Suppose an individuals "health preferences" are represented by the utility function u(s, e, h) = s(2-e) + h, where s is sugar, e is exercise, and h is its general state of health. In turn, h is adversely (i.e., negatively) affected by s and positively affected by e. Speciffically, suppose h(s, e) = ln(e) - (s2-2). Overall, s and e are subject to the respective constraints 0 ≤ s ≤ 2 and 1/2 ≤ e ≤ 2.
(a) How much s and e would the individual choose to consume? Verify that this is indeed a maximum.
(b) Next, suppose that while the direct effects of s and e on u are certain, the effects of these on health are not. Rather, each of them may or may not have the purported (i.e., claimed) effect on health. For x = s or e, let πx denote the probability that factor x does have the purported health effect and (1-πx) denote the probability that x does not, and assume πs and πe are independent. In the event that s does affect health but e does not, then h is given by h(s) = -s2/2. Similarly, if e is effective but s is not, then h(e) = ln(e); and if neither is effective, then h = 0. For each case, determine the optimal choices of s and e if the individual knew whether or not the purported health efects were true.
(c) Returning to the case in which the true effects on health are uncertain, write the decision problem facing the individual and characterize an interior solution assuming the individual maximizes expected utility. What, if any, restrictions on s and e ensure that an interior solution exists?