In his first press conference after winning re-election in 2004, George W Bush made it clear that he would pursue an ambitious conservative agenda in his second term.

"I earned capital in the campaign, political capital," he said. "Now I intend to spend it."

Barack Obama did not show his hand. On the night of his comprehensive electoral college victory, he was content to play a favourite record, by now badly scratched: "We are not as divided as our politics suggest." Last Friday, at the White House, he again presented himself as a conciliator. "I'm not wedded to every detail of my plan. I'm open to compromise. I'm open to new ideas," he said.

The response from some Republicans has been to deny that Obama's re-election was a victory at all, because the popular vote was so evenly split and Democrats failed to recapture the House of Representatives. "It's not a mandate. It's a second chance," concluded the New York Post.

This is either spin or denial. Despite unemployment close to 8%, Obama won all of the swing states, albeit by narrow margins. Democrats increased their majority in the Senate when many had forecast they would lose it. New Senators Elizabeth Warren in Massachusetts and Tammy Baldwin in Wisconsin are both unabashed liberals.

In Maine and Maryland, measures legalising same-sex marriage were approved by voters. Colorado and Washington voted to regulate and tax marijuana. A younger, more diverse electorate delivered a string of small victories to the Democratic party's liberal wing.

The House of Representatives results are misleading because of partisan redistricting after Republicans swept the board in the 2010 midterm elections. In Ohio, Virginia and Pennsylvania, gerrymandered boundaries packed the Democratic vote into a few deep blue districts. Obama carried all three states, but Republicans took 12 of 16 seats in Ohio, 11 of 14 in Virginia and 13 of 18 in Pennsylvania. In Illinois, where Democrats drew the new map, they won 12 of 18 seats. Nationally, the Democratic party won half-a-million more votes than the GOP in elections to the House, but is still in a minority, 192 seats to 233.

The morning after, the right- wing National Review was in no mood to sugar-coat: "Conservatives suffered a terrible defeat last night, and there is no point pretending otherwise. President Obama won with an improving but still weak economy, and while running a campaign that was quite liberal by historical standards."

In short, Obama earned political capital. How will he spend it? He must do so soon, because re-elected presidents typically lose significant numbers of House and Senate seats in their second midterm election, and by the summer of 2014, Congress will be pre-occupied with the campaign.

Bush chose to reform social security but was thwarted by implacable Democratic opposition. "If I had to do it all over again, I would have pushed for immigration reform, rather than social security, as the first major initiative of my second term," he wrote in his memoir, Decision Points.

Obama's first priority will be the impending crisis variously referred to as "taxmaggedon" or the "fiscal cliff," in which the Bush tax cuts and a payroll tax cut expire on the same day $1 trillion of spending cuts take effect. The Congressional Budget Office has estimated that the combination of higher taxes and 10% cuts to every government programme apart from Medicare and social security will reduce GDP by 4% and throw the economy back into recession.

To avert this, the president and Congress must strike a deficit reduction deal before January 1. This is something they have tried and failed to do several times – the automatic cuts result from the collapse of a previous attempt. The question is whether Republicans are more likely to compromise in the post-election political environment.

"It was a very close race and it showed a country that's still very divided," said Will Marshall, president of the Progressive Policy Institute. "But I think that the president does have a specific mandate for an end to tax breaks for the rich." Taxation was the one area in which Obama struck a more confrontational pose during his first post-election press conference. "This was a central question during the election. It was debated over and over again," he said. "And on Tuesday night we found out that the majority of Americans agree with my approach."

There is reason for scepticism. Obama has been promising to raise taxes on the rich for five years. His position is much stronger now, though, than it was the last time the tax cuts were extended. Then his approval rating was at 45%, unemployment was threatening to climb above 10% and his party had just suffered an overwhelming loss in the midterm elections.

Most importantly, if congressional Republicans are unwilling to sanction tax rises as part of the "grand bargain" Obama favours, taxes will go up in the New Year automatically. The spending cuts that will take effect at the same time are severe but not instantaneous. In these conditions, Obama would certainly get a much better deal than the one leading House Republicans John Boehner and Eric Cantor walked away from last year, in which the president agreed to deep cuts to Medicare and social security for just $800 billion in additional revenue.

"If Speaker Boehner's attempts to sober up his troops in the House fail and they refuse to countenance a tax hike of one penny for deficit reduction, the president will have to use his leverage – and he's got a lot of it," Marshall said.

