Austin Frakt has a very insightful post on the growth of health care premiums for low and high deductible plans. He shows in a simple case that the percentage growth in the health care premium for a high deductible plan is higher than for a low deductible plan.
His framework is to assume fixed dollar deductibles and a growth in the underlying cost of health procedures. He avoids use of probability and works in a world of certainty with perfect foresight by consumers and companies.
The deductibles are 250 and 1000 dollars. The procedure costs 5000 initially. The procedure’s cost grows 6.3 percent per year. The premium on each plan equals the procedure’s cost minus the deductible. There is no interest and no expenses.
This example comes from the company Aon Hewitt.
The graph of premium growth rates shows the two growth rates of the plans (which vary by deductible) converging.
This problem can be used in algebra class in a variety of ways. It can also be used for admission tests to schools like Thomas Jefferson High School since it is simple but counter-intuitive.
Using algebra to prove the result is an interesting use of letters to prove a theorem.
Probability can be introduced by making the chance of needing the procedure be random. The cost of the procedure also could be made random, so a frequency and severity model. The impact of earning interest can be introduced. Premium payments at one year or one month can be compared and the timing of the payment considered. This interacts with the interest rate.
Students can vary the problem by finding what growth rate combinations of the two deductibles will result in the same growth rate in their premiums. This too could be an admissions test problem at Thomas Jefferson High School. It is simple yet tricky. Just the sort of problem desired for creative math types.