So while we will sadly never know how a debate between Paul and Obama would look like, here is how Paul would have likely responded to the debate questions and actually provided Americans a real alternative.
What are the major differences between the two of you about how you would go about creating new jobs?
Obama responded with an argument that he uses frequently. The president likes to remind us that he inherited a terrible mess (which is entirely true), and that his policies helped prevent a disaster, created millions of jobs, saved the auto industry, and that the housing industry is getting back on its feet.
Obama then criticized Romney for supposedly wanting to undue his progressive policies and legislation by cutting taxes, deregulating, and blowing holes in the budget, and Romney predictably ran away from this (false) caricature.
Paul would undoubtedly take his time to criticize the president's policies but most importantly explain that without a proper diagnosis of the disease, a cure is impossible. The cause of the "worst economic crisis since the Great Depression" are precisely the policies that the president and virtually every administration before him have pursued.
Specifically, it was the Federal Reserve's artificially lower interest rates combined with congressional mandates to flood this cheap money into the housing industry that created a predictable bubble that finally burst. Generally, it has been the policy of the U.S. government, in combination with the Federal Reserve, to finance debt and deficits by the creation of money, pyramiding debt, and dumping the cost on the taxpayer.
What will you do about the federal deficit/debt?
Obama bragged about his supposed "deficit reduction" plan that would cut $4 trillion and his trimming of waste from the military budget. Even if one were to believe that the president honestly plans to cut a penny anywhere, Paul would fire back with a bit of truth to counter the president's propaganda. Paul has been in Congress for decades, and he sees firsthand the accounting tricks and numbers that are used by both sides to fit their agendas.
Paul always points out how supposed "cuts" are really just cuts on the rate of increases in the federal budget. If a healthy person is supposed to consume 2,000 calories a day, and one promises to eat 9,995 calories a day after years of 10,000 per day, can that really be considered a diet? Same goes for the budget and any promises of fiscal responsibility.
For example, Republicans attack the president for "gutting the military" while Democrats attack Republicans for their supposed cuts to social welfare programs. Both sides are wrong because a real cut would mean spending less money than the previous year. Paul would also likely point out that Obama's military budgets have increased every single year of his administration.
Paul often says that he thinks politicians in Washington in denial about the fiscal and economic mess the U.S. is in. With $16 trillion in on-the-books debt, $221 trillion in future unfunded liabilities, and a dollar that plunges in value because the Fed keeps printing up the money to finance this nonsense, Paul may have a point.