The Times reports the sad and scary case of the spread of a superbug at UCLA as an invasive piece of medical equipment was used on multiple patients without being properly sterilized after each use. Economic risk analysis can be used here to analyze the tradeoffs between costs incurred in purchasing durable equipment and the expected loss in life from reusing the equipment.
Suppose that it costs $F to purchase one of these devices and that it has a marginal cost of $c to clean it each time. Suppose that this device will be used n times before being thrown away. Assume the interest rate = 0%. Under these assumptions, the total cost of using this device n times = F + c*n.
Note the economies of scale here that the more it is used the average fixed cost declines.
Now define p(n) as the probability of a superbug infection that kills an individual and assume that p(0) = 0 and that p(n) is an increasing function of n such that the more the equipment is used that this raises the probability of a superbug infection disaster for the next patient. Suppose that each person for who the equipment is used on values her life by $V.
What is the optimal number of times to use the equipment if the patients at the hospital are risk neutral?
Suppose you set n = 1, in this case --- nobody dies from the superbug infection but the hospital bears a cost of $F each time and for N total procedures; the total cost is N*F
suppose you set n = 2, then the total cost to society of treating the two patients = F + 2*c + p(1)*V
the first person the new piece of equipment is used on has no chance of dying but the 2nd person does have a chance to die;
suppose you set n = 3 , then the total cost to society of treating the three patients = F + 3*c + p(1)*V + p(2)*V
The general problem:
choose n to minimize F + n*c + sum from 0 to n of p(n-1)*V
UPDATE: A reader named "Glenn" correctly pointed out that I need to be clear about the aggregate demand. For example, if the hospital must conduct 100 procedures, what is the optimal configuration here to minimize the cost of achieving that goal.
A researcher who knows the parameters; F, c, p() and V could solve this economy and figure out the optimal reuse.
Note the comparative statics, as the value of a statistical life goes up (V), the hospital should not reuse equipment. As demand for 1 time use equipment rises, some firm would create this and this would lower F. Of course, the ideal would be to set p(n) = 0 for all n but that is asking for a free lunch.
This is a classic Gary Becker risk tradeoff model similar to his crime deterrence work.
Now, you might ask; does the hospital know the p(n) function? If they knew that they did not know the function, what issues does that raise?