I apologise for the length of this answer, but for some reason, I was extremely fired up by it (don’t worry @daloon, I’ve GQed you =) )
I don’t like the idea of a world like this at all. Not only would there be possible manipulators of the system as @SuperMouse said, there also are three incredibly glaring flaws in this world from what I can see.
1. How do you decide what is to be valued? How much one is loved? What one’s relative social importance is? There are too many things which determine who a person is, and how much you’re loved is only one of them. Also, how do you qualify what social importance is? So many factors can go into one’s social importance, be it how much they’re loved, how much they donate to charity, how much they love, how close they are to their friends, how much of a go-getter they are etc. etc. The list goes on and on. To try to construct such a comprehensive list would, in my opinion, inevitably leave out something which still adds to one’s social worth.
Let’s just use your example of “how much one is loved” as a scenario (since your question seems to be focusing quite heavily on this). So what if our world does allow for such a measure? Don’t you think it’s bad enough that people are judged purely based on how much money they earn? Now they’re going to be judged by others based on how much they’re loved? By making one part of a person’s character so obvious to everyone else, it makes it only so easy to ignore all the other parts of that person’s character which makes him who he is, especially since now you’re providing a direct window into his soul, where his monetary worth only allows for very indirect inference.
2. Even if we were able to truly take everything which determines a person’s social worth and put it into a quantifiable measure, I think that it would spell the end of personal privacy. As earlier said, in order to truly know what one’s social worth is, you’ll have to take everything in one’s life into account, and then present that as a measure to practically everyone. This means that this person, from the data provided, will be able to see how much I care for my family, how well I’m doing with my friends, how much I love them, how much I’m loved by them, whether I’m a go-getter or whether I prefer to take orders. Sure, some of these things may already be obvious to the sharp observer, but to make it such that they’re so blindingly obvious to the whole world is just a horrible thought.
Furthermore, have you thought about how anyone would be able to gather such information? It would have to involve the technology which pries into every single aspect of your personal life. I don’t think any normal person would be able to stomach a thought like that.
3. The last, and perhaps most important criticism, is that it’s unfair, no matter how accurate it is. Why? Because humans have the capacity for change. So what if my social-o-meter reads that I can only contribute ten sociobels to my city? That is only a measure of how much I can contribute at that exact moment. People are not stagnant. Who I am today does not determine who I am tomorrow, no matter how strong a link there may be. Tell me, who has the right to pigeonhole me, perhaps permanently, based on what my abilities are at a certain point in time? Your world would probably be filled with individuals who can never reach their true potential because of this measure.
And if your measure can tell me what the absolute social potential of a person is for his whole life.. Well.. Then that’s a very worrying world we’ll be living in. People might start being segregated from a young age. It’ll be no different from tyranny where free choice is eliminated (@Dr_Lawrence has very helpfully listed out various examples of eras which would share the characteristics of this world).
In conclusion, such a world is one which I hope never comes to pass, for that would be the end of the life we all love and know.