Here's an idea. Before the House of Representatives shuts down the government and threatens to default, they ought to actually bother to vote on, and preferably pass all twelve appropriations bills. Then they would fulfill one of their unique and primary Constitutional duties - to initiate government spending. If they want to cut entitlement programs, then they should pass bills that specifically reform the programs. Then we'd all know specifically what Republicans want, and could evaluate it.
Republicans are playing a game where they have the right to veto spending without the responsibility to propose spending. Like it or not, that is not their Constitutional role.
They only brought to the floor and passed 4 full appropriations bills: Defense, Homeland Security, Veterans, and Energy. They also passed half of the Agriculture bill, the half that provides assistance to farmers rather than the half that assists the poor with food stamps. These are the parts of government that Republicans like, and they tended to spend more than the Democrats wanted on these programs, and in some cases more than the Defense Department wanted.
The rest of the appropriations didn't make it to the floor, not because of Democrats but because Republicans couldn't agree among themselves, or perhaps couldn't disclose to constituents exactly what they were going to cut. It is easy to agree to cut very high level Department expenses in a budget resolution, but much tougher when you have to name the exact programs. For example, Republicans couldn't agree on a Transportation bill because although they agreed they wanted to cut it, they couldn't agree on what would be cut. The choices would create real pain to voters, businesses, and lobbyists.
So rather than governing, the House Republicans are blaming the Democrats. Two years ago they would default over excessive spending. This year it is over Obamacare, although that is now being switched to some nebulous demand to reduce spending. Meaning that if the shutdown wasn't over Obamacare, it would be over something else. As I've said before, I'm dubuious a default will occur past a day because delaying senior's Social Security checks is a non-partisan way to get un-elected.
Sequestration has stopped the growth of discretionary government spending, so that is now the baseline. Political contributions are up. The minority's tactics have worked up to now, at least until Senator Cruz demanded his TV close-up.
The problem with prolonged shutdowns is that people and businesses will discover that eventually they really do need the thousands of little things that government does. As these accumulate it will remind voters that they don't want to reduce government to the libertarian, minimal ideal. And every attempt by Republicans to open just a few more programs that people need will underscore that quite a bit of government may be annoying, but is in fact useful if not necessary.