A driver can choose a level of care to exert, which is denoted by x, which must be finite and non-negative. Her cost of exerting x is given by wx. That is, her marginal cost of exerting care is w. The probability p(x) that she causes harm of A to someone else is given by 1/[1+x]. So p'(x) = -1/[1+x]².

Suppose A = 2000 and w = 5. What is the socially optimal level of care?