As such, the President can use his Executive power to order the Treasury to take measures to cover its obligation eg. selling assets, lay public workers, freeze checks to retirees, veterans, businesses etc. Tough, unpalatable and politicall risky, but he can use the "bully pulpit" to try and explain that his actions are as a result of the recalcitrance and rigidity of the Republicans , especially the "no tax pledgers 234 of 240 Republicans in the House and 40 of 47 in the Senate, who are playing politics and have boxed themselves in and thus are unwilling to compromise. To do nothing would result in the US failing to meet its financial obligations, would see its credit ratings drop, which would lead to high interest rate, higher cost of borrowing for everyone, slow down in business and hiring and economic growth, reduction of corporate profits, govt. bonds being seen as not "risk-free", inflation, jobless growth,inability to pay soldiers, pensions for seniors, veterans benefit, govt. contracts with thousands of businesses etc.
The Founding Fathers in 1787, established what would be the basis of American governance....Compromise. They wanted to limit excesses of government powers and created a document, the Constitution with "separation of powers and checks and balances". This was not to create gridlock, because of jockeying for positions by the three branches of government or playing politics with rigid positions. It was to ensure a debate and eventually a compromise/ equilibrium. It has worked well in the past. Since 1962, the debt limit has been raised 75 times eg. 17 under Reagan and since 2001, 7 times under Bush ( in 2006 Senators Obama and Biden voted against raising it).
Why is Pres. Obama having such a difficult time. Reagan and O'Neill made deals on Social Security and deficit reduction. So did G.W. Bush and Tom Foley on deficit-cutting, with spending caps and tax increases, and Clinton and Gingrich produced a balance budget. Republicans and Democrats, working out compromises, even though they did not like each other.
Those were different times, for two major reasons. There was no Tea Party and there is much malice towards the first African-American president.