### How much money will "Hunger Games: Catching Fire" make?

The new "Hunger Games" movie is opening this Friday. Get ready for more Jennifer Lawrence killing everything in sight:

 Credit: Lionsgate

The first "Hunger Games" film made \$408 million at the US box office. Almost half of that total was made in just the opening week! Let's take a look at the graph of how much money the first film made on a weekly basis, to see if we can predict how the second film will do.

This looks a lot like a decaying exponential function, which can be written in the form Ce-t/τ, where C is the amplitude of the function, and τ is the time constant. Another way to think about this is that each week, the film makes a fraction of the money it made the week before. That means we can treat the weekly data as a geometric series!

The formula for the sum of a geometric series is A/(1-r), where A is the first term in the series (i.e., how much money the film makes in the first week), and r is the ratio from one week to the next. For example, if a film makes \$100 million in its first week, and \$60 million in its second week, then r = 0.6, which is pretty high for a blockbuster release. Larger values of r mean that a film has a longer "lifetime" in the theaters -- people are still buying tickets later into the film's run. After "Catching Fire" is out for a week, we'll know exactly what A is. But how can we determine what r is?

To estimate r for "Catching Fire," let's look at how some similar movies did in the past. Let's specifically take the total amount of money each film made in its first week and divide that by how much it made in total, and let's call this ratio w (for opening week). So then:

So using a film's opening week and total intake (or "gross"), we can estimate r. Let's see how a few different film franchises compared:

For each of these graphs, there seems to be a downward trend, meaning that later films in each franchise have shorter lifetimes than earlier films. But each of these franchises behaves a little differently. For example, The Harry Potter franchise steadily declined over its 8 films, while the Twilight films plummeted after the first one. Maybe after the first film, only the "Twihards" saw the remaining movies, often in the first few weeks of release (resulting in a low value of r).

With more study and more data from other films, we could generate a probability distribution for the value of r for "Catching Fire." But for now, we can guess that it'll probably be less than r for the first "Hunger Games" film, which was ~0.53. Given the positive buzz the new movie is getting, let's say r is a relatively healthy 0.5. So to figure out the total amount of money this second film makes, take the gross from its opening week, and divide it by (1-0.5). In other words, double it!

Here's our best guess for "Catching Fire":

Opening weekend (first 3 days) gross: \$160 million
Opening week (first 7 days) gross: \$220 million
Total gross: \$440 million (double the opening week)

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