The debt ceiling is a law that that limits the amount the US government can borrow. Yes, it is a law. We have a budget , it is a law (or a series of laws). Medicare and Social Security are also laws. We have a Constitution that says the says that the President “shall take Care that the Laws be faithfully executed.” (Article 2, Section 3).
So the President is there to carry out the laws but there are many laws. And there is one law (the debt ceiling) that would block his ability to carry out the other laws. So what is a President to do? Ignore the debt ceiling law (which itself may be unconstitional) so that the other laws passed by congress are executed and fulfill his Constitutional duty or obey the debt ceiling law and fail in that duty.
It is a difficult situation but would be made much easier if there was an agreement to raise the debt ceiling. It is certainly a dilemma if there is no debt ceiling increase since it is not possible to obey all laws but if it were my choice I’d go with most laws and the Constitition.
Here are 2 articles that discuss these issues in more depth
http://www.newyorker.com/online/blogs/hendrikhertzberg/2011/07/the-debt-limit-faithful-execution-1.html#entry-more
http://www.thefiscaltimes.com/Columns/2011/07/08/Obama-May-Have-to-Break-the-Law-if-Debt-Talks-Collapse.aspx
