effects. We may consider a straight discount as a coupon without the clipping, dropping and processing costs, and a rebate as a coupon without the retailer processing cost.
Therefore, we have decided to model the dealing as the somewhat more complex and encompassing couponing practice. It must be recognized at the outset that only a few
features of the couponing phenomena are critical for the unified modeling of dealing with a limited number of coupons per customer. Yet, there is a long list of couponing variables
which typically appear in strictly empirical rather than analytical couponing studies. We refer to several of these studies in the next section.
Dealing with a limited number of coupons per customer is a form of multipart pricing which has received attention in the economics and business literature [see, for example,
Alchian and Allen 1977; Hirshleifer 1984]. Yet, the only analytical model of multipart pricing we are aware of is that of La Croix 1981, 1983. La Croix’s purpose was confined
to that of deriving welfare implications. He thus ignored several concerns which are critical when advocacy of managerial action is the purpose, but which are secondary when
deriving general welfare implications. Specifically, he [La Croix 1981, 1983] was not concerned with the role of coupons in inducing voluntary third-degree price discrimina-
tion [see Levedahl 1984, 1986 and Sweeney 1984]. He did not derive explicit optimal coupon values formulae, and his analysis of changes in the optimal coupon value as a
function of the shape of the market response function was limited.
The plan of our paper is as follows: In the next section, we review some relevant literature affecting our modeling approach. It is followed by Section III where we develop
the model of optimal discount couponing. In Section IV, we conclude with implications and directions for future research.
II. Selected Couponing Literature
Typically, dealing studies have emphasized the role of coupons in inducing voluntary third-degree price discrimination, based on factors such as differential inventory cost [see
Blattberg et al. 1981], differential coupon processing costs [Narasimhan 1984; Leve- dahl 1986], or both differential reservation prices and inventory costs [Jeuland and
Narasimhan 1985].
If a higher coupon value is directly related to the size of the order quantity, then, in the post-deal period, we should expect a lower probability of repurchase. The empirical
findings of several studies have supported this hypothesis and, in turn, the inventory- shifting hypothesis [Shoemaker and Shoaf 1977; Dodson et al. 1978; Jones and
Zufreyden 1981; Guadagni and Little 1983]. Other studies have emphasized differen- tial exposure to information [Varian 1980], differential shopping tendencies [White
1983], taste or quality differences [Narasimhan 1988], or brand loyalty through differential reservation prices [Raju et al. 1990]. Empirical support for the hypothesis of
negative correlation between brand loyalty and coupon usage has been given by Webster 1965, Montgomery 1971 and Bawa and Shoemaker 1987a. Several explanations
were offered as support for this observation. They range from the effect of couponing on reference pricing [Monroe 1973; Winer 1986; Shindler 1992], to cognitive disso-
nance [Doob et al. 1969], to self-perception theory [Dodson et al. 1978].
Bawa and Shoemaker 1987b and Krishna and Shoemaker 1992 further clarified the intricate relationships between brand loyalty and coupon usage. They found that custom-
ers who are brand loyal to a specific brand possess a higher probability of redeeming a
Optimal Face Value of a Discount Coupon
161
coupon for that brand than customers who are not loyal to that brand. Accordingly, suppose the segment of full-price, brand-loyal customers is sufficiently large, and the
degree of price discrimination is mild. The firm may not be able to rely on voluntary price discrimination induced by coupon distribution to bar a sufficient number of old customers
from entering the market, thus encountering a loss, unless it restricts the number of coupons per customer.
Following this traditional modeling emphasis, we extended the function set of the coupon beyond that of strict multipart pricing considered by La Croix, and allowed for
coupon inducement of price discrimination. Our approach thus emphasizes complemen- tarity rather than substitution between the role of the coupon in multipart pricing and its
role in price discrimination.
Although the results of our analysis can be quite general, for simplicity of exposition, the mathematical presentation has been simplified by restricting our analysis to the case
where the number of coupons per customer is limited to one. In practice, one can observe periodical general discounts for all customers, and cases where the discount is limited to
new customers only or to all customers who are restricted by the number of coupons per customer. In this paper, we analyze the last case.
Coupon Value Formulae
The prime decision variable of a marketer administering couponing policy is the optimal coupon value. Hence, the derivation of an explicit expression for the coupon value as a
function of other parameters is highly desirable. Most previous couponing models have provided only implicit expressions for the optimal coupon value, and their analysis of the
change of the value is limited. The only exception we are aware of is that of Blattberg et al. 1981, who also derived an explicit solution for the optimal coupon value, but only for
the case where the new customer share is insensitive to the coupon value. In addition, Jeuland and Narasimhan 1985 also provided an approximation of the optimal deal value.
In our study, we have derived implicit solutions for the coupon value, thus providing the marketer with a clearer instrument for evaluating the couponing policy. Our approach is
shared with Shaffer and Zhang 1995, who also explicitly calculated coupon values and performed some comparative statics in a competitive spatial framework with a linear
market response function to coupon value. We have extended their analysis to some other questions, such as the question of how the coupon value is sensitive to the distribution of
customers.
1
Aggregate Demand and Market Composition
Most dealing studies have not formally restricted the number of coupons per customer in any manner; studies by La Croix 1981, 1983 being the exception. Hence, the traditional
objects of analysis are the aggregate demands of the different segments.
1
We will inquire into the relationship between the shape of the market response functions and the optimal coupon value. We thus will admit any general market response function that is monotonic in the coupon value,
derive some explicit coupon solutions for non-linear cases and demonstrate that the shape of the market response function carries immediate and critical consequences for the feasibility, value, and sensitivity of the optimal
coupon value. [Ben-Zion et al. 1990, 1992, 1993].
162
U. Ben-Zion et al.
Generally, previous analytical normative models have assumed that each segment is comprised of consumers with identical demand of some particular shape. This assumption
has been only somewhat relaxed in competitive models [Narasimhan 1988; Raju et al. 1990]. However, these competitive models have assumed the consumer demand at the
product category level to be perfectly inelastic.
A study by Levedahl 1986 is an exceptional normative study, which did not restrict parametrically the demand shape of the individual consumer. Rather, he studied the effect
of a general aggregate demand of the deal-prone segment on the optimal coupon value. However, Levedahl 1986 assumed no restriction on the number of coupons per customer
and did not deal with the customer composition of the aggregate demand.
In contrast with most prior models, we used a different approach. Our study allowed for a continuous distribution of reservation prices in the deal-prone segment. This allowed
us to endogenize the number of deal-prone customers as a function of the coupon’s discount value.
As is common, we assumed that a consumer whose reservation price is above the discounted price purchases the product. Thus, the cumulative number of consumers in the
deal-prone segment whose reservation price is above the discounted price is defined by us as the market response function. It is a function of the coupon value.
Our modeling focuses on the market response rather than on individual consumer motivation
for coupon usage, and follows La Croix 1981 and the econometric models of Ward and Davis 1978a and Neslin 1990. The shape of the market response function
critically affects couponing feasibility and coupon value. A prime aim of this paper is to study this effect.
III. The Model
