The Betfair Prof / Leighton Vaughan Williams / 20 May 2009 /

The Betfair Prof looks at what would have happened if John McCain made it to the White House...

On September 4th, 2008, Senator John McCain of Arizona delivered his acceptance speech at the Minnesota-St Paul Republican convention as the party's nominee for President of the United States. For a fleeting few days thereafter it looked as if he and his Alaskan running-mate might actually win the keys to the White House. Indeed, for a short while around this time, the respected political and election forecasting web-site, www.fivethirtyeight.com, had Mr. McCain as slight favourite to succeed George Bush, and the prediction markets were never to see it closer. For the record, Betfair traders were never quite as convinced as most others in the world of election forecasting, but it was getting closer.

It was quite reasonable, therefore, that on September 9th, Greg Mankiw, Professor of economics at Harvard University, should ask the prediction markets to reveal what kind of tax policy the US would get if John McCain should be elected President. We knew the Republican candidate's stated policy, which was that of making the Bush tax cuts for the wealthy permanent. But what would he actually do if given the opportunity?

For this Professor Mankiw looked to an esoteric Intrade market about the likelihood of the top US income tax rate in 2011 exceeding 38%. He calls this the "Probability of a Tax Hike", i.e. P (tax hike). It was already part of Barack Obama's policy platform to introduce this scale of taxation, and politicians do not tend to under-state how much they are likely to increase taxes, and so given the very likely shape of the Congress, Mankiw attributed the probability of such a tax hike in an Obama administration as a sure thing, i.e. P (tax hike) = 1. Using a complementary estimate of the probability of Mr. Obama being elected President of 53% (0.53), he is able to create two new probability scores, i.e. P (Obama) = 0.53 and P (McCain) = 0.47. Now comes the interesting part of the exercise, which is the use of conditional probabilities to estimate the likelihood of the tax rise under a McCain Presidency.

The formula is straightforward. Probability of a tax hike , i.e. P (tax hike) = Probability of a tax hike GIVEN THAT Obama wins times the Probability of an Obama win, plus the Probability of a tax hike GIVEN THAT McCain wins times the Probability of a McCain win. Using conventional symbols, this can be written as: P (tax hike) = P (tax hike I Obama) P (Obama) + P (tax hike I McCain) P (McCain). We have numbers for all of these variables except for the probability of a tax hike by a President McCain. It's a matter of simple arithmetic, therefore, to deduce what the prediction market says about the likelihood of this. It can be calculated as 0.87 minus 0.53 all divided by 0.47, or 72%. If we reduce the probability of an Obama tax hike from 1 to, say, 0.9, this simply increases the implied probability of a tax increase by McCain (in this particular case, from 72% to 84%).

So would John McCain have raised the top rate of income tax if he'd been elected? Well, the conditional probabilities have been consulted and they've given their answer. Yes he could have, and in all likelihood, yes he would have!

Professor Leighton Vaughan Williams is the Director of the Political Forecasting Unit and Betting Research Unit of Nottingham Business School, Nottingham Trent University

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