A Choice Model for Packaged Goods: Dealing with Discrete Quantities and Quantity Discounts

Solution processes of two groups of 9‐10 year old children to a common set of fraction problems were contrasted at the conclusion of a teaching experiment. One group was taught meanings for fractions using popsticks as discrete quantity material; the other group used paper strips as continuous quantity material. The study suggests that a knowledge of fractions based upon discrete quantities may have certain advantages over fraction knowledge based on continuous quantities. The Continuous group children were less successful with problems cast in discrete quantity settings. They also displayed unstable strategies when using continuous material. Children of both groups preferred discrete material to solve problems cast in either discrete or continuous contexts. Discrete quantity materials could have a valuable role in the development of fraction knowledge in the elementary school.

The lack of conceptual analysis within cognitive science results in multiple models of the same phenomena. However, these models incorporate assumptions that contradict basic structural features of the domain they are describing. This is particularly true about the domain of mathematical cognition. In this paper we argue that foundational theoretic aspects of psychological models for language and arithmetic should be clarified before postulating such models. We propose a means to clarify these foundational concepts by analyzing the distinctions between metric and linguistic compositionality, which we use to assess current models of mathematical cognition. Our proposal is consistent with the scientific methodology that determines that careful conceptual analysis should precede theoretical descriptions of data. Scientific theories must not only be true, but also coherent and systematic. Part of their being coherent and systematic rests on the conceptual clarity of the most fundamental terms that these theories explain. This means that scientific theories are not only determined by experimentation and quantification, but also by conceptual analysis. Actually, it is conceptual clarity that distinguishes a set of random facts from a systematic scientific explanation. Cognitive psychology has produced an impressive corpus of data. However, few attempts are made to clarify the meaning of the concepts that are being used in interpreting these data, whether the use of concepts in different theories is consistent and how these concepts relate to one another. One of the central theoretical issues in psychology is the nature of measurement in psychological experiments and its relation to cognitive capacities. Some of the most influential psychologists have addressed this issue directly by defining mapping relations between mental content and units of measurement. For instance, Stevens (1951) proposed different scales (ex: nominal, ordinal, interval, ratio) and distinguished them in terms of their formal properties. A fascinating aspect of the \\server05\productn\T\THE\27-1\THE107.txt unknown Seq: 2 4-JAN-08 8:01 54 Journal of Theoretical and Philosophical Psy. Vol. 27, No. 1, 2007 topic of measurement in psychology is that certain cognitive computations appear to preserve metric and scale structure, while others remain qualitative and are not metrically structured. Recently, mathematical, temporal and spatial cognition gained interest in different areas of psychology, and it is certainly a fundamental concern in cognitive science. However, the models that have been proposed to account for mathematical cognition and the representation of quantities reveal serious conceptual confusion, a situation that could be alleviated by revisiting the fundamental assumptions of the theory of measurement. For instance, these models differ with respect to their approach to the acquisition of numerical concepts and the representational structure underlying discrete quantities. According to one model, the content of discrete quantities is determined by linguistic representations in the course of psycholinguistic development (Carey, 2004). However, other models favor a visual-attentional mechanism that automatically picks out the discrete quantities provided that there is a limit to the size of the perceived set, independently of linguistic representations (Xu & Spelke, 2000). These two models assume that the representation of small and large quantities has different formats and are processed differently in the mind. This assumption has specific predictions concerning the psychophysical properties of discrete quantity representations and it is another dimension in which these models of mathematical cognition differ. Another model, which differs from the visual-attentional and psycholinguistic accounts, assumes a common metric underlying the numerical representations of all sizes. This model assumes that all numbers are represented in the same format (on the same mental continuum) and differs from the previous models with respect to the development of mathematical cognition. For instance, according to the ‘common metric’ account, the innate structure of magnitude based representations (Gallistel and Gelman, 1992) allows cognitive agents to represent the numerical attributes of the physical world with no functional dependence on their linguistic or visual-perceptual capacities. As a consequence of the linguistic independence postulated by this account, the ‘common metric’ model generalizes to mathematical cognition processes in human and non-human animals. But even within the models that propose a common metric, there are divergences concerning the type of mapping between the physical and mental quantities. Some theorists (Dehaene, 2001) assume a logarithmic mapping with constant variability, while others (Gallistel & Gelman, 1992) assume a linear mapping with scalar variability. These views also have different predictions about the psychophysical properties of the discrete quantities. In this paper, we evaluated the assumptions of these models by critically assessing their implications with respect to the theory of measure\\server05\productn\T\THE\27-1\THE107.txt unknown Seq: 3 4-JAN-08 8:01

