Abstract
We study the dynamic pricing problem faced by a monopolistic retailer who sells a storable product to forward-looking consumers. In this framework, the two major pricing policies (or mechanisms) studied in the literature are the preannounced (commitment) pricing policy and the contingent (threat or history dependent) pricing policy. We analyse and compare these pricing policies in the setting where the good can be purchased along a finite time horizon in indivisible atomic quantities. First, we show that, given linear storage costs, the retailer can compute an optimal preannounced pricing policy in polynomial time by solving a dynamic program. Moreover, under such a policy, we show that consumers do not need to store units in order to anticipate price rises. Second, under the contingent pricing policy rather than the preannounced pricing mechanism, (i) prices could be lower, (ii) retailer revenues could be higher, and (iii) consumer surplus could be higher. This result is surprising, in that these three facts are in complete contrast to the case of a retailer selling divisible storable goods (Dudine et al. in Am Econ Rev 96(5):1706–1719, 2006). Third, we quantify exactly how much more profitable a contingent policy could be with respect to a preannounced policy. Specifically, for a market with N consumers, a contingent policy can produce a multiplicative factor of \(\Omega (\log N)\) more revenues than a preannounced policy, and this bound is tight.
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Notes
A firm may wish to reduce the lifespan of a durable good to avoid a Coasian outcome, that is, one where the firm loses all monopoly power. Indeed, Amazon has recently been granted a US patent (8,364,595) for a second-hand market in digital goods that restricts the number of times a good can be resold, that is, limits the longevity of the goods. Apple has applied for a similar patent (US 20130060616 A1).
A durable good is a long-lasting good that can be “stored” and consumed repeatedly over time. Examples include digital goods, land and, to a large degree, housing, metals, electronic goods, etc.
These storable goods models are also referred to as dynamic inventory models.
The reason the profit is not 32 as in the case where goods are not divisible is simply because of the scaling: the demand in this example is scaled to a unit per period, which is half of the example from Sect. 2.3.
We remark that the model in Dudine et al. (2006) contains additional assumptions that are not required here, such as the retailer revenue being a concave function of the price in each time period.
Essentially, we can view the all the consumers in the Bagnoli et al. (1989) model as only wishing to consume the durable good in the final period. In those circumstances, it does not matter whether the good is durable or simply storable.
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The authors would like to thank to Gustavo Vulcano and Jun Xiao for helpful discussions.
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Berbeglia, G., Rayaprolu, G. & Vetta, A. Pricing policies for selling indivisible storable goods to strategic consumers. Ann Oper Res 274, 131–154 (2019). https://doi.org/10.1007/s10479-018-2916-x
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DOI: https://doi.org/10.1007/s10479-018-2916-x