Optimal Bundle Pricing

Hanson, Ward; Martin, Richard R.

doi:10.1287/mnsc.36.2.155

Cited by 264 publications

(123 citation statements)

References 17 publications

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“…McAdams (1997) found that the existing analytical machinery for analyzing mixed bundling could not be easily generalized to even three goods, because of the interactions among sub-bundles. In general, price-setting for mixed bundling of many goods is an NP-complete problem, requiring the seller to determine a number of prices and quantities that grows exponentially as the size of the bundle increases (Hanson and Martin 1990). …”

Section: The Bundling Literaturementioning

confidence: 99%

Bundling Information Goods: Pricing, Profits, and Efficiency

1999

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W e study the strategy of bundling a large number of information goods, such as those increasingly available on the Internet, and selling them for a fixed price. We analyze the optimal bundling strategies for a multiproduct monopolist, and we find that bundling very large numbers of unrelated information goods can be surprisingly profitable. The reason is that the law of large numbers makes it much easier to predict consumers' valuations for a bundle of goods than their valuations for the individual goods when sold separately. As a result, this "predictive value of bundling" makes it possible to achieve greater sales, greater economic efficiency, and greater profits per good from a bundle of information goods than can be attained when the same goods are sold separately. Our main results do not extend to most physical goods, as the marginal costs of production for goods not used by the buyer typically negate any benefits from the predictive value of large-scale bundling.While determining optimal bundling strategies for more than two goods is a notoriously difficult problem, we use statistical techniques to provide strong asymptotic results and bounds on profits for bundles of any arbitrary size. We show how our model can be used to analyze the bundling of complements and substitutes, bundling in the presence of budget constraints, and bundling of goods with various types of correlations and how each of these conditions can lead to limits on optimal bundle size. In particular we find that when different market segments of consumers differ systematically in their valuations for goods, simple bundling will no longer be optimal. However, by offering a menu of different bundles aimed at each market segment, bundling makes traditional price discrimination strategies more powerful by reducing the role of unpredictable idiosyncratic components of valuations. The predictions of our analysis appear to be consistent with empirical observations of the markets for Internet and online content, cable television programming, and copyrighted music.

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Section: The Bundling Literaturementioning

confidence: 99%

Bundling Information Goods: Pricing, Profits, and Efficiency

1999

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“…Although not directly related to our study, see also Ansari et al (1996) for the determination of the optimal number of items to be included in a service bundle, Ben-Akiva and Gershenfeld (1998) for customer choice behavior for bundles with correlated demand, Carbajo et al (1990) for incentives for bundling under imperfect competition, Hanson and Martin (1990) for the calculation of optimal bundle prices in a deterministic setting, using mixed integer linear programming, Ernst and Kouvelis (1999) for the effect of selling product bundles (as opposed to price bundles in our case) on inventory decisions, and Stremersch and Tellis (2002) for a clear discussion of bundling terms which are used in marketing, economics and law literature in a somewhat unclear way. Finally, we note the growing literature on bundling of information goods (see, for example, Bakos and Brynjolfsson (1999)).…”

