilibrium sampling. For example, if a product being sampled in-store greatly increases immediate purchases, an increase in will increase the firm’s long-run equilibrium sampling of this product. Table 1 also shows how the steady state is affected by changes in the other parameters, which we discuss in less detail beginning with the remaining parameters that build long-run goodwill ( and ). A higher increases sales. It has a negative impact on sampling effort Z ∗ because by increasing the purchasing effect of those who do not get samples, a higher implies that sampling is less necessary. While a higher directly increases goodwill, the new lower sampling level has the opposite effect, making changes in X ∗ indeterminate. Increasing the goodwill-building factor when not buying after sampling () increases both goodwill X ∗ and sales Y ∗ at the new steady state as sampling helps customers to understand more about the product even though they do not buy it immediately. Its effect on Z ∗ at the new steady state is indefinite. A higher immediate purchasing effect of sampling ( ) increases both goodwill X ∗ and sales Y ∗ . While 541
HEIMAN, McWILLIAMS, SHEN, AND ZILBERMAN Learning and Forgetting: Optimal Product Sampling Over Time
we would expect sampling to increase in most situations, we are unable to sign the change in sampling Z ∗ (see discussion of ). A higher forgetting rate () decreases goodwill X ∗ and sales Y ∗ at the new steady state but has an indeterminate impact on the steadystate sampling level Z ∗ . Both the discount rate (r) and exogenous price () have unambiguous impacts on X ∗ , Y ∗ , and Z ∗ at the new steady state. When r increases, the opportunity cost of sampling becomes more expensive, reducing the sampling level. The lower sampling level lowers both goodwill and sales at the new steady state. When increases, the value of marginal productivity of both sampling effort (Z) and goodwill (X) increases. Therefore, both X ∗ and Z ∗ are higher at the new steady state. Sales also increase because of both higher sampling efforts and higher goodwill. Although is treated as a fixed parameter in our model, our analysis can extend to situations where changes exogenously over time. If increases over time (price steadily increases or production costs decrease with learning by doing), then the steadystate sampling, goodwill, and sales levels will also increase with time. We may also consider a more general environment in which consumers are heterogeneous. Heterogeneity may arise in consumers’ (1) responsiveness to sampling or product purchase level X or ; (2) likelihood of forgetting X; (3) cost of targeting CZ; or (4) willingness to pay for the product . If the firm can identify and target groups of consumers separately according to these characteristics, they can solve the sampling problem for each group separately, considering possible intergroup word-of-mouth overlap. For example, we would expect firms to initially target groups that are responsive to sampling efforts and building goodwill so that they can quickly build a customer base. In the long-run steady state, the firm would focus a greater portion of its sampling efforts on groups with greater forgetting potential (e.g., influence from competing brands or close substitutes) or with high impulse purchases (immediate sales responses). The firm may data mine or invest in strategies to learn the characteristics of groups most likely to purchase their product (see Bult and Wansbeek 1995 for 542
developing sampling methods to identify the optimal selection of target groups in the context of direct mail campaigns). Alternatively, or additionally, the firm may introduce self-revealing mechanisms to identify individuals with strong purchase potential; for example, requiring interested customers to call a phone number to receive a free sample. If customers are differentiated continuously, the firm cannot create discretely segregated groups for the analysis. In this case, the model presented in this paper becomes more complex because the firm will need to solve the problem using double integrals that reflect the proportion (density) of customers at each sampling response level. We expect that the firm would target those with high response and goodwill-building potential first, and in the steady-state equilibrium target greater ongoing sampling to individuals with higher forgetting and/or impulse buying rates. We leave this analysis to future research.
4.
