# Why is ANNUAL exam cost per life-year saved not three times higher than TRIENNIAL?

In an introductory text on micro- and macroeconomics, Economics: Principles and Applications, 5th ed. by Hall and Lieberman, at the end of Ch. 2 I've found the following question:

One might think that performing a mammogram once each year—as opposed to once every three years—would triple the cost per life saved. But according to Table 6 [see below], performing the exam annually raises the cost per life-year saved by about 40 times. Does this make sense? Explain.

Table 6:

• Mammograms: Once every three years, for ages 50–64: Cost per life-year saved: 2,700
• Mammograms: Annually, for ages 50–64: cost per life-year saved: 108,401

Note, $$\frac{108,401}{2,700} \approx 40$$.

#### Official solution:

I've been trying to wrap my mind around this, and despite the solution for the 5th ed. provided by the authors, I still don't get it. The solution:

Yes. If women are screened every year, instead of every three years, total mammogram costs triple. For the cost per life-year saved to remain constant, the extra screening would have to triple the number of cancerous, life-threatening tumors found. A 40% [sic] increase in the cost per life-year saved indicates that the extra screening leads to the detection of fewer than triple extra cancerous, life-threatening tumors.

#### My attempt (stuck):

Assume a medical exam costs $$\\\100$$, and $$100$$ people get examined during an interval of $$15$$ years (ages $$50$$ to $$64$$). $$20$$ of these people have a tumor, which is discovered on the first test, and all the following years (assume they each live an extra $$25$$ years) will be counted towards life-years saved. They all get cured, and they all keep getting tested. (I'm ignoring the fencepost error here, and assuming there are 5 exams, not 6, every 3 years within a 15 year period, and 15 exams, not 16, within the same period every year.)

Then,

• for annual exams, $$100~ \text{people} \times 15~ \text{exams} \times 100 = 150,000$$ in total costs
• for triennial exams, $$100~ \text{people} \times 5~ \text{exams} \times 100 = 50,000$$ in total costs.

Dividing by the life-years saved, we get the cost per life-year saved:

• annual exam: $$\frac{150,000}{25\times 20} = 300$$
• triennial exam: $$\frac{50,000}{25\times 20} = 100$$

If the triennial exam saves fewer people, e.g. out of the $$20$$ assumed saved above, only $$10$$ are, the cost per life-year saved increases, since $$\frac{50,000 - 10\times 500}{25\times 10} = 180$$, and if the annual exam saves more lives, its cost per life-year saved decreases. Either way, the ratio decreases from the current 3.

It would seem that more screening results in more missed tumors (in which case the cost per life-year saved would increase for annual exams, and the ratio might be higher than 3), which makes no sense.

The reason why there's a $$4000\%$$ (i.e. $$\times~ 40$$) increase in the cost per life-year saved is presumably that more frequent screenings do not allow for malignant tumors to develop, and that removing/monitoring/treating benign ones is not counted towards the life-years saved count. (On the other hand, less frequent screenings most often catch later-stage, malignant tumors, which when removed directly count towards the life-years saved.)
Interestingly, the (unsubstantiated) $$40\%$$-increase figure given in the official answer would be plausible in the fictitious case in which only malignant tumors existed. There, one might expect that annual exams come up short of finding three times more of such tumors compared to triennial exams, which would account for a figure between $$0$$ and $$300\%$$ increase in the cost per life-year saved (with $$300\%$$ meaning triennial exams find all the tumors, and $$0\%$$ meaning annual exams find three times more).