Nonlinear Pricing of Software with Local Demand Inelasticity

The human intraparietal sulcus (IPS) is implicated in processing symbolic number information and possibly in nonsymbolic number information. Specific IPS activity for discrete quantities (numerosities) as compared with continuous, analogue quantity has not been demonstrated. Here we use a stimulus-driven paradigm to distinguish automatic estimation of "how many things" from "how much" and "how long." The discrete analogue response task (DART) uses the perception of hues which can change either abruptly (discrete, numerous stimuli) or smoothly (analogue, nonnumerous stimuli) in space or in time. Subjects decide whether they saw more green or more blue. A conjunction analysis of spatial and temporal conditions revealed that bilateral IPS was significantly more active during the processing of discrete stimuli than during analogue stimuli, as was a parietal-occipital transition zone. We suggest that processing numerosity is a distinct process from processing analogue quantity, whether extended in space or time, and that an intraparietal network connects objects' segmentation to the estimation of their numerosity.

Although price promotions have increased in both commercial use and quantity of academic research over the last decade, most of the attention has been focused on their effects on brand choice and brand sales. By contrast, little is known about the conditions under which price promotions expand short-run and long-run category demand, even though the benefits of category expansion can be substantial to manufacturers and retailers alikeThis chapter studies the category-demand effects of consumer price promotions across 560 consumer product categories over a 4-year period. The data describe national sales in Dutch supermarkets and cover virtually the entire marketing mix, i.e., prices, promotions, advertising, distribution, and new-product activity. We focus on the estimation of main effects (i.e., the dynamic category expansive impact of price promotions) as well as the moderating effects of marketing intensity and competition (both conduct and structure) on short-and long-run promotional effectiveness.The research design uses modern multivariate time-series analysis to disentangle short-run and long-run effects. First, we conduct a series of unit-root tests to determine whether or not category demand is stationary or evolving over time. The results are incorporated in the specification of vector-autoregressive models with exogenous variables (VARX models). The impulse-response functions derived from these VARX models provide estimates of the short- and long-term effects of price promotions on category demand. These estimates, in turn, are used as dependent variables in a series of second-stage regressions that assess the explanatory power of marketing intensity and competition. Several model validation tests support the robustness of the empirical findings.We present our results in the form of empirical generalizations on the main effects of price promotions on category demand in the short and the long run and through statistical tests on how these effects change with marketing intensity and competition. The findings generate an overall picture of the power and limitations of consumer price promotions in expanding category demand, as follows.Category demand is found to be predominantly stationary, either around a fixed mean or a deterministic trend. Although the total net short-term effects of price promotions are generally strong, with an average elasticity of 2.21 and a more conservative median elasticity of 1.75, they rarely exhibit persistent effects. Instead, the effects dissipate over a time period lasting approximately 10 weeks on average, and their long-term impact is essentially zero. By contrast, the successful introduction of new products into a category is more frequently associated with a permanent category-demand increase.Several moderating effects on price-promotion effectiveness exist. More frequent promotions increase their effectiveness, but only in the short run. The use of nonprice advertising reduces the category-demand effects of price promotions, both in the short run and in the long run. Competitive structure matters as well: The less oligopolistic the category, the smaller the short-run effectiveness of price promotions. At the same time, we find that the dominant form of competitive reaction, either in price promotion or in advertising, is no reaction. Short-run category-demand effectiveness of price promotions is lower in categories experiencing major new-product introductions. Finally, both the short- and long-run price promotion effectiveness is higher in perishable product categories.The paper discusses several managerial implications of these empirical findings and suggests various avenues for future research. Overall, we conclude that the power of price promotions lies primarily in the preservation of the status quo in the category.

By employing order-theoretic comparative statics, we provide necessary and sufficient conditions on the direct utility function (alternatively, on the indirect utility function) to guarantee that the compensating and equivalent variations are monotonically increasing in income. We also show that these conditions are sufficient to guarantee normal demand. Our results do not depend on the smoothness or strict quasiconcavity of the utility function, and can be extended to a setting with non-linear prices.

