In 2009 Apple increased the price of its best-selling iTunes tracks from 99 cents to $1.29, triggering a more enduring than expected sales decline. What did the company do wrong?

For one thing, it may have gone too far too soon in increasing prices, according to research by Javad Nasiry and Ioana Popescu. They looked at dynamic pricing when there are loss-averse consumers who anchor on the lowest price. Convincing such consumers to keep buying when the price requires careful handling.

"In repeat-purchase markets, consumers form price expectations known as 'reference prices'. Prices are perceived as discounts or surcharges relative to these reference prices and this perception affects demand and profitability," they say.

"For example, while a price promotion might have a short-run positive impact on sales, the lowered price might result in the installation of a low price in consumers' memory, eroding price expectations and willingness to pay and thereby negatively affecting profitability in the long run."

"This suggests it's important for a firm to understand first, how its pricing policy and history affect consumer's price expectations and purchase decisions, and second, how to maximize profitability in this context."

Previous scholars have assumed consumers' price expectations follow a fast-decaying memory process whereby the consumer anchors on a weighted average of all past prices. "Despite its prevalence in analytical pricing research, this model is not very well grounded in behavioral evidence," the authors say.

They explore a different model based on the peak-end rule which is supported by extensive psychological research. It proposes that consumers remember the lowest and most recent prices, and the reference price is a weighted average of the two. Research on frequently-purchased experience goods, such as music and food, suggests this is indeed what is happening.

The authors develop a new model that brings these ideas together and supports their hunch that firms need to take into account peak-end memory when pricing.

"Our results show peak-end anchoring processes interact with loss aversion to affect the structure of optimal pricing strategies," they say.

"Behavioral regularities lead prices to converge monotonically over time and induce a range of optimal constant pricing strategies supporting everyday low pricing. This range is wider the more loss averse consumers are and the more they anchor on the lowest price."

If consumers have an initial expectation of a low price, this affects the steady-state price, which means the effect of low historical prices is more enduring the previously understood. This in turn can lower optimal prices and profits.

"Overall our results suggest that behavioral regularities, and in particular low price anchors, limit the benefits of varying prices," the authors say

As for iTunes, they suggest the 99-cent price is anchored in consumer memory and this is having long-term effects on prices and profits. "These effects are best mitigated by gradually changing prices to manage expectations," they say.