# The GM scandal: a little math

The House Committee on Energy and Commerce is holding a hearing on whether GM delayed too long before recalling cars with defective ignition switches. It was revealed at the hearing that the cost of the replacement part is only 57 cents, and this has led to calls for the heads of those responsible. This, from Michael Moore, is typical:

I hope the criminals at General Motors will be arrested and made to pay for their premeditated decision to take human lives for a lousy ten bucks. The executives at GM knew for 13 years that their cars had a defective ignition switch that would, well, kill people. But they did a “cost-benefit analysis” and concluded that paying off the deceased’s relatives was going to be cheaper than having to install a \$10 part per car.

The \$10 figure is from a Washington Post article; the 57 cent figure can be found in other sources.

I was curious about whether it is true that GM’s decision to install the switch (rather than replace it with a better switch) or not to recall the car in 2007 really would have failed a cost-benefit analysis. Here are some back-of-the-envelope calculations, based on (still murky) details from a House report and newspaper stories.

Let’s suppose that back in 2005, which was the model year for the first car with the ignition switch problem, GM knew that it would install that switch in 2.6 million cars (the eventual number that would be recalled). Those cars were sold over the next 9 years. If we assume a constant number of cars sold per year, we get 290,000 cars sold per year. A little additional math shows that this amounts to 13,050,000 car-years, by which I mean the number of years in which a car with the ignition switch was owned and driven by someone.

During this period, the switch problem caused accidents that caused 13 deaths, or 0.000001 death per car-year. It is worth noting parenthetically that the probability of an average person in an average car being killed in any year is 0.0000556. So the effect of buying a GM car with an ignition switch problem increased the risk from 0.0000556 to 0.0000566.

Still, no one wants to take an unnecessary risk of death, however small. But how much would you be willing to pay to reduce the risk of dying from a car accident by 0.000001 in one year? One way to get the answer is to use the U.S. government’s valuation of a statistical life, which was around \$6 million when these decisions were made. (That number is based on data that measures how much people are willing to pay to avoid very small risks of death, as in our example.) Another is to look at tort awards, which at the time averaged about \$3 million for deaths. Let’s call it \$4.5 million. And assume that the average buyer drives the car for 10 years. Multiplying these numbers together, a person should be willing to pay about \$45 to avoid this risk of death (\$4.5 million * 0.000001 per year * 10 years).

Now if GM could have fixed the problem by using a 57 cent or \$10 switch at the time of manufacturing, it should have. But it appears not to have recognized the problem until most of the cars were on the road. The Washington Post says that repairing the car would take less than an hour. Let’s call it half an hour, and assume \$100 per hour in labor costs, which is roughly consistent with the CEO’s testimony that it would have cost GM \$100 million to  recall the 2.6 million cars on the road. If all this is true, the decision not to recall–saving \$50 to avoid an expected cost of \$45 per car–seems reasonable, although as we have seen the proper measurement of a loss of life is controversial. The bottom line is that GM could very well have complied with a reasonable cost-benefit analysis; if not, it was probably close. To know the truth, we need more data than are publicly available. So why the headlines? (That’s a rhetorical question.)