The House of Representatives now finds itself in the position of the proverbial dog who caught the car.

Under pressure from its conservative Tea Party wing, and having been backed into a corner by firebrand Ted Cruz and his GOP allies in the Senate, House leaders have drawn a line in the sand that, for House leaders to save face, seems to require some concessions from the White House on Obamacare delays or changes.

But the White House and the Senate Democrats have shown no inclination to negotiate, and to all appearances seem content to let the government shut down today.

"They're not doing me a favor by paying for things that they have already approved for the government to do," President Obama told NPR. "That's part of their basic function of government; that's not doing me a favor. That's doing what the American people sent them here to do, carrying out their responsibilities."

The President seems confident that history will repeat itself. The last time a conservative House tried to stare down a Democrat in the White House, it led to a rejuvinated Clinton presidency, as the GOP was held responsible by the news media and, ultimately, the general public.

Last night the GOP House defeated a revolt by GOP moderates, who sought to send a "clean" continuing resolution to the Senate.

Earlier this Spring the Senate passed a formal budget for the first time in four years, after in 2012 unanimously rejecting President Obama's proposed budget. But the House and Senate have made little progress on reaching a budget compromise, and as a result the government continues to stumble along from one make-do "continuing resolution" to the next.

In the absence of actual budget negotiations, Republicans have sought to leverage the continuing resolution votes and the now-routine hikes of the debt limit to force debate and change on major fiscal concerns.

In 2011 Congress passed a budget deal that expanded the debt limit conditioned upon automatic spending cuts that would take effect if specific reduction targets were not met. No one expected the cuts to take effect, but no deal was reached to avert them.

When the sequester took effect in January of this year, it was widely expected that it would have devastating impacts on government functioning and the economy. But the effect proved obscure, and the issue largely disappeared. Those automatic cuts now seem likely to survive the current budget crisis, most observers expect.

"This is a huge victory that nobody has talked about," Rep. Mario Diaz-Balart (R., Fla.), a member of the House Appropriations Committee, told the Wall Street Journal. "The spending portion of the spending bill we have won."

The House has now passed two resolutions, both rejected by the Senate. A game of hot potato seems to have developed, with neither House or the Senate wanting to be left holding it when the shut down takes effect.

Plan C for the Republicans, it now seems, may be to try to force Congress to live by the rules it set out when Obamacare passed. During the original debate on the Affordable Care Act, as Ezra Klein outlines at the Washington Post, the GOP had offered, as a kind of poison pill, an amendment that required Congressional staff to be covered under the exchanges.

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The proposal slipped into the bill and became law, but it was later waved in the rule making process by the Obama administration, in a move of disputed legality.

A Plan C that forced the Democrats to implement the law as written would have the upside of saving face, while forcing Capitol Hill to live with the rawest side of the new health care law. But it would have the downside of injuring GOP staffers as well.

Democrats would probably have a hard time voting against such a move. And short of that, it is hard to see a viable exit strategy that allows the GOP House to save face.