Someone I was talking to the other day was speculating: stuff goes online and becomes ad-supported. But it's still all ads that end in purchases. Company X wants you to see their ads so that sooner or later you buy their stuff. What if, eventually, the money were taken out of the picture? Ads became just a way to get you to look at more ads, to get inside your head more.
The most obvious application of this is political ads: people want you to spend money on candidate X, so they can buy more ads for candidate X, so more people will... like candidate X. Eventually it's all about votes.
Besides the political ad situation, it's hard to think about how this would actually work. Who would actually pay for attention if it doesn't end in a purchase? I am confident, however, that someone will solve this question. People are good at finding new ways to profit. For better or worse, this is our future.
One nice property: we all start with a more-or-less equal amount of attention, unlike money.
However, a few bad properties, which make the attention-economy world kinda dystopian:
1. you can take my attention for free. I might be reading a magazine, I turn the page, and bam! your ad has taken my attention! Even if it's something I don't want! It'd be like if people walking past you on the street could force you to buy things. Very small things, but still. See also: junk mail, spam.
2. you can't really solve problem #1. What are you going to do, say "do you want to look at an ad for company X?" At that point, you've already gotten me to think about company X and taken my attention. Worse: you've taken more attention, because you've forced me to make a judgment and answer a question.
3. most importantly: your attention is one of the most precious things you have. If you run out of money, that can be fixed (hopefully). Ask friends or family for a loan, get a new job, etc. (let's set aside all the class privilege reflected in this statement for a second.) The point is, if you run out of money, or just have a small amount of money, you're not necessarily miserable. If you have no attention, you probably are.