Concluding Remarks, Managerial Applications, and Future Research
Sampling differs from advertising in that it has a strong effect on immediate sales. We therefore developed a model that decomposes sampling into two distinct effects: its immediate effect on sales and its long-term effect on goodwill formation (learning). This decomposition allows us to better understand the sampling decision and to examine the dynamics of optimal product sampling. This approach implies that even if potential consumers who sample do not immediately purchase the product, they can be influenced by sampling in a manner that affects their longrun behavior. The primary findings are: 1. Although sampling efforts will decline over the life-cycle of a product, it may continue for mature products. Firms can sample in the steady state to increase profits from immediate sales and to counteract the effects of forgetting that reduce the goodwill level. As given in Equation (6), firms should invest in sampling until the marginal profits from immediate sales, plus the future value of marginal learning from Management Science/Vol. 47, No. 4, April 2001
HEIMAN, McWILLIAMS, SHEN, AND ZILBERMAN Learning and Forgetting: Optimal Product Sampling Over Time
sampling, equal the marginal cost of additional sampling. 2. Increasing either the goodwill-building effect () or the purchasing effect ( ) will increase the equilibrium goodwill (X ∗ ) and sales (Y ∗ ). However, their effect on the equilibrium sampling level (Z ∗ ) is positive when the direct effect (increasing the marginal productivity of sampling) due to a higher or more than offsets the indirect effect (reducing in the marginal productivity of sampling due to a higher X), and negative when the reverse is true. The need to address sampling for mature products is brought home by an advertising agency president’s statement that “traditional brands are losing market share to generics and increased competition, so sampling is not just for new products anymore” (Eisman 1993), and by empirical evidence that, on average, firms invest over 50% of their sampling budget on established product (Donnelly 1992). Ongoing sampling of mature products arises in our model for two reasons. First, when forgetting is significant, a firm will continue sampling to maintain product goodwill. Second, when there are strong immediate sales impacts (i.e., bX hX, we may observe continued sampling even if the forgetting effect is small. Depending on the characteristics of the sampling being employed, management can use sampling to target immediate sales or long-run (goodwillbuilding) goals. For example, Pepsi samples have been given out at General Cinema theaters and Dentyne gum samples have been provided at Cineplex Oden theaters (Eisman 1993). Both are well-known brands but face heavy competition, which implies high forgetting potential. Sampling here can promote immediate sales and, in the specific context, may help develop long-term patterns of consumption at movie theaters. When the rate of forgetting is high because of competing brands, close product substitutes, or a long time between purchases, sampling may be used primarily to increase immediate purchases. Sampling is an effective promotional tool in various environments. Extensive sampling of Cadbury Schweppes’s Diet Sunkist brand at stores and special events increased sales by 700% (Eisman 1993). Management Science/Vol. 47, No. 4, April 2001
American Online for several years has been providing diskettes and now CD-ROMs throughout the United States, offering consumers a one-month free trial of their services (Nakache 1998). Five years after a launching in 1987, Purina adopted a strategy of offering free trials to people who called their 800 number or mailed in business reply cards. In addition to allowing customers to test their products, this provided them a database to keep in touch and maintain relationships with their customers. Marketing segmentation can also lead to sampling of mature products. Procter & Gamble, RJR, Nabisco, and Quaker Oats use the services of Segmented Marketing Services, Inc. (SMSi) to target African-American and Hispanic consumers. SMSi puts the samples in a bag with illustrations of ethnic heroes and twice a year distributes 1.5 million samples through its Church Family Network of 7,000 African-American and Hispanic participating churches (Eisman 1993). Established companies also frequently provide free samples of brand name products to college students at the beginning of the academic year. Students tend to be receptive to free hand-outs, increasing the likelihood of building goodwill. We have argued that this model was designed primarily to fit the sales of products taste-tested