But can they outfox the newly empowered consumers?

J. C. Penney has introduced a streamlined system: daily prices, lower monthlong specials and clearance prices. Mango, the fashion retailer, has cut all prices by one-fifth. Stein Mart, the specialty chain, has reduced its coupons. Supervalu, the grocery chain, has sworn off heavy promotions and lowered some prices. Even Walmart has pledged to match competitors’ prices if it sets its own too high.

“The customer knows the right price,” said the chief executive of J. C. Penney, Ron Johnson. “We can raise the price all we want; she’s only going to pay the right price. And why is that? Because she’s an expert.”

Penney’s argues its proof is in the math. Over the last decade, higher prices did not make customers spend more, said Mr. Johnson, who took over the top job last year after running the retail operations at Apple.

An item that cost Penney’s $10 in 2002 was typically marked up to $28. By 2011, a $10 item had been marked up to $40. But the price the customer actually paid for the $10 item increased only 5 cents during that period — to $15.95, from $15.90.

Until 40 percent off, “the customer doesn’t even pay attention,” Mr. Johnson said.

Ronald S. Friedman, the head of the retail group at Marcum L.L.P., an accounting firm that advises department stores and manufacturers, said the average markup for apparel at a department store began around 65 percent. Over 10 weeks, the stores will go to 25 or 30 percent off, then 50 percent off, 60 percent, and finally 70 percent or more, a discount so deep that the stores sometimes sell below cost.