As I was driving to school today, there was a story on NPR about an upcoming ballot initiative in Missouri. Proponents want to raise the cigarette tax from $0.17 per pack (the lowest rate in the nation) to $0.90 per pack. The idea is to generate some revenue and at the same time discourage smoking.

Missourians love their cancer sticks.

Now, I’m not going to go into the pros and “cons” of the initiative. I would be *for* it, based on some pretty common-sense data, but I don’t live in Missouri so I guess my opinion doesn’t matter. However, I’d like to comment on some squishy reasoning put forth on the radio by an opponent of the measure.

In a nutshell: “The measure,” the opponent says, “wouldn’t raise money at all. It would actually lower revenue. That’s because any gains made by raising the tax would lower the amount of regular sales tax accrued.” [Note: this isn’t a verbatim quote but a recreation based on my imperfect memory]

You see, Missouri has a 4.225% sales tax which also applies to cigarettes. And, the “logic” goes, if less people are buying cigarettes, the less sales tax revenue would be generated.

Really? Is this person completely innumerate? Let’s say that a pack of cigarettes costs $6.00. Without the ballot initiative, a Missourian would pay $6.00, plus $0.25 in sales tax, plus $0.17 in cigarette tax, for a total of $6.42. The state gets $0.42 cents a pack.

Under the new plan, a smoker pays $6.00, plus $0.25 in sales tax, plus $0.90 in cigarette tax, for a total of $7.15. The state gets $1.15 cents a pack, obviously an increase.

To be fair, the state *does* actually lose money if the gain in revenue per pack is offset by the loss in the number of sales. But when would this occur? Let’s use algebra to find out. Suppose that *N _{1}* is the number of packs of cigarettes bought in Missouri in a given year. Under the current system, and for a $6.00 average price per pack, the yearly intake for the state is just

*N*x 0.42. Now let

_{1 }*N*be the number of packs bought under the new plan. It’s easy to see that if the proposal passes, the yearly intake for the state is

_{2 }*N*x 1.15.

_{2 }The problem is that *N _{2}* <

*N*(presumably). So the proposal loses money if

_{1}*N*x 0.42 >

_{1 }*N*x 1.15, which, after one line of 8

_{2 }^{th}-grade algebra, is the same as

*N*> 2.74. That is, translated into English,

_{1}/ N_{2}**the new proposal loses money compared to the current system if state cigarette purchasing goes down by a factor of almost 3.**

I don’t know about you, but I seriously doubt this initiative will cut smoking in Missouri by that much.

The numerically savvy will notice that there is a subtlety. This result is for a $6.00 pack of cigarettes. What if the price is much different? This matters, because the cigarette tax is a flat number added to a pack, whereas the sales tax is a percentage. If cigarettes cost the same as houses the ballot initiative would be ridiculous, since 4.225% of a huge number is much, much greater than a measly $0.90. No one would buy cigarettes at all, and the state would lose a lot of revenue.

Is there a tipping point? That is, is there a price for a pack of cigarettes for which the proposal loses money for *any *decrease in purchasing? Surprisingly, the answer is no. Mathematically, we would say that *N _{1}/ N_{2}* = 1 only when the price of a pack is infinite.

But setting the bar at *N _{1}/ N_{2}* = 1 is unrealistic. Any tax hike will cause

*some*smokers to buy less. So let’s make an educated guess: looking at the first graph at this site, I can see that a price hike of $0.73 produced a corresponding decrease in consumption of about 25%. (The actual drop would probably be less than this, since in the 1980’s and 1990’s $0.73 represented a higher

*percentage*of the cost of one pack). So we let

*N*= 1/0.75 = 1.33, and solve for the price of cigarettes.

_{1}/ N_{2}You get $48.33. For one pack of cigarettes.

Conclusion: **the proposed tax increase would increase revenue even with a plausible decrease in sales, unless cigarettes cost around $48 or more per pack**.

Yay algebra!

Some close analogue to Mark Twain or E. B. White (with so many misquotes on the web, you can’t be very sure these days) said that analyzing humor is like dissecting a frog: you don’t learn much, and the frog dies. So why did I dissect this frog, and show all the gory math details?

I wanted to point out that there is a lot more mathematical detail in most stories you hear, most issues you examine, than you suspect. And I want to emphasize: if you don’t know math, if you are not mathematically literate, then you don’t know much of anything.

There. I said it.

Go study your algebra.

on November 8, 2012 at 10:22 AM |David MartinoExcellent analysis. I defer to your math acumen, but how is the analysis changed if you factor in that, particularly on the east coast (smaller states), at a certain price, people may cross borders to buy a $6 pack of cigarettes than spend $47 in their home state? I guess this is a question about black markets.

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