Multiproduct Pricing and Product Line Decisions with Status Externalities

III. The Effect of Changes in Externalities on Pricing

This 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, β1 = β2 = 0, and neither the positive spillover β2 q1 nor the negative spillover β1 q2 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 β1 and maximize the magnitude of the positive externality β2.

We perform comparative statics using the implicit function theorem. The equilibrium prices p1 and p2 are implicitly defined as functions of β1 and β2 in the two equations that are yielded through the first order conditions.

We restrict F1 and F2 to the level set where F1,F2 = 0, values of p1, p2, β1, and β2 such that F1, F2 = 0, and denote the restriction Fr1 and Fr2. The partial derivatives of Fr1 and Fr2 with respect to p1, p2, β1, and β2, are equal to the partial derivatives of F1 and F2 with respect p1, p2, β1, and β2, 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 β1, β2. 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, ∂Fr1/∂p2 = ∂Fr2/∂p1. 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 β1, β2 > 0, but the derivatives are evaluated at β1 = 0. Hence, we evaluate the impact of the externalities at the point where β1 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 β1,β2 > 0, but the derivatives are evaluated at β2 = 0. Hence, we evaluate the impact of the externalities at the point where β2 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, β1, β2 > 0, but the derivatives are evaluated at β1 = β2 = 0. Hence we evaluate the impact of the externalities at the point where both β1 and β2 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 β1 = β2 = 0, we get:

Therefore, ∂p1/∂β1 < 0, ∂p1/∂β2 < 0, ∂p2/∂β1 > 0, and ∂p2/∂β2 > 0. All else equal, an increase in either β1 or β2 is associated with a decrease in Fr1 and an increase in Fr2. We have shown that in the neighborhood of β1 = β2 = 0, the restoration of equilibrium necessitates a decrease in p1 and an increase in p2. An increase in β1 and β2, starting at β1 = β2 = 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 β1 implies, in equilibrium, a decrease in p1 and an increase in p2. All else equal, an increase in β1 implies that the first market is hurt more by sales of product 2. Therefore, to restore equilibrium, an increase in β1 is associated with an increase in p2, since this leads to a decrease in D*2p2 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 p2 shifts up demand for market 1, the demand in market 1 is nevertheless lower than before the increase in β1 and therefore, to restore an optimum, p1 is decreased.

We explain the intuition behind why an increase in β2 implies, in equilibrium, a decrease in p1 and an increase in p2. All else equal, an increase in β2 implies that market 2 benefits more by sales of product 1. To restore equilibrium, an increase in β2 is associated with a decrease in p1, since this leads to an increase in D*1p1 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 p2.

IV. Discussion and Conclusions

This 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 p1 increases the brand is associated with even more status, whereas if p2 decreases the brand status is even further diminished. Below is a possible model, where if p1 = p2 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.

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Endnotes

1. I would like to thank Professor James D. Dana, Jr., Northeastern University, for all his support. His ideas and advice were most helpful and inspiring. I am very grateful for the privilege I had to learn so much from him. I also would like to thank Christina M. Kompson and Professor G¨unther K.H. Zupanc for their helpful comments on the manuscript.