at retail outlets, where 82% of people who sample products buy at least one as a result (Eisman 1993). However, if the sample is consumed away from the place of purchase, the immediate sales effect may be lost, and the model can be represented by traditional goodwill models. For example, if a toothpaste firm distributes free samples, these samples will not be consumed immediately and therefore will not affect immediate sales. However, after consuming the sample at home, the consumer’s preferences may change in favor of the sampled toothpaste, and long-term sales and profits can increase. Sampling has an element of investment. Sometimes short-term losses from sampling can be justified if a firm builds long-term goodwill. However, if the firm is unable to successfully build goodwill, sampling will not be profitable. Finally, we note that sampling may increase the probability of purchase but can harm goodwill 543
HEIMAN, McWILLIAMS, SHEN, AND ZILBERMAN Learning and Forgetting: Optimal Product Sampling Over Time
formation if it creates over-expectation. For example, consumers may decide to purchase a product after sampling it when it is cooked professionally or served nicely, or simply because they are in a good mood. However, at home they may fail to recover the sampling outcome, become disappointed, and decide to not purchase the product again. In these cases, firms may want to increase the educational aspect of sampling, demonstrating how to cook and prepare the product. Alternatively, the firm may still want to sample the product if profits from immediate sales are positive and if the negative goodwill depreciates. A limitation of our model is that we assume pricetaking behavior that does not allow prices to vary. Clearly, sales depends on prices as well as sampling effort and goodwill. Nevertheless, the relative simplicity of the model allows us to focus on the role of immediate sales as well as learning and knowledge (goodwill) accumulation, and to examine the optimal dynamic evolution of product sampling. An interesting direction of research would be to expand the model by allowing for monopolistic competition so that prices can reflect changes in quantity and/or sampling effects. This model extends to cases where the market can be segmented into homogeneous groups that are solved separately. However, optimal sampling when consumers are continuously differentiated provides an area of future research. Furthermore, there are clearly trade-offs and complementarities between the use of sampling and advertising, and in the long run the firm can employ both. Further research could model the substitution and interrelationship between these two methods of product promotion. The conceptual framework developed here can provide the basis for an empirical investigation. Part of the challenge is to develop practical measures of goodwill, perhaps based on either sales or interviews with consumers as well as on quantitative measures of sampling. Within this empirical framework, it would be interesting to quantify both the immediate and long-run effects of sampling activities. Acknowledgments
The authors are very grateful to the reviewers and editors for their substantial insights and comments. We also benefited from participant discussions at presentations at Tel Aviv
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University and the Haas Business School at UC Berkeley. The authors gratefully acknowledge the support of BARD for research.
Appendix A
To prove that the isocline of X˙ = 0 is positively sloped, i.e., dZ > 0 dX X=0 ˙ take the partial derivative of Equation (3) with respect to X and Z around X˙ = 0: dX dZ + fZ = 0 fX dt dt where fX = LX − FX < 0 and fZ = LZ > 0, because LX < 0 FX > 0, and LZ > 0. Therefore, f dZ = − X > 0 dX fZ The proof that the isocline of Z˙ = 0 is negatively sloped, (i.e., Z/X Z=0 < 0, around the steady state (near X˙ = 0) is similar. Taking the partial derivatives of Equation (11) with respect to X and Z yields
gX = A − LXX +
+ X˙
ZX L + ZX LX − FX C Z − Z LZ
ZX X
CZ − Z + ZX 2 + CZ − Z 2
LZX X
LZ − LZX 2 LZ 2
XX LZ + X LZX CZ − Z + Z LZ ZX −A >0 CZ − Z 2 around X˙ = 0 because A = LZ X Z > 0. Furthermore,
CZ −Z CZZ
> 0 LZZ ZX LZX < 0, and
X LZ CZZ ˙ XZ CZZ gZ = A >0 − X CZ − Z 2 CZ − Z 2 around X˙ = 0 because CZZ X , and LZ > 0. Then, totally differentiating Z˙ = 0 yields gX
dX dZ + gZ = 0 dt dt
and g dZ = − X < 0 dX Z˙ gZ around X˙ = 0. If X˙ = 0 and X is large, dZ/dX of isocline Z˙ = 0 might change sign. When X → 1, LZ → 0, X/LZ2 will dominate the other terms in gX . Because X˙ < 0 when X → 1, gX < 0 when X → 1. On the other hand, as X → 1, gZ becomes positive since X˙ < 0 when X → 1. Therefore, as X → 1, dZ/dX become positive.
