As long as sex is still sex, there is going to be competition regardless of material scarcity.
Not if all people have the opportunity to be attractive, have access to desirable partners and are not afflicted with issues of self-esteem.
Indeed much progress in arts and sciences has been driven by various forms of unrequited love, or as elrond said, "to impress the opposite sex" (with some exceptions such as da Vinci). And as mind said, we must rely on capitalistic drives for a while longer.
So an interesting question is what would happen to innovation and creativity in a "Utopian" society? Would they just dry up?
On the contrary - creativity and innovation would flourish. I'll explain. Amongst us, there are some individuals who are driven to explore and to innovate solely for the pleasure of finding things out. Not for money, love or fear. I call this type of person a 'searcher'. Some of them become scientists or engineers but invariably they tinker, read and contemplate because they enjoy those activities, generally over other pleasures, in and of themselves.
In a society of asymmetrical resources those with the most powerful drives - capital acquisition, sex and control of others - invariably identify the searchers and recruit them for their own ambitions. Consequently a symbiosis of sorts takes place (although often, as many scientists and engineers will attest, rather than symbiotic such relationships can be parasitic) where the searchers innate skills are harnessed.
Therefore, a Utopian society would see searchers flourish uninhibited and unfettered by the recruitment of those with other drives.
Unfortunately, John S didn't come across as a searcher and that's particularly sad because he probably is one.
PS Give a searcher some personal space and eventually the problem is solved. Most grand discoveries are not made in the labs of pharma companies with million dollar budgets. In fact very few key discoveries were made under such circumstances.