Section: Introductionmentioning

confidence: 99%

Bundle pricing of inventories with stochastic demand

Bulut

Gürler

Şen

2009

European Journal of Operational Research

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We consider a retailer selling a fixed inventory of two perishable products over a finite horizon. The products are sold individually or as part of a bundle. The customers arrive following a Poisson Process and each customer makes a purchasing decision based on her reservation prices of individual products: she either buys one of the individual products, buys the bundle or leaves without a purchase. Assuming that the customer reservation prices follow a bivariate distribution, we determine the optimal product and the bundle prices that maximize the expected revenue. The performances of three bundling strategies (mixed bundling, pure bundling and unbundling) under different reservation price distributions, demand arrival rates and starting inventory levels are compared. The impact of the shape of the reservation prices is also investigated by considering both the bivariate normal and bivariate gamma densities, the latter of which has not been considered before in the literature. Our numerical results based on the bivariate normal reservation prices indicate that the performances of the policies heavily depend on the parameters of the demand process and the initial inventory levels. Bundling is observed to be most effective with negatively correlated reservation prices and when the starting inventory levels are high. When the starting inventory levels of two products are equal, most of the benefits of bundling can be achieved through pure bundling in case of excess supply. However, the mixed bundling strategy clearly dominates the other two when the starting inventory levels are not equal. Our numerical results show that bundling becomes more effective and the expected revenues increase as the products become less substitutable and more complementary. We also observe that the correct modeling of the reservation price distribution is important, and the use of sub-optimal prices resulting from assuming bivariate normal density when in fact the appropriate distribution is bivariate gamma may result in significant losses in the expected revenue especially if the reservation prices are negatively correlated. Finally, the model is extended to allow for price changes during the selling horizon using a dynamic programming formulation. It is shown that offering price bundles mid-season may be a more effective mechanism than changing individual product prices.

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“…However, this problem is known to be computationally intractable and difficult to solve in closed form except for small numbers of goods (Hanson and Martin, 1990). Recent work (Bakos and Brynjolfsson, 1999) has shown that when the marginal costs of goods are sufficiently low and customers share a common probability distribution for valuation of different goods, pure bundling (offering all goods for a fixed price) is optimal, greatly simplifying the bundling and pricing problem.…”

Section: Introductionmentioning

confidence: 99%

Bundling with Customer Self-Selection: A Simple Approach to Bundling Low-Marginal-Cost Goods

2005

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With declining costs of distributing digital products comes renewed interest in strategies for pricing goods with low marginal costs. In this paper, we evaluate customized bundling, a pricing strategy that gives consumers the right to choose up to a quantity M of goods drawn from a larger pool of N different goods for a fixed price. We show that the complex mixed-bundle problem can be reduced to the customized-bundle problem under some commonly used assumptions. We also show that, for a monopoly seller of low marginal cost goods, this strategy outperforms individual selling (M = 1) and pure bundling (M = N) when goods have a positive marginal cost or when customers have heterogeneous preferences over goods. Comparative statics results also show that the optimal bundle size for customized bundling decreases in both heterogeneity of consumer preferences over different goods and marginal costs of production. We further explore how the customized-bundle solution is affected by factors such as the nature of distribution functions in which valuations are drawn, the correlations of values across goods, and the complementarity or substitutability among products. Altogether, our results suggest that customized bundling has a number of advantages-both in theory and practice-over other bundling strategies in many relevant settings. Bundling with Customer Self-Selection: A Simple Approach to Bundling Low Marginal Cost Goods AbstractThe reduction in distribution costs of digital products has renewed interest in strategies for pricing goods with low marginal costs. In this paper, we evaluate the concept of customized bundling in which consumers can choose up to a quantity M of goods drawn from a larger pool of N different goods (N>M) for a fixed price. We show that the complex mixed bundle problem can be reduced to the customized bundle problem under some commonly used assumptions. We also show that, for a monopoly seller of low marginal cost goods, this mechanism outperforms individual selling (M=1) and pure bundling (M=N) when goods have a positive (even small) marginal cost, when users face costs of evaluating goods in the bundle, or when customers have heterogeneous preferences over goods. Comparative statics results also show that the optimal bundle size for customized bundling decreases in both heterogeneity of consumer preferences over goods and marginal costs of production. Altogether, our results suggest that customized bundling provides an efficient and mathematically tractable pricing scheme for selling information and other low-marginal cost goods, especially when consumers have heterogeneous preferences over goods or are budget or attention constrained.1

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Optimal Bundle Pricing

Cited by 264 publications

References 17 publications

Bundling Information Goods: Pricing, Profits, and Efficiency

Bundling Information Goods: Pricing, Profits, and Efficiency

Bundle pricing of inventories with stochastic demand

Bundling with Customer Self-Selection: A Simple Approach to Bundling Low-Marginal-Cost Goods

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