Consider the probability functions p(a)= min{1/a,1} and p(a)=1+a(a-2). Calculate the socially optimal levels of care for the
simple model outlined in section 10.1.

Data from section 10.1]

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Consider a context in which a firm's activities have the potential to cause an environmental accident that leads to external damages. The accident occurs with probability p(a), where a is the level of accident prevention undertaken by the firm. Throughout the chapter we will refer to a as the care level or abatement level. We assume the probability of an accident decreases with additional care at a decreasing rate, and that it converges to zero when care is infinite. Formally these assumptions imply that p'(a) <0, p"(a)>0, and lim(a)=0. Finally, we assume for most of the chapter that external damages are fixed and denoted by D. This allows us to write expected damages as p(a)xD. To link the firm's activities to our familiar model of market benefits and costs we denote x as the quantity of output associated with the hazardous activity and P(x) as the inverse demand function faced by the firm. Let C(x,a) be the cost of production as a function of the output quantity and level of abatement. We assume C()>0 and Ca()>0, so that both additional output and additional care have positive marginal cost. Furthermore we assume that C()0, Caa() 0, and Car()=Cra()0, so that the cost function is weakly convex in both its arguments. With this the social optimum is found by selecting x and a to maximize