Question: Assume that there exists an unlimited number of different approaches to developing a new drug, each costing $1. The probability that the drug will be discovered by at least one of the approaches is increasing in the number of approaches undertaken. In particular the probability is given by:

P = 1 - e^-0.5085*n

where n is the number of approaches undertaken.

If the drug is developed, it will generate total surplus if $25.

From a social welfare perspective, how many research approaches should be undertaken? Hint: define the expected social gain as the probability (P) multiplied by $25 and the cost as being the number of approaches (n) multiplied by $1. Use calculus to find the optimal value of n, i.e. maximize the function expected gain minus cost.