As consumers are often exposed to price-related advertising in purchase situations and as, among other factors, both prices (Chang and Wildt 1994) and product attitudes (Spears and Singh 2004) affect purchase decisions, it is important to examine possible negative effects of price promotions on product attitudes. Some previous studies provide evidence for positive effects of price promotions on purchase intentions (Ailawadi and Neslin 1998; Grewal et al. 1998), but other studies show that discounts can negatively affect future sales (Kalwani et al. 1990; Lattin and Bucklin 1989). A possible explanation for such opposite findings might be that sales promotions have direct positive effects on purchase intentions but indirect negative effects through more negative product attitudes on purchase intentions (Davis et al. 1992). The objective of this study is to focus on effects of price promotions on product attitudes and to examine in which conditions such effects are more or less negative. This study goes beyond previous studies that analyzed possible negative effects of price promotions (Dodson et al. 1978; Raghubir and Corfman 1999) by considering conditions in which price-related advertising can have more or less negative effects on product attitudes. The conditions are the number of consumers’ ad contacts (1 vs. 5), the saving format used to display the price reduction (percentage off vs. amount off), and the salience of the regular price in the ad (yes vs. no) as well as the perceived levels of discount information credibility (low vs. high). We simultaneously analyze direct and interaction effects of marketer-controlled factors and consumers’ perceptions that play a role in the context of consumers’ responses to price promotions in terms of product attitudes. An analysis of effects of marketer-controlled factors helps marketers to choose the combination of factors that produce the most positive consumer responses. The effects of price promotions on consumers’ attitudes could be moderated by as how credible consumers perceive the price claims displayed in the ad. Perceived ad credibility is the extent to which consumers perceive the claims as believable (MacKenzie and Lutz 1989). When consumers perceive a discrepancy with prior perceptions, they might perceive price claims as less credible (Lutz et al. 1983). The findings provide marketers with insights in which conditions a sales promotion leads to comparatively negative product attitudes and thus enables them to avoid such constellations when launching a sales promotion.

Retailers' marketing objectives can be classified into three broad categories: attraction effects that focus on consumers' store-entry decisions, conversion effects that relate to consumers' decisions about whether or not to make a purchase at a store they are visiting, and spending effects that represent both dollar value and composition of their transactions. This paper proposes a framework that incorporates all three of these effect categories and examines their influence on store performance. Specifically, store sales are broken down into four components: front traffic, store-entry ratio, closing ratio, and average spending. Using inexpensive and readily available infrared and video imaging technology, it is possible to measure these four components in a wide variety of retail environments, allowing retailers to obtain a richer understanding of the effectiveness of promotional activities on store sales. A set of twelve hypotheses based on the economics of information and promotion literatures is proposed. These hypotheses relate the presence of various promotions (price, clearance, and new product), promotion scope, and the type of out-of-store communication vehicle used by retailers to each of the four store sales components. The proposed approach is then applied in two different empirical settings, both to test formally the hypotheses and to demonstrate more generally the richness of the information the approach can provide. The first application involves a Canadian apparel store that sells ladies' casual wear. The second application is based on a U.S. sporting-goods retail chain that sells a variety of sporting goods, including sportswear, sports shoes, and sports equipment. A joint model of four simultaneous equations using front traffic, store traffic, number of store transactions, and store sales as the endogenous variables is then formulated for the applications. Promotional factors are used as explanatory variables, along with a number of additional control variables (including length of operation, day of week, holidays, seasonality, and weather). Seemingly unrelated regression is used to estimate the model efficiently. A comparison model that includes only store sales as the endogenous variable is estimated for comparison with the joint model. Results from these applications indicate that the proposed framework provides more detailed information about promotional effectiveness than more traditional models of store performance. The effects of specific promotional decisions on store performance are described. Specifically, price promotions have little impact on front traffic, but positively affect store entry and likelihood that a consumer will make a purchase. The effect of price promotion on consumers' spending in a store is also significant, but varies in sign with the type of promotion employed. Second, while greater promotional scope enhances store entry, promotions with narrow scope seem to have negative impact on store traffic. The effects of promotion scope on store performance also seem to be moderated by the scope of merchandise carried by the retailer. Increased promotional scope appears to have a greater effect on store traffic and consumers' spending for a multicategory retailer than for a more focused seller. Third, clearance promotions have a weaker effect on store entry when compared to other multiple-category promotions, while new-product promotions have a positive impact on conversion. Finally, newspaper advertisements, when compared to targeted coupons, have a stronger effect on store attraction but a weaker effect on spending. In addition to understanding the key drivers of store sales, retailers are also interested in determining whether or not their promotions affect store profitability. An assessment of the profit impact cannot be based on the change in overall store sales because promotions may affect various items or product categories inside a store differentially, and gross margins may not be the same for all items or categories. Although gross margin and item- or category-specific sales were not available for the two applications studied, the paper describes how such information can be integrated with the output of the proposed joint model to arrive at a richer understanding of how promotions affect overall store profitability. Finally, managerial and academic implications of this work are described, and potential extensions of the joint model are suggested.