Boehner sounded a conciliatory note after the election: "Mr President, this is your moment. We're ready to be led – not as Democrats or Republicans, but as Americans." He suggested he was open to "revenue increases," but not higher taxes. "The American people have also made clear that there is no mandate for raising tax rates," he said.

Senate Minority leader Mitch McConnell said "we're there to meet him half way," but added that "voters have not endorsed the failures or excesses of the president's first term".

Obama has often expressed a belief that "the fever may break" once his adversaries are confronted with the political cost of obstruction "because there's a tradition in the Republican party of more common sense than that". This may be wishful thinking. The day after the election, Republican Senator James Inhofe wrote that "President Obama's far-left agenda must be stopped, and it's going to be up to Congress to apply the brakes to a runaway bureaucracy."

Many, nonetheless, believe a deal is inevitable. "It doesn't seem like there's much of a will to not raise taxes on upper-income people," said Adam Brandon, executive vice-president of FreedomWorks, a Tea Party pressure group. "I think you start with 'everything's on the table,' and while Americans want to see compromise they also want us to stand on principle. If we're going to vote to raise taxes you'd better believe that we want to see spending cuts that are happening right now."

There will be a bitter debate among congressional Republicans about what is and is not negotiable. Before the election, conservative talk radio host Rush Limbaugh predicted that "if Obama wins, it's the end of the Republican Party". Moderate Republicans blame hardliners for alienating swing voters, while the far right contends that a pure conservative candidate would have had a clearer case to make than Romney.

The next flashpoint is likely to be immigration reform. Obama has said he will promote legislation creating a "path to citizenship" for undocumented immigrants. To the nativist wing of the Grand Old Party, this is an "amnesty" to be resisted at all costs. But after a presidential election in which the votes of Latinos were decisive in Florida, Nevada and Colorado, both parties have an incentive to address the issue.

"Republicans realise that having systematically alienated Hispanic voters they've painted themselves into a demographic corner," said Marshall. "Conversely, having won 71% of the Latino vote, President Obama is going to feel absolutely obliged to pass comprehensive immigration reform. And again, the burden will be on Republicans to decide whether they will let their extremists call the play."

Since the election, influential conservative voices have called for the party to embrace reform. "Must have sweeping, generous immigration reform, make existing law-abiding Hispanics welcome," tweeted Rupert Murdoch.

Boehner told ABC news that "a comprehensive approach is long overdue, and I'm confident that the president, myself, others can find the common ground to take care of this issue once and for all". This is anathema to many members of his caucus, but political necessity will prevail.

The one other area in which a grand legacy project might tempt the president is energy policy. In the aftermath of Hurricane Sandy, New York's Mayor Michael Bloomberg reluctantly endorsed Obama, citing climate change as the defining issue. Obama funded alternative energy projects through the economic stimulus package, but his much-mocked hubris after winning the Democratic nomination – "this was the moment when the rise of the oceans began to slow and our planet began to heal" – has not been matched by aggressive policy to reduce greenhouse gas emissions.

"I think the Democratic party feels that people just don't care about climate change," said Daphne Wysham of the Institute for Policy Research. "What we've learned about Obama after four years of dealing with him is that he's not prepared to go out on a limb unless people are pushing him very strongly."

During the election campaign, Obama stopped talking about climate change entirely and instead expressed his strong support for the natural gas industry. Environmentalists hope that now he no longer has to worry about protecting votes in the mining belt of Ohio and Pennsylvania, the president will start to think about his environmental legacy. They are likely to be disappointed.

"There just isn't support for a standalone climate change bill that's going to raise energy costs and provide no benefit other than reduction in greenhouse gas emissions," said Marshall. "Voters care about the issue but it's not the salient one now."

Obama has a daunting to-do list in the first few weeks of his second term. He must replace key members of his team, including Secretary of State Hillary Clinton, Treasury Secretary Tim Geithner, Defence Secretary Leon Panetta and political adviser David Plouffe, all of whom have said they will step down. On Friday, CIA director David Petraeus resigned unexpectedly after confessing to an extra-marital affair, adding to the list.

The day after his victory, Obama addressed staff in Chicago. "Even before last night's results, I felt like the work that I'd done in running for office had come full circle," he told them, his voice cracking with emotion, a tear in his eye. He has the four more years that he fought so hard for, and no time to rest.