What would happen to the world if Bitcoin were the only allowed currency? Could a world economy be based solely on Bitcoin? I mean the only mean to pay would be bitcoin.

From the point of view of economy would this be possible or would all the economy collapse?

Could we live only on Bitcoin, to make payments to work and so on, while this currency is so volatile?
It's possible, but with the current block size and lack of lightning network or side chains, no. Most likely people would do what they've done throughout history and use other forms of currency for daily use.