Steve is driving his truck and can choose a level of care to exert, which is denoted by x, which must be finite and non-negative. His cost of exerting x is given by wx. That is, Steve's marginal cost of exerting is w. The probability p(x) that he causes harm of A to someone else is given by 1/[1+x]. So p'(x)= -1/[1+x]^2.
Suppose A=256 and W=4. What is the socially optimal level of care?