Résumé Cet article analyse la tarification non linéaire optimale pour un monopole offrant un package comprenant un bien et un service attaché au bien, lorsque ce dernier augmente l’utilité marginale que les consommateurs accordent au bien. Nous examinons les mécanismes d’incitation de révélation des préférences dans le cas où l’information privée est représentée par un paramètre unidimensionnel. Sous l’hypothèse de coûts constants nous retrouvons le résultat standard où il est profitable pour les consommateurs d’acheter le bien et le service attaché sous forme de package plutôt que séparément. Cependant, avec des coûts croissants, lorsque le degré de complémentarité entre le bien et le service est suffisamment élevé, le prix marginal du bien séparé est plus faible que comme élément du package. Contrairement à Martimort (1992), lorsque le bien et le service sont parfaitement complémentaires, nous ne pouvons pas conclure qu’il est plus coûteux pour les consommateurs de signer deux contrats aux différents magasins que d’acquérir le package. Du fait des asymétries prises en compte dans la fonction d’utilité, la profitabilité d’une stratégie de ventes liées dépend, d’une part du degré de complémentarité entre le bien et le service attaché, et d’autre part du degré d’optionalité du service lié. Classification JEL : D42, L12, Q4.

Using a Hicksian approach to cost-benefit analysis in discrete choice: An empirical analysis of a transportation corridor simulation model

Nonlinear Pricing with Product Customization in Mobile Service Industry

We consider a nonlinear pricing problem faced by a dominant firm competing with a minor firm. The dominant firm offers a general tariff first, and then the minor firm responds with a per-unit price, followed by a buyer choosing her purchases. By developing a mechanism-design approach to solve the subgame perfect equilibrium, we characterize the dominant firm’s optimal nonlinear tariff, which exhibits convexity and yet can display quantity discounts. In equilibrium the dominant firm uses a continuum of unchosen offers to constrain its rival’s potential deviations and extract more surplus from the buyer. Antitrust implications are also discussed. (JEL D21, D43, D82, K21, L13, L42)

The format of pricing contracts varies substantially across business contexts, a major variable being whether a contract imposes a fixed fee payment. This paper examines how the use of the fixed fee in pricing contracts affects market outcomes of a manufacturer-retailer channel. Standard economic theories predict that channel efficiency increases with the introduction of the fixed fee and is invariant to its framing. We conduct a laboratory experiment to test these predictions. Surprisingly, the introduction of the fixed fee fails to increase channel efficiency. Moreover, the framing of the fixed fee does make a difference: an opaque frame as quantity discounts achieves higher channel efficiency than a salient frame as a two-part tariff, although these two contractual formats are theoretically equivalent.To account for these anomalies, we generalize the standard economic model by allowing the retailer's utilities to be reference dependent so that the up-front fixed fee payment is perceived as a loss and the subsequent retail profits as a gain. We embed this reference-dependent utility function in a quantal response equilibrium framework where the retailer is allowed to make decision mistakes due to computational complexity. The key prediction of this behavioral model is that channel efficiency decreases with loss aversion for sufficiently Nash-rational retailers. Consistent with this prediction, the estimated loss-aversion coefficient is 1.37 in the two-part tariff condition, significantly higher than 1.27 in the quantity discount condition. At the same time, loss aversion dominates contract complexity in explaining the data. Lastly, we conduct a follow-up experiment to confirm the central role of loss aversion as a behavioral driver. In one condition, the retailer becomes less loss averse when we temporally compress the fixed fee payment and the realization of retail profits, which supports the loss aversion theory. In the other condition, the retailer's contract acceptance rate does not decline when we reward the manufacturer a higher cash payment for each experimental point earned, which rules out the competing hypothesis that the retailer rejects contract offers due to fairness concerns.

The purpose of this article is to determine the optimal nonlinear pricingrules for water services. The model presents a standard water distributionsystem and a separate wastewater collection and treatment system. Technicaland financial constraints are considered. The optimization process of awelfare utility function subject to different kind of constraints providesthe optimal pricing rules for drinking water and sewerage services. Theseprices reflect efficiently the costs of systems constraints, the cyclicityof demands, the time-of-use, heterogeneity types of consumers and the realvalue of water resources scarcity. They provide important guidance in thedesign of efficient water rates and in the development of different waterutilities planning strategies. New technologies in water metering(telemetry) should permit applying this model.

Promotions are used in marketing to increase sales and drive profits by temporarily decreasing the price per unit of a good. Some price promotions apply to all quantities (20% off), some have limits on the number of units that can be purchased at a reduced price, and others only offer the discount if the volume purchased is sufficiently high. We develop a model of price promotions in the context of a direct utility model where its effects are incorporated through the budget constraint. Price promotions complicate the estimation and analysis of direct utility models because they induce kinks and points of discontinuity in the budget set. We propose a Bayesian approach to addressing these irregularities and demonstrate the ability of the direct utility model to be used in counterfactual analyses of price promotions. We investigate the stability of utility function estimates for consumers under alternative price promotions, and find that the majority of the effect of a price promotion is through the budget set, not through changes in the utility function. We also investigate the economic value of customized price promotions where the customization includes the value and format of the offer. Data, as supplemental material, are available at http://dx.doi.org/10.1287/mksc.2015.0948 .