From The Developing Economist VOL. 2 NO. 1Multiproduct Pricing and Product Line Decisions with Status ExternalitiesIII. The Effect of Changes in Externalities on PricingThis section examines how prices change due to changes in the magnitude of the spillover parameters. However, we first explore possible reasons for the existence of and changes in the externalities. We implicitly assume that the externalities exist due to a link between the low-end and the high-end products, as well as a link between the two groups of consumers. Due to the links consumers associate the low-end product with the high-end product. Therefore, a change in the sales for the one product will result in a change in the demand of the other product. An increase in the links is associated with an increase in the size of the externalities. The links are based on the interaction of the two consumer groups during which the product, which jointly branded with the product that can the purchased by the other consumer group, is displayed. Marketing can be used to alter the intensity of the link. We assume implicitly that the link between the two groups of customers is affected by the extent to which the two groups of customers have information about each other's purchases. This depends on how frequently the two groups interact, since through interaction the purchase decisions are exhibited. The interaction is not limited to direct in-person interaction, but can occur indirectly via various media. The frequency of the interaction may be limited due to geographic or social barriers. In addition to the frequency of the interactions between the two groups, we consider the extent to which the products are displayed throughout these interactions. Therefore, an additional factor that has an effect on the magnitude of the externalities is whether the products are consumed privately or publically. Products that are frequently displayed in public, such as accessories or smartphones, are associated with larger status externalities in absolute value than products that are usually consumed in private, such as furniture. Furthermore, a link is established between the low-end and high-end products if the firm sells the two products under the same brand. If the products have recognizable similar features, such as name, logo, and design, the products will have a shared identity and therefore influence each other's reputation. In the case of status externalities, if the brand is associated with exclusiveness, a lower priced product may diminish the exclusivity and possibly decrease the value of the brand, since high-end exclusive customers want to dissociate themselves from the other group. If the firm brands the products separately, we assume that the link does not exist, β_{1} = β_{2} = 0, and neither the positive spillover β_{2} q_{1} nor the negative spillover β_{1} q_{2} would be experienced. The firm's decision to brand the products jointly or separately depends on the relative importance of the markets. The firm could also have sub-brands or luxury and regular product lines. Furthermore, marketing can communicate to consumers the extent to which the products are similar or dissimilar, and therefore alter the link. The firm can, for example, advertise the exclusiveness of the high-end product. The firm can also advertise the products together and underline their similarities. In addition, high volumes of advertisement and prominent branding are associated with larger externalities because this results in the brand being more known and recognizable by the public. The firm may consider utilizing advertisement to minimize the magnitude of the negative externality β_{1} and maximize the magnitude of the positive externality β_{2}. We perform comparative statics using the implicit function theorem. The equilibrium prices p_{1} and p_{2} are implicitly defined as functions of β_{1} and β_{2} in the two equations that are yielded through the first order conditions. We restrict F_{1} and F_{2} to the level set where F_{1},F_{2} = 0, values of p_{1}, p_{2}, β_{1}, and β_{2} such that F_{1}, F_{2} = 0, and denote the restriction F^{r}_{1} and F^{r}_{2}. The partial derivatives of F^{r}_{1} and F^{r}_{2} with respect to p_{1}, p_{2}, β_{1}, and β_{2}, are equal to the partial derivatives of F_{1} and F_{2} with respect p_{1}, p_{2}, β_{1}, and β_{2}, however, the manipulation is simplified. We want to determine the sign of each component of the matrix of partial derivatives of price with respect to β_{1}, β_{2}. To do so we calculate an expression for each component. Based on our current assumptions, the signs of two of the below partial derivatives cannot be determined. In each of the entries of the matrix below the sign of one of the partial derivatives cannot be determined. The two partial derivatives whose sign cannot the determined are equal, ∂F^{r}_{1}/∂p_{2} = ∂F^{r}_{2}/∂p_{1}. Therefore, their product is either zero or positive. However, if their product is a nonzero positive we cannot sign the determinant. To build intuition we will examine to the case where β_{1}, β_{2} > 0, but the derivatives are evaluated at β_{1} = 0. Hence, we evaluate the impact of the externalities at the point where β_{1} starts with no impact. Given this assumption, the signs of the other partial derivatives remain unchanged and we sign: However, under this assumption, we are still unable to sign the determinant. We now examine the case where β_{1},β_{2} > 0, but the derivatives are evaluated at β_{2} = 0. Hence, we evaluate the impact of the externalities at the point where β_{2} starts with no impact. Given this assumption the signs of