Skip to main content

Pricing policies for selling indivisible storable goods to strategic consumers


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.

This is a preview of subscription content, access via your institution.


  1. 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).

  2. 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.

  3. These storable goods models are also referred to as dynamic inventory models.

  4. 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.

  5. 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.

  6. 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.


  • Aviv, Y., & Pazgal, A. (2008). Optimal pricing of seasonal products in the presence of forward-looking consumers. Manufacturing & Service Operations Management, 10(3), 339–359.

    Article  Google Scholar 

  • Bagnoli, M., Salant, S., & Swierzbinski, J. (1989). Durable-goods monopoly with discrete demand. Journal of Political Economy, 97, 1459–1478.

    Article  Google Scholar 

  • Berbeglia, G., Sloan, P., & Vetta, A. (2014). Bounds on the profitability of a durable good monopolist. In Web and Internet Economics (pp. 292–293). Springer.

  • Chiarolla, M., Ferrari, G., & Stabile, G. (2015). Optimal dynamic procurement policies for a storable commodity with Lévy prices and convex holding costs. European Journal of Operational Research, 247(3), 847–858.

    Article  Google Scholar 

  • Coase, R. (1972). Durability and monopoly. Journal of Law and Economics, 15, 143–149.

    Article  Google Scholar 

  • Correa, J., Montoya, R., & Thraves, C. (2016). Contingent preannounced pricing policies with strategic consumers. Operations Research, 64, 251–272.

    Article  Google Scholar 

  • Dasu, S., & Tong, C. (2010). Dynamic pricing when consumers are strategic: Analysis of posted and contingent pricing schemes. European Journal of Operational Research, 204(3), 662–671.

    Article  Google Scholar 

  • Drexl, A., & Kimms, A. (1997). Lot sizing and scheduling—Survey and extensions. European Journal of Operational Research, 99(2), 221–235.

    Article  Google Scholar 

  • Dudine, P., Hendel, I., & Lizzeri, A. (2006). Storable good monopoly: The role of commitment. American Economic Review, 96(5), 1706–1719.

    Article  Google Scholar 

  • Elmaghraby, W., Lippman, S. A., Tang, C. S., & Yin, R. (2009). Will more purchasing options benefit customers? Production and Operations Management, 18(4), 381–401.

    Article  Google Scholar 

  • Gul, F., Sonnenschein, H., & Wilson, R. (1986). Foundations of dynamic monopoly and the coase conjecture. Journal of Economic Theory, 39(1), 155–190.

    Article  Google Scholar 

  • Güth, W., & Ritzberger, K. (1998). On durable goods monopolies and the coase-conjecture. Review of Economic Design, 3(3), 215–236.

    Article  Google Scholar 

  • Hendel, I., & Nevo, A. (2006). Sales and consumer inventory. RAND Journal of Economics, 37(3), 543–561.

    Article  Google Scholar 

  • Herbon, A. (2014). Optimal piecewise-constant price under heterogeneous sensitivity to product freshness. International Journal of Production Research, 54(2), 365–385.

  • Heskett, J. L., & Earl Sasser, W. (2010). Southwest airlines: In a different world. Harvard Business School Entrepreneurial Management Case No. 910-419.

  • Jans, R., & Degraeve, Z. (2008). Modeling industrial lot sizing problems: A review. International Journal of Production Research, 46(6), 1619–1643.

    Article  Google Scholar 

  • Koenigsberg, O., Muller, E., & Vilcassim, N. J. (2008). easyjet\({\textregistered }\) pricing strategy: Should low-fare airlines offer last-minute deals? QME, 6(3), 279–297.

    Google Scholar 

  • Kuo, C., & Huang, K. (2012). Dynamic pricing of limited inventories for multi-generation products. European Journal of Operational Research, 217(2), 394–403.

    Article  Google Scholar 

  • Li, M. (2001). Pricing non-storable perishable goods by using a purchase restriction with an application to airline fare pricing. European Journal of Operational Research, 134(3), 631–647.

    Article  Google Scholar 

  • Liberali, G., Gruca, T., & Nique, W. (2011). The effects of sensitization and habituation in durable goods markets. European Journal of Operational Research, 212(2), 398–410.

    Article  Google Scholar 

  • Liu, Q., & van Ryzin, G. (2008). Strategic capacity rationing to induce early purchases. Management Science, 54(6), 1115–1131.

    Article  Google Scholar 

  • Nie, P. (2009a). Commitment for storable goods under vertical integration. Economic Modelling, 26(2), 414–417.

    Article  Google Scholar 

  • Nie, P. (2009b). Commitment for storable good duopoly. Nonlinear Analysis: Real World Applications, 10(3), 1838–1845.

    Article  Google Scholar 

  • Pesendorfer, M. (2002). Retail sales: A study of pricing behavior in supermarkets. Journal of Business, 75(1), 33–66.

    Article  Google Scholar 

  • Robinson, P., Narayanan, A., & Sahin, F. (2009). Coordinated deterministic dynamic demand lot-sizing problem: A review of models and algorithms. Omega, 37(1), 3–15.

    Article  Google Scholar 

  • Stokey, N. (1979). Intertemporal price discrimination. Quarterly Journal of Economics, 93, 355–371.

    Article  Google Scholar 

  • Su, X. (2007). Intertemporal pricing with strategic customer behavior. Management Science, 53(5), 726–741.

    Article  Google Scholar 

  • Surasvadi, N., Tang, C., & Vulcano, G. (2017). Using contingent markdown with reservation to profit from strategic consumer behavior. Production and Operations Management, 26(12), 2226–2246.

  • Tereyağoğlu, N., Fader, P. S., & Veeraraghavan, S. (2017). Pricing theater seats: The value of price commitment and monotone discounting. Production and Operations Management, 26(6), 1056–1075.

    Article  Google Scholar 

Download references


The authors would like to thank to Gustavo Vulcano and Jun Xiao for helpful discussions.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Gerardo Berbeglia.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Berbeglia, G., Rayaprolu, G. & Vetta, A. Pricing policies for selling indivisible storable goods to strategic consumers. Ann Oper Res 274, 131–154 (2019).

Download citation

  • Published:

  • Issue Date:

  • DOI:


  • Price optimization
  • Storable goods
  • Monopoly
  • Price discrimination
  • Price commitment
  • Dynamic programming
  • Profit bounds