Economics questions

The confidence interval for the individual job is wider (27) than the confidence interval for the

average cost (9). So there is greater uncertainty over an individual result than an average

a. Calculate E[X] and Var(X). b. Consider a proportional policy where I(x) = kx 0 Var[X - I,(x)]. endix Establish the lemma by using an analytic rather than a geometric argument. [Hint: Expand u(w) in a series as far as a second derivative remainder around the point z and subtract w(2).] Adopt the hypotheses of Theorem 1.5.1 with respect to B and insurance con- tracts I(x) and assume E[X] = p. Prove that Var[X - I(X)] = E[(X - I(X) - u + 3)?] is a minimum when I(x) = I (x). You will be proving that for a fixed pure premium, a stop-loss insurance contract will minimize the variance of re- tained claims. [Hint: we may follow the proof of Theorem 1.5.1 by first prov- ing that x2 - 22 2 (x - z)(2z) and then establishing that [x - I(x)] - [x - I (x)P = [I(x) - I(x)][2x - 21,.(x)] 2 2[1 (x) - I(x)]d*. The final inequality may be established by breaking the proof into three cases. Alternatively, by proper choice of wealth level and utility function, the result of this exercise is a special case of Theorem 1.5.1.] Adopt the hypotheses of Theorem 1.5.1, except remove the budget constraint; that is, assume that the decision maker will pay premium P, 0 0) = 1. In words, the probability of their joint distribution is concentrated on a line of pos- itive slope. In part (b), the correlation coefficient of X and X - I(X) was found to be 1. Thus, X - I(X) = aX + b, which implies that I(X) = (1 - a)X - b. To be a feasible insurance, 0 = I(x) = x or 0 s (1 - a)x - b s x. These inequalities imply that b = 0 and 0 s 1 - a = 1 and Osa= 1. d. To determine a, set the correlation coefficient of X and X - I(X) equal to 1, or equivalently, their covariance equal to the product of their standard deviations. Thus, show that a = VV/Var(X) and thus that the insurance that minimizes f(Var[X]) is I(X) = [1 - VV / Var(X)]X