It doesn't see liquidity because you don't sell enough of yourself. Put more than 50% of you on the marketplace, and set up laws that say you need to do whatever the plurality dictates, and then you have a different situation. Liquidity comes from the exchange, not from the issuing company. I don't understand the details of how the exchanges work, but I know they have mechanisms to ensure liquidity (specialists). You won't get listed on an exchange if you only sell 1% of yourself.

Alternatively, part of the contract of me buying 1% of you could stipulate that all of your income will be distributed via dividend and you are not allowed to retain any earnings.

The bigger point, though, is that companies don't just issue 1% of their equity. Enough equity is sold on the secondary market that there is liquidity.