Một kiến thức của đại số được yêu cầu quen thuộc với tính toán và xác suất giới thiệu là hữu ích, nhưng không cần thiết. Sách giáo khoa phổ biến này là hữu ích hơn do sự bao gồm các chủ đề sau: phương pháp tiếp cận để đạt kết quả số bằng cách sử dụng công nghệ máy tính, khái niệm cơ bản về tài chính, chẳng hạn như tỷ lệ lợi nhuận thực sự của, thời hạn đầu tư và giới thiệu một chức năng bảo hiểm đơn giản đội ngũ | Utilitarian Pricing of Annuities 111 Figure . First-best allocation of utilities. while Vh 1 ph ph L y Eh 1 1 ph Thus the utilitarian first best has inequality in expected utilities but may have equality in consumption levels Arrow 1992 . This result is similar to Mirrlees 1971 optimum income tax model where individuals differ in The first best allocation provides higher expected utility to those with a higher capacity to produce utility. In the appendix to this chapter it is shown that 0 1 pn h -1 ch dph while dch dpj Ệ 0 for j h h j 1 2 . . . n. 2 In Mirrlees model with additive utilities the first best has all individuals with equal consumption and those with higher productivity having a lower disutility for generating a given income are assigned to work more and hence have a lower utility. 112 Chapter 13 Concavity of u and imply d V dứ u ch 1 pn u cn Ị 0 Ơ ph d pn while d V. dