Of Presidents and Differential Equations

I was talking with a friend the other day and he started complaining about how everyone that was for the war in Iraq is now against it, mainly because the media is against it. To him, the way the media was able to control the opinions of Americans is wrong. Which it is. Really though you have to wonder who is actually in control of the beast.

It is kind of fascinating, as a person who has always been against the war (and Bush for that matter) watching opinion turn against him. If you look at the polls there has been a fairly constant and steady decline in Bush's approval ratings ever since 9/11. It would be great fun to fit an exponential decay curve to the data. I sense that there is some sort of fundamental truth wrapped up in the decay constant of the curve. In fact, lets do this.

There was a fairly big bump around the 2004 election but immediately afterword it continued downward and is only now starting to level out. Immediately following 9/11 his approval rating was roughly 87%. It reached 45% on 5/20/2004. That is roughly 2.75 years for roughly one half decrease in popularity giving a decay constant of 0.25 (years^-1). Following the election his popularity again started falling starting at about 50% on 1/20/2005 and reaching an average of 28% on 5/20/2007 giving a drop of 22% over 2.4 years. The calculation is a little more difficult but it leads to a decay constant of .24 (years^-1). Pretty dang close.

Whats great is that we now have a model for the average approval rating over time which is given by the following equation: A = 50*e^(-(date - 1/20/2005)*.25). And with a model we can predict future approval ratings. Lets look at his approval rating in November when General Potreas is giving his report on the effectiveness of the Surge in Iraq A = 50*e^(-2.8*.25) = 25%. Looking even further towards the primary season would give an approval rating of only 23%.

The question is, will this trend continue? To answer this we have to look at our model and ask what could be driving this behavior and if it will continue.


What the models are great for is giving an abstract way of looking at relationships between quantities. What is interesting about exponential decay is it corresponds to a differential equation where the rate of change of a quantity is equal to the the quantity times a constant. In other words, the rate at which Bush's popularity is decreasing is directly proportional to his current popularity.

What popularity is a measure of is the number of people who approve of his performance. So in this sense it makes sense that the rate of change of his popularity is proportional to the number of people who approve. The more people there are who do approve of him means the more people who are able switch and disprove of him. What is hidden in this discovery is that, in reality, there is no one who should approve of president Bush. Given enough time, everyone will come around to the fact that he is a bad president and his approval rating will approach 0%.

Indeed, this is likely the case. Bush's popularity is almost entirely derived from his perception as the boss. I remember a good friend who is very smart saying prior to the Iraq war that we must support our president. And, in a sense he was right. The country thought we were at war with a dangerous and amorphous enemy. Divided as a country we may fall to this threat and Bush did an excellent job of placing himself as a rallying point for people with his strong language, determination, and apparent self confidence. Ultimately though, this is a feeling based on the perception of a threat which has not materialized. Our way of life has continued and indeed has been more threatened by things that Bush has done nothing about such as increasing oil prices, global warming, Katrina, health care costs, apparent increasing gap between rich and poor, and so on. It is hard to name a group of people that actually have benefited from his presidency.

These next few months then should be interesting. Barring any major event that will cause fear of an external enemy his poll numbers will serve as a test of our model. Specifically, if his poll numbers continue to fall at this rate then we can become more confident that ultimately everyone, even the thick headed ones, will come to believe that Bush is truly the worst president we have ever had.