the other partial derivatives remain unchanged and we sign: However, under this assumption, we are still unable to sign the determinant. Given the previous two assumptions separately, we could not sign the determinant. Hence, we explore the case where, β_{1}, β_{2} > 0, but the derivatives are evaluated at β_{1} = β_{2} = 0. Hence we evaluate the impact of the externalities at the point where both β_{1} and β_{2} start with no impact. Given these assumptions, the signs of the other partial derivatives remain unchanged and we sign: We derived earlier that: Given the assumption where the partial derivatives are evaluated at β_{1} = β_{2} = 0, we get: Therefore, ∂p_{1}/∂β_{1} < 0, ∂p_{1}/∂β_{2} < 0, ∂p_{2}/∂β_{1} > 0, and ∂p_{2}/∂β_{2} > 0. All else equal, an increase in either β_{1} or β_{2} is associated with a decrease in F^{r}_{1} and an increase in F^{r}_{2}. We have shown that in the neighborhood of β_{1} = β_{2} = 0, the restoration of equilibrium necessitates a decrease in p1 and an increase in p_{2}. An increase in β_{1} and β_{2}, starting at β_{1} = β_{2} = 0, is representative of moving from the case in which the firm sells two products with different brands to the case in which the firm sells two products under the same brand. Therefore, jointly branding products that were previously sold with different brands is associated with a decrease in the price for the high-end product and an increase in the price for the low-end product. We first explain the intuition behind why an increase in β_{1} implies, in equilibrium, a decrease in p_{1} and an increase in p_{2}. All else equal, an increase in β_{1} implies that the first market is hurt more by sales of product 2. Therefore, to restore equilibrium, an increase in β_{1} is associated with an increase in p_{2}, since this leads to a decrease in D*_{2}p_{2} so that less of product 2 is sold, so that demand for product 1 remains high and the damage is dampened in market 1, at the loss of less profit in market 2. Whilst an increase in p_{2} shifts up demand for market 1, the demand in market 1 is nevertheless lower than before the increase in β_{1} and therefore, to restore an optimum, p_{1} is decreased. We explain the intuition behind why an increase in β_{2} implies, in equilibrium, a decrease in p_{1} and an increase in p_{2}. All else equal, an increase in β_{2} implies that market 2 benefits more by sales of product 1. To restore equilibrium, an increase in β_{2} is associated with a decrease in p_{1}, since this leads to an increase in D*_{1}p_{1} so that more of product 1 is sold, so that demand for product 2 is further increased and profit in market 2 is further increased, at the loss of less profit in market 1. Given our model, an increase in demand for product 2 implies an increase in p_{2}. IV. Discussion and ConclusionsThis paper investigates a multiproduct monopolist's product line and pricing decisions of two differentiated status products, under the explicit assumption of two externalities. Specifically, whilst the sales of the high-end product positively affect the demand for the low-end product, the sales of the low-end product negatively affect the demand for the high-end product. We find that jointly branding products, which were previously sold with different brands, is associated with a decrease in the price for the high-end product and an increase in the price for the low-end product. Whilst it necessitates empirical tests of the model to investigate its value of representing observable reality, we will outline ways in which the model and analysis can be improved and extended. First, the model could be improved by making the assumptions explicit and deriving the demand functions from assumptions on preferences. Demand could be derived as a function of the consumer's wealth, the quality of the product, and the status of the brand, which could be the average wealth of the consumer who purchases from the brand. In addition to making the status externalities explicit, an improvement would be not assuming that markets are completely segmented and allowing spillage, which allows a low-priced product to cannibalize the high-priced product. This would better depict reality, where some wealthy consumers purchase low-end products and some non-wealthy consumers purchase high-end products. Modeling relative price differences is an improvement of the model that does not involve assumptions about quality. The intuition is that if p_{1} increases the brand is associated with even more status, whereas if p_{2} decreases the brand status is even further diminished. Below is a possible model, where if p_{1} = p_{2} the externalities do not exist. Furthermore, additional externalities, such as an advertising effect (Qian, 2011) or network externalities that affect the products own demand, could make the model represent reality more accurately. The model could also be generalized to an array of products that vary in quality and an array of market segments that vary in size. A further consideration is to model competition, where firms react to each other's price changes. The interaction of costs could also be modeled. Further analysis might also consider maximization of total welfare. References
Endnotes
Suggested Reading from Inquiries JournalInquiries Journal provides undergraduate and graduate students around the world a platform for the wide dissemination of academic work over a range of core disciplines. Representing the work of students from hundreds of institutions around the globe, Inquiries Journal's large database of academic articles is completely free. Learn more | Blog | Submit Latest in Economics |