I found some early data regarding the impact on site traffic of the New York Times' new paywall. Looks like both traffic and page impressions are down, which is no surprise, of course. The author notes that there is no apparent change in traffic from Google or social media sites, which allow one to access the Times without paying and without it counting against your 20-article per month limit. However, I doubt people are so highly motivated to read the times that they hunt for links to stories at these channels. One commentor makes the case that people may not aware these articles don't count against your limit, and may avoid the site as a result. The other concern I would have were I the New York Times is that some readers may have heard about the paywall but aren't aware that it doesn't kick in immediately.
Having a smaller audience isn't necessarily a bad thing, even leaving aside the revenue generated from subscriptions. A smaller readership makes it easier for the Times to tease out demographics and other audience characteristics, and can sell advertisers on tailoring content appropriately. The Times should be concerned, as at one least one commentator writes, that its content is not unique enough to maintain subscribers over time. I wonder if they've given thought to approaching advertisers about creating content specific to the Times that adds value, such as short films that can be seen only via subscription at the Times. (Or that count toward the 20-article limit, so that interested readers get a tease of what they might be missing.) If they can generate subscription dollars and increase advertising revenue by delivering readers more likely to click on sponsored content, then they can make skeptics like me look like fools.