"It makes more sense to pay today's bills with the money one collects today;"
Not pensions, it doesn't, OG. The system you describe can be seen as a glorified "Ponzi" scheme and as you can see this model is not very stable:
http://uk.ask.com/wiki/Ponzi_scheme?lang=en
Pension schemes should be funded by individuals (and their employers if appropriate) and benefits paid out in direct relation to the funds paid in. Those funds should be invested (thus increasing their value) and be used to pay the individuals when the time comes. It is unlikely that current receipts alone will properly fund the benefits (especially if the membership is declining, though this is not the case with the State scheme).
Funding the State scheme suffers from a major drawback: "pensions" are paid to people who have contributed far too little (very often nothing) towards the benefits paid. As such these payments are not pensions at all but simply retirement age benefits. If the State Pension scheme is to be retained the "proper" State pension receipts and payments should be separated from the payments which have not been funded by lifetime payments. These receipts and payments can be properly calculated and payments to individuals properly related to funds paid in. If that is not to be so then the pension scheme should be would up.