## Abstract

Consumers often stockpile goods to store for future consumption. The existing theoretical literature has focussed on a price-based explanation where stockpiling arises due to temporary price reductions. In contrast, this paper explores a transaction-cost-based explanation where consumers stockpile to avoid the need to incur future transaction costs. It shows how transaction costs lead to positive consumer stockpiling in an oligopoly equilibrium even when future prices are expected to fall. Relative to a no-stockpiling benchmark, such stockpiling lowers profits, but improves consumer and total welfare. Our results extend to the case of quantity discounts where stockpiling consumers pay relatively lower per-unit prices than non-stockpiling consumers, when purchasing multi-unit bundles.

## Acknowledgements

We are very thankful for comments from the editor (Hendrik Juerges), two anonymous referees, Tobias Wenzel, and from various conference and seminar audiences including EARIE (2017), Jornadas de Econom\’ıa Industrial (2017), Nottingham University Business School and Loughborough University. Li acknowledges the support from "The Fundamental Research Funds of Shandong University (2019HW008)".

# Appendix

## Proof of Lemma 1.

From eqs. (5) and (6), consumer *m* will stockpile if (a) *m*’s relative brand preference for firm *i*,

## Proof of Lemma 2.

If *i* rather than *j* in period 1 if *i*’s total period 1 demand equals

If, instead, *i* rather than *j* in period 1 if *i* in period 1. Of these,

## Proof of Lemma 3.

If *i* rather than *j* in period 2 if *B* and a positive measure of consumers with *A*. As in eq. (8), this implies that firm *i*^{ʹ}*s* total period 2 demand is equal to the benchmark demand, *i* in period 1,

## Proof of Lemma 4.

Suppose *i*’s period 2 profit function as *i*’s period 2 best response for given stockpiling levels, *i*, *j* ≠ *i* ∈ {*A*, *B*}. After repeating for firm *j* and solving simultaneously one obtains the unique period 2 equilibrium prices,

## Proof of Proposition 1.

Having derived period 2 equilibrium prices, we first consider consumers’ stockpiling decisions, before deriving the equilibrium levels of stockpiling demand as a function of period 1 prices, in eq. (10).

First, consider consumers’ stockpiling decisions and initially suppose that each firm has positive period 2 demand with *i*, with (ii) the net marginal benefits of waiting to buy from firm *i*, rather than firm *j*, in period 2. From eq. (6), this implies

We are now in a position to derive the equilibrium levels of stockpiling demand as a function of period 1 prices. First, consider the top line of eq. (10). Here, *i* = {*A*, *B*}, and so this case occurs when

Second, consider the bottom line of eq. (10). Here,

Third, consider the middle line of eq. (10). Here, there exists a unique level of equilibrium stockpiling,

After deriving a similar equation for *i*, *j* ≠ *i* ∈ {*A*, *B*}. For

Finally, note that the levels of stockpiling and associated conditions in eq. (10) are continuous as (i)

## Proof of Proposition 2.

From eq. (10), *κ* > 0. □

## Proof of Proposition 3.

First suppose that period 2 is active with *i*’s profit function from eq. (11) can then be rewritten as:

where

where *κ* > 0, this case requires

Second suppose that period 2 is inactive with *i*’s profit function then equals

However, to ensure

## Proof of Proposition 4.

The proof proceeds by initially deriving the equilibrium, before then comparing the level of stockpiling to that in the main model. First suppose that period 2 is active, with *i* must then choose *κ* > 0, this case requires

Second, suppose that period 2 is inactive such that

Finally, for any *κ* > 0, one can show that the level of stockpiling under quantity discounts is weakly higher than that in the main model because i) the level of stockpiling within the interior solution is relatively higher,

## Proof of Proposition 5.

Proceed by contradiction. Suppose no consumers stockpile in period 1. Building on the benchmark, the firms then receive a demand of *κ* > 0 would then have an incentive to stockpile in period 1 as

## References

Anton, J. J., and G. Das Varma. 2005. “Storability, Market Structure, and Demand-Shift Incentives.” *RAND Journal of Economics* 36: 520–43.10.2139/ssrn.412601Search in Google Scholar

Armstrong, M. 2006. Recent Developments in the Economics of Price Discrimination. In: Blundell R., Newey W.K., and Persson T., editors. *Advances in Economics and Econometrics: Theory and Applications, Volume II* 97–141. Cambridge, UK: Cambridge University Press.10.1017/CBO9781139052276.006Search in Google Scholar

