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Now that Republicans are set to control both houses of Congress, the great question is: What will they do with it?

The message from Washington (and Louisville, Ky., where the incoming Senate majority leader Mitch McConnell spoke) on Wednesday was of constructive desire to find areas for common ground. The question is what concrete legislation that might lead to, and whether it will dissolve the first time that concrete legislation makes its way to the Senate floor.

As we wrote earlier, moments like this — when a president in his final two years of an eight-year run faces a sharply hostile Congress — are certainly not when big, ideologically polarizing legislation is likely to be enacted. Republicans may have a comfortable majority in both the House and Senate, after all, but not enough votes to override a presidential veto. My colleague Carl Hulse reports that Republicans are eager to show they can be a governing party and seek to move legislation that many Congressional Democrats might object to but which Mr. Obama is likely to sign.

When there has meaningful lawmaking in this environment, it has tended to be on more below-the-radar issues in which the party leading Congress has greater enthusiasm. Analysts who try to handicap what will happen in Washington have been grappling with what, if anything, that might consist of in the 114th Congress, which begins in January. Here’s a rundown of the possibilities.

Keeping the government running.

Republicans will have to decide how aggressively they will use the need to fund the government to extract policy concessions from the Obama administration. Will these be the Republicans of 2011 through 2013, when a hard-edged approach culminated in a two-week government shutdown driven by demands that the president agree to a repeal of his signature health reform law?

Or will they be the Republicans of 2014, who presented themselves as a responsible governing party while looking to chip away at Obamacare and other White House priorities piece-by-piece and without a grand, disruptive showdown?

One way or another, the debt ceiling will need to be raised sometime in the first part of 2015 to prevent a government default, and deals will need to be made to continue funding the federal government without the high drama of standoffs like that over the debt ceiling in 2011, the “fiscal cliff” of late 2012 and the shutdown in 2013.

It will pit two sides of the G.O.P. against each other. One team of Republican leadership seeks to bolster the party’s brand in the run-up to 2016 elections and is responsive to business interests who prefer stability. The other is the harder-right contingent of the caucus that wants to try to shut down Obamacare at any cost. Presidential politics may come into play as well, as potential candidates like Ted Cruz of Texas and Rand Paul of Kentucky push for a harder-edged approach than that preferred by the majority leader Mitch McConnell. (“Let me be clear: There will be no shutdowns and no default on the national debt,” Mr. McConnell said Wednesday in his appearance in Louisville).

Sean West, who heads the United States practice of the Eurasia Group, a political risk consultancy, argues that the odds of a return to disruptive showdowns of the recent past is only about 25 percent. “The Republicans have no incentives to play games,” he said. “They’re focused on looking like a party that can govern.”

A trade deal on the way?

The likelihood of a major trade deal with Pacific Rim nations is, if anything, higher after the Republican victory than before. The soon-to-be-former Senate majority leader, Harry Reid, and many other Democrats have been skeptical of further American trade agreements because of union opposition and a fear of the impact of continued globalization on American wages. The Obama administration has pursued the deals, viewing them both as positive for American exporters and as part of a geopolitical strategy of creating stronger ties with Asian nations that can counter Chinese influence.

So the Obama administration may have better luck with the Republican Senate than it would have with a Democrat-led chamber in obtaining passage of the Trans-Pacific Partnership, a sweeping trade agreement with 11 countries stretching from Japan to New Zealand to Chile.

That said, there are many hurdles to getting a complicated deal done. They need to finesse a two-step process in which Mr. Obama first wins “trade promotion authority” to fast-track negotiations and then separately secures passage of a finished agreement, which has many moving parts. Early signs are that the Republican Congress could move on that authority early next year.

Republicans will probably demand greater heed to businesses’ wishes than the organized labor and environmental concerns that are priorities for Obama. And with just over two years left in his presidency, the clock is ticking.

A second trade agreement, with Europe, known as the Transatlantic Trade and Investment Partnership, is moving more slowly and will be even more challenging for Mr. Obama and congressional Republicans to enact in the coming two years.

Is it time to overhaul the corporate income tax?

The tax system for American corporations is, by all accounts, a mess. It places higher tax rates on businesses than most other advanced nations, yet raises less revenue relative to the economy thanks to a byzantine set of carve-outs and special deductions. It makes possible so-called corporate inversions, in which American firms cut their tax bill by buying a smaller company overseas and relocating their headquarters. The tax code hasn’t been systematically reformed since 1986.

In other words, it would seem ready made for a bipartisan compromise to cut the rate, cut the deductions, and make the system overall more fair and efficient. It is easy in theory to imagine a bill that a Republican Congress might pass that President Obama would happily sign. “The political noise on inversions obscures the broad similarity of the political parties’ approach to tax reform,” wrote Terry Haines, head of political analysis at ISI Evercore, in a report. “If anything, the continued inversions controversy continues to provide a powerful stimulus for both parties to address the fundamental noncompetitiveness of the U.S. tax code.”

The challenge is not the theory, but the practical dimensions.

Tax reform legislation is almost by definition a hard slog even when the parties broadly agree, pitting lobbyists for companies that win from the changes against those who would lose. Things are also complicated by Republicans’ desire to keep reform of the individual income tax code on the table as part of the process, with an eye toward reducing tax rates — most likely a nonstarter for Mr. Obama and other Democrats.

Some less likely policies.

There are other areas that superficially would seem to fit the model of a policy area where there is room for bipartisan compromise, but in which legislation appears unlikely to make it to President Obama’s desk before he leaves office.

First, immigration reform. Although the Senate passed a bipartisan immigration bill last year, it has been bogged down in the House as conservative members object to what they see as amnesty for illegal immigrants. While the bill, or one like it, has strong support in the business community, as a practical matter it’s not clear that anything has happened that would change that dynamic. If anything, Congress has become more conservative.

Second, housing finance reform. Fannie Mae and Freddie Mac have been under government control since 2008, leaving the government managing a vast proportion of the mortgage market.

But Democrats and Republicans remain far apart on how much government support should exist for home lending in any deal, and neither side shows much interest in compromising. Throw in the fact that the status quo is working O.K., making a lot of money for taxpayers and keeping a flow of cheap money into housing that makes the politically powerful real estate industry happy, and it’s not clear what would force a compromise.

Given those realities, writes Brian Gardner, an analyst at Keefe, Bruyette & Woods, a financial services consultancy, “we remain confident that mortgage finance legislation is unlikely to pass Congress until 2017 at the earliest.”

That’s what everyone wants to hear: confidence in Washington.