Anna Gelpern, Stephen Lubben and I have an article in The American Prospect entitled The Debt Limit Is Unconstitutional—but Not for the Reason You Think. Various commentators—and members of Congress—have suggested that the President “invoke the 14th Amendment” to declare the debt limit unconstitutional. They're right to argue that the debt limit is unconstitutional, but the constitutional problem isn't the 14th Amendment. Instead, it's Article I of the Constitution, namely Congress's power to enter into contracts. The tl;dr version is that Congress has a power to make binding commitments for the United States and the President is constitutionally obligated to perform those commitments. If the Treasury lacks the funds, then the President must borrow. No specific authorization is needed. Instead, it is implicit every time Congress appropriates funds to perform a binding commitment.
Relocating the constitutional problem with the debt limit isn't merely an academic exercise. It has two implications.
First, it changes the nature of the legal debate and puts the administration on much, much firmer legal footing. The 14th Amendment argument is weak because it simply is not a prohibition on defaulting. It's a prohibition on repudiation, and a default is not a repudiation. An Article I argument reframes the issue as being about the validity of the debt ceiling, rather than the ability to default. In other words, it goes to question of whether the House GOP has holdup power, rather than whether the administration is under some cryptic constitutional limitation that it must affirmatively "invoke."
Second, it means that the President not only can, but must disregard the debt limit in order to fulfill his own constitutional duty to "Take Care" that the laws are faithfully executed. In other words, breaching the debt limit is not merely an option, but a legal requirement if Treasury is short of funds. Once Congress has appropriated funds, the President must carry out the authorized spending.