Armstrong, M. 2017. “Ordered Consumer Search.” *Journal of the European Economic Association* 15: 989–1024.10.1093/jeea/jvx017Search in Google Scholar

Bell, D. R., G. Iyer, and V. Padmanabhan. 2002. “Price Competition under Stockpiling and Flexible Consumption.” *Journal of Marketing Research* 39: 292–303.10.1509/jmkr.39.3.292.19103Search in Google Scholar

Boizot, C., J. M. Robin, and M. Visser. 2001. “The Demand for Food Products: An Analysis of Interpurchase Times and Purchased Quantities.” *Economic Journal* 111: 391–419.10.1111/1468-0297.00613Search in Google Scholar

Erdem, T., S. Imai, and M. P. Keane. 2003. “Brand and Quantity Choice Dynamics under Price Uncertainty.” *Quantitative Marketing and Economics* 1: 5–64.10.1023/A:1023536326497Search in Google Scholar

Gangwar, M., N. Kumar, and R. C. Rao. 2013. “Consumer Stockpiling and Competitive Promotional Strategies.” *Marketing Science* 33 (1): 94–113.10.1287/mksc.2013.0814Search in Google Scholar

Guo, L., and J. M. Villas-Boas. 2007. “Consumer Stockpiling and Price Competition in Differentiated Markets.” *Journal of Economics and Management Strategy* 16 (4): 827–58.10.1111/j.1530-9134.2007.00159.xSearch in Google Scholar

Hendel, I., and A. Nevo. 2006a. “Sales and Consumer Inventory.” *RAND Journal of Economics* 37 (3): 543–62.10.1111/j.1756-2171.2006.tb00030.xSearch in Google Scholar

Hendel, I., and A. Nevo. 2006b. “Measuring the Implications of Sales and Consumer Inventory Behavior.” *Econometrica* 74 (6): 1637–73.10.1111/j.1468-0262.2006.00721.xSearch in Google Scholar

Hendel, I., and A. Nevo. 2013. “Intertemporal Price Discrimination in Storable Goods Markets.” *American Economic Review* 103 (7): 2722–51.10.1257/aer.103.7.2722Search in Google Scholar

Hong, P., R. P. McAfee, and A. Nayyar. 2002. “Equilibrium Price Dispersion with Consumer Inventories.” *Journal of Economic Theory* 105 (2): 503–17.10.1006/jeth.2001.2890Search in Google Scholar

Hosken, D., and D. A. Reiffen. 2007. “Pricing Behaviour of Multiproduct Retailers.” *B.E. Journal of Theoretical Economics* 7 (1): 1–43.10.2202/1935-1704.1354Search in Google Scholar

Marshall, G., and T. Pires. 2018. “Measuring the Impact of Travel Costs on Grocery Shopping.” *Economic Journal* 128: 2538–57.10.1111/ecoj.12523Search in Google Scholar

Perloff, J. M., and S. C. Salop. 1985. “Equilibrium with Product Differentiation.” *Review of Economic Studies* 52 (1): 107–20.10.2307/2297473Search in Google Scholar

Perrone, H. 2017. “Demand for Nondurable Goods: A Shortcut to Estimating Long-Run Price Elasticities.” *RAND Journal of Economics* 48: 856–73.10.1111/1756-2171.12194Search in Google Scholar

Pesendorfer, M. 2002. “Retail Sales: A Study of Pricing Behavior in Supermarkets.” *Journal of Business* 75 (1): 33–66.10.1086/323504Search in Google Scholar

Salop, S., and J. E. Stiglitz. 1982. “The Theory of Sales: A Simple Model of Equilibrium Price Dispersion with Identical Agents.” *American Economic Review* 72 (5): 1121–30.Search in Google Scholar

Sobel, J. 1984. “The Timing of Sales.” *Review of Economic Studies* 51 (3): 353–68.10.2307/2297428Search in Google Scholar

Wang, E. Y. 2015. “The Impact of Soda Taxes on Consumer Welfare: Implications of Storability and Taste Heterogeneity.” *RAND Journal of Economics* 46 (2): 409–441.10.1111/1756-2171.12090Search in Google Scholar

Zhou, J. 2017. “Competitive bundling.” *Econometrica* 85 (1): 145–72.10.3982/ECTA14251Search in Google Scholar

**Published Online:**2019-05-24

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