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I seem to recall reading of a design by Robert Heinlein (but maybe Sheffield, Clarke or another scientific writer), where the energy cost of a space elevator was virtually nil, as descending loads were used to recoup the energy cost of the lift.
Looking for the story,author and if possible the quote where it is described.
If you think it might have been Clarke, I would try The Fountains of Paradise. I don't remember the specific passage you're talking about, but it predates (and in fact probably caused) the era of science fiction in which space elevators are a common trope, so as I recall it spends more time describing them than most.
Edit
Chapter 10 "The Ultimate Bridge"
Quote delivered by the hologram of Vannevar Morgan, Chief Engineer of Terran Construction's Land Division.
"Capsules for passengers, freight, fuel would ride up and down the tubes, at several thousand kilometers an hour. Fusion power stations at intervals would provide all the energy needed: since ninety percent of it would be recovered, the net cost per passenger would be only a few dollars. As the capsules fall earthward again, their motors will act as magnetic brakes, generating electricity. Unlike re-entering spacecraft, they won't waste all their energy heating up the atmosphere and making sonic booms, it will be pumped back into the system. You could say that the down trains will power the up ones. So even at the most conservative estimate, the Space Elevator will be a hundred times more efficient than any rocket."
As Mike Scott mentions, Sheffields book The Web Between the Worlds fits the bill. Here is the relevant passages, as gleaned from amazon's "look inside" feature:
..if we send a mass all the way to the end of the beanstalk, far beyond geostationary orbit, the we will also have a free launch system. A mass released from 100,000 kilometers out can be thrown to any part of the solar system. The energy for this is, incidentally, free. It is provided by the rotational energy of the Earth itself [...] since any energy used in the drive train to take mass up the beanstalk can be recovered by making the same mass do work as it comes down, a remarkably efficient system is possible. And by using the beanstalk as a slingshot, we have the energy-free launch system for payloads going to destinations anywhere in the solar system
But note that this is from the book's second appendix, "space elevators in fact and fiction" where the author goes into technical detail. I'm not sure if it's elaborated on in-story or if the appendix was present in the first edition. In any case, and with apologies to Heinlein, there is no such thing as a free launch.
