They're afraid to let us live forever because we'll breed a lot and some people will use the magic of compound interest to gain huge fortunes at the expense at everyone else.
It's kind of silly, though, because I expect that the real value of lending money is measured as a proportion of your life that you won't have access to it while it's lent out (and vice-versa). In the end, this will put interest rates (and other investment returns considered as a whole market) as a dependent of mean human lifespan.
So the immortals won't really get compounded wealth and power unless they make sure that there's a LOT of people who die young. Which means, of course, that life-extended people will find ways to just exactly that. Perhaps by instigating wars. Perhaps by killing them off directly. Perhaps by denying life extension treatments to most people. But certainly it will happen.
Denying treatments to lots of people may work for the first few decades, but it has no long term stability -- tech will eventually be available to everyone. Instigating wars have the unfortunate side effect of blowing up useful stuff too, so I expect it will eventually revert to institutionalized direct killing.
So the assumptions are 1) everyone has access to life extension technology and 2) power and wealth will depend on how long you're alive. Design the society to include lots of stuff like dangerous sports, duels-to-the-death, and ritualized suicides. Perhaps a status scheme depending on a massive competition where your access to technological interventions is dependent on how far up the duel-to-the-death ladder you make it. (I'd add a religious aspect to it for an added touch of irony, but that's just me.)
Kind of like a blood-thirsty variant of rationed health-care, where powerful people get preferential treatment (and this is true of socialized systems as well as pay-as-you-go systems).
A few hundred generations of high attrition evolution and what results? homo killedeveryoneelseius