“In China, people don’t pay for the software,” Bill Gates told a crowd at the University of Washington back in 1998. “As long as they’re going to steal it, we want them to steal ours. They get sort of addicted, and then we’ll somehow figure out how to collect sometime in the next decade.” Gates was suggesting that there’s such thing as productive piracy, or, at least, that it could be worthwhile for a company to stand down instead of combating every single intellectual-property violation.

It’s often thought that counterfeits and knockoffs eat into sales and demand legal attention, and people who try to limit the reach of copyright law are often said to be anti-innovation. There’s no shortage of evidence for this claim: Just two weeks ago, a paper examining Italian operas from between 1770 and 1900 found “a significant increase in the number of new operas” after copyright law was adopted and strengthened.

While counterfeits do take away from the sales of the originals they parrot, they at the same time spread awareness of the imitated brand. This positive pull has typically been thought to be negligible when compared to the magnitude of sales lost, but a study published in August in Management Science by University of British Columbia business professor Yi Qian suggests that, under the right circumstances, that positive effect can outweigh the losses.

When knockoff versions of high-end footwear started popping up in China, Qian saw that sales of the authentic products increased by more than 60 percent—in part because people were made aware that the authentic brand is desirable enough to emulate. Spikes in sales were especially pronounced for brands that weren't as well-known—which makes sense, because counterfeits in these cases essentially serve as free advertising. The opposite was true for low-end brands, whose potential customers stuck with the much cheaper fakes.

It’s not a coincidence that Qian picked China and its footwear to study this phenomenon: footwear, because it’s consistently the counterfeit good most frequently seized by U.S. border patrol, and China, because a 1995 anti-counterfeiting policy paradoxically opened the door for any would-be fake-shoe makers. Using 31 Chinese manufacturers’ data on counterfeit reports from before and after this influx of knockoffs—which Qian told me in an email was enough of a jolt to resemble an experiment done in a lab—she was able to isolate the effects of fake footwear on legitimate sales.

Qian’s study provides the strongest evidence that counterfeit goods can boost sales, but it isn’t the only one out there. A study by MIT’s Renee Gosline a few years ago found that people exposed to a knockoff good eventually develop affinities for the real thing. Shoppers effectively use knockoffs as trial versions, and nearly half of counterfeit purchases were found to lead to buying the authentic products. Seeing, talking about, and wearing fake purses even led some to buy from high-end stores that they’d previously never entered.

But the positive effects of bootlegging don’t transfer over to other product categories as cleanly. Products associated with lower-end brands, for example, don’t enjoy these benefits. And the piracy of recorded music, another type of good entirely, might stimulate a disproportionate amount of ticket sales for concerts, but the literature is mixed on the value of the net result.

So, it’s the wealthiest brands—which probably are devoting the most resources to fight copyright infringement—that are actually better off backing down. Besides, fighting infringements is a Sisyphean task, even in richer countries with robust legal protections. Perhaps the best tactic, for high-end brands that aren't as well-known, is to do as Gates said: Let the market fill itself with knockoffs to accumulate brand awareness, and then start mailing out those cease-and-desists.