The ex-New York mayor and former Massachusetts governor each claimed to be more committed to lowering taxes, as they competed for conservative support.
The home of the auto industry, Michigan has the highest unemployment in the US.
Former actor and Tennessee Senator Fred Thompson made his presidential debate debut and was given the first question.
Better known to many for his role in the television series Law & Order, Mr Thompson only joined the race for the nomination in September, several months after his rivals.
He was criticised for choosing to announce his candidacy on the Tonight Show with Jay Leno on the day that his competitors were taking part in a debate in New Hampshire.
'Baloney'
Tuesday's debate was expected to provide a platform for Mr Thompson to promote his candidacy, but he remained a bystander during a heated confrontation between two of his rivals.
| | REPUBLICAN CANDIDATES Sam Brownback Rudolph Giuliani Mike Huckabee Duncan Hunter John McCain Mitt Romney Ron Paul Tom Tancredo Fred Thompson |
The clash between Mr Giuliani and Mr Romney began when the Republican frontrunners were asked to discuss their differences over economic issues.
"I cut taxes 23 times. I believe in tax cuts," Mr Giuliani said, referring to the two terms he served as mayor of New York.
Mr Giuliani said he had lowered taxes by 17% during this period, whilst Mr Romney had let them increase by 11% when he was governor of Massachusetts.
"The point is that you've got to control taxes. I did it, he didn't... I led, he lagged," he added.
Mr Romney dismissed the claim, insisting it was "baloney".
"Mayor, you've got to check your facts. I did not increase taxes in Massachusetts - I lowered taxes," he said.
While Mr Giuliani leads in national polls, Mr Romney is ahead in the key early voting states of Iowa and New Hampshire.
Thompson debut
When asked about the economy, Mr Thompson warned that while it was strong today, the government was "spending money of future generations and those yet to be born".
| | [Like Law & Order, the Republican contest] has a huge cast, the series seems to go on forever, and Fred Thompson shows up at the end Mitt Romney |
"Everyone knows that we have to address that, and it's the fundamental and foremost challenge, I think, facing our country economically," he added.
Mr Thompson also criticised recent calls for the imposition of blanket restrictions and duties on Chinese imports.
He said he favoured limiting imports of potentially dangerous materials but "turning our back on free trade, that's not the direction to go in".
Mr Thompson was later the target of a joke by Mr Romney, who likened the former actor's late entry into the Republican presidential contest to Law & Order.
"It has a huge cast. The series seems to go on forever. And Fred Thompson shows up at the end," Mr Romney said.
"And to think I was going to be the best actor on the stage," Mr Thompson retorted.
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