There have been some alternatives to the completely volunteer system we have had and the payment method. One involves an exchange of donated kidneys between friends of people who need a kidney transplant but who are not compatible with their friend (or relative). Here is how it would work:
"Over the past decade, however, a more promising trade has become possible. In its simplest form, known as “paired exchange,” incompatible pairs are matched. For example, a husband with type A blood wants to give a kidney to his wife, who has type B blood. Meanwhile, a mother with type B wants to donate to her son with type A. So the mother gives to the wife, and the husband gives to the son. As originally developed at Johns Hopkins, paired exchanges involved not only swaps but simultaneous surgeries, so that no one could back out."
But it can be hard to find two pairs of people where the donors match for the exchange. There can be all kinds of medical reasons why one person can't donate a kidney to another. Here is the problem:
"Bartering kidneys has the same problems as bartering anything else. What each side needs has to match perfectly. “In economics, for maybe a hundred years, people have talked about why money is important—because of the difficulties of barter. The phrase that’s come to signify that in economics is The trouble with barter is, you have to find a double coincidence of wants,” says Alvin Roth, a Harvard economist who has designed algorithms for kidney exchanges. “When you think about regular kidney exchange, what that means is, not only do we have to want your kidney; you have to want our kidney—a double coincidence of wants.” Worse, the need for simultaneous operations, requiring four operating rooms and four transplant teams, severely limits where such exchanges can be done. Paired exchange can’t significantly reduce the waiting list."
So there is another system, called "donor chains." It starts with an altruistic person donating a kidney to be used for anyone who is a match. Then a friend or relative of the recipient donates a kidney that goes into the pool of available kidneys. Then a friend or relative of the next recipient donates a kidney and so on. It helps, but it is logistically very hard and the chain can be broken. And these other methods still leave us way short on kidneys. So we still have the question of allowing payments to get more people to give up a kidney. The Economics of Public Issues mentions that Iran has had a system of payments for 20 years and seems to work well. So maybe it is worth looking into.
Update Sept. 17, 10:22 am: A poll of economists included the question "The U.S. should allow payments to organ donors and their families." Here are the results:
STRONGLY DISAGREE(1) 4.7%
STRONGLY AGREE (5) 25.0%
This is from the article The Policy Views of American Economic Association Members: The Results of a New Survey by Robert Whaples.