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Appendix B
To prove that the system of differential equations X˙ and Z˙ is unstable around the steady state and there exists a saddle path: Find a linear approximation to the nonlinear autonomous system around X˙ = 0 and Z˙ = 0 ˙ X X X f X fZ ≈ = H Z˙ Z Z gX gZ where fX fZ gX , and gZ are partial derivatives of the differential equations. According to the Stability Theory, the equations X˙ and Z˙ are asymptotically unstable if the Hessian matrix H has positive eigenvalues. Because the product of two eigenvalues is equal to H , which is negative, H = fX gZ − gX fZ < 0 one eigenvalue must be positive and the other must be negative. Therefore, the steady state is asymptotically unstable and there exists a saddle path towards the steady state. Next we prove that the saddle path can be positively sloped when X is close to 1, i.e., dZ/dX > 0 as X → 1. Using Equations (3) and (11), the slope of the saddle path is: dZ Z˙ = = dX X˙
CZ −Z CZZ
r+
ZX LZX X LZ X+ X− − LX − FX CZ − Z LZ C Z − Z LX Z − F X
Since X˙ < 0 and LX − FX < 0 as X → 1, four of the five terms in the numerators are positive: r ZX /CZ − Z X LZX /LZ X and −LX − FX . If X˙ becomes sufficiently large in absolute value as X → 1 so that the positive terms dominate the negative term −X LZ /CZ − Z , the slope of the saddle path would change sign from negative to positive.
Here we prove the comparative statics results for changes in . The proof for the remaining parameters in Table 1 are similar and may be requested from the authors. To prove: if the goodwill effect of samples increases (d > 0) and X is not large, then in a new steady state X ∗ will be higher, but the sampling level Z ∗ might be either higher or lower. Sales Y ∗ will be higher. Formally, dZ ∗ > 0 d ≤
and
dY ∗ > 0 d
Totally differentiating equations f X Z = 0 and gX Z = 0 yields:
X ∗
f X fZ = −f gX gZ Z ∗ −g
g =
g X Z
=
C Z − Z CZZ
L ZX f + ZX f CZ − Z LZ −
since
X LZ LX − CZ − Z
< 0
LX 1 − Xb Z = >0 X N
when X is not large, and 1 LZ = 1 − Xb > 0 N Applying Cramer’s Rule: X ∗ 1 = −f gZ + g fZ > 0 H 1 Z ∗ = −g fX + f gX H =
f g −g /gX + f /fX > 0 0, gX > 0, and gZ > 0, and H < 0. To prove that dY ∗ /d > 0, consider the current Hamiltonian H ∗ = ∗ + #∗ L∗ − F ∗ According to the equivalent of the “Envelope Theorem” in optimal control theory (Kamien and Schwartz 1991, p. 169), the derivative of the object with respect to is equal to its partial derivative with respect to , dH ∗ L∗ − F ∗ L∗ − F ∗ = + #∗ = #∗ >0 d Since ∗ / = 0. Therefore, signdY ∗ /d = signdH ∗ /d > 0.
Appendix C
dX ∗ > 0 d
and
where f = f X Z/ = 1 − XbZ/N > 0
Management Science/Vol. 47, No. 4, April 2001
Appendix D
To prove that when the slope of the X˙ = 0 isocline is relatively flat (steep), dZ ∗ /d is more likely to be negative (positive). This can be easily proved utilizing the comparative statics equation in footnote 12: −g /gz fx /fz f /fz gx /gz dZ ∗ = + d fx /fz − gx /gz fx /fz − gx /gz Take the example where X˙ = 0 becomes flatter (steeper), which is equivalent to be bigger (smaller) fX /fZ (the slope of X˙ = 0 is −fX /fZ , where fX /fZ is negative). It is easy to show that a higher (lower) fX /fZ decreases (increases) both the first and second terms on the right hand side of the above equation. Therefore dZ ∗ /d is more likely to be negative (positive). The proof is similar for showing that when the slope of the Z˙ = 0 is relatively flat (steep), dZ ∗ /d is more likely to be positive (negative) .
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Accepted by Dipak Jain.
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