President Obama appears to have rejected the "Constitutional option" to raise the debt ceiling without Congressional approval.[1] I agree.
There is another way to avoid default without Congress that would be legal. It's undoubtedly a loophole, a mistake by Congress in drafting the law.
The President cannot unilaterally just print enough money to pay our debts, because Congress has put limits on the amount of money that can be printed, similar to the limit on the debt ceiling. There is no limit on the amount of coins that can be minted, but there are limits on the denominations of coins, so it would probably be impossible to mint enough quarters before next Tuesday. Here's the loophole: Congress has authorized the Treasury to mint platinum coins, and Congress did not limit either the amount or the denomination of such coins.[2] At the direction of the President, the Treasury could print two $1 trillion platinum coins. We just deposit those in the government's bank account, pay our debts and go on our merry way. We wouldn't have to round up $2 trillion worth of platinum. All we need is enough platinum to make coins big enough for the word "trillion."
I think the public would be aghast if Obama did it. But it would be perfectly legal. And it would be better, at least in the short run, than defaulting on our debt obligations.
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[1] http://volokh.com/2011/07/22/president-obama-rejects-authority-to-violate-debt-ceiling/
[2] http://thinkprogress.org/yglesias/2011/07/28/282471/the-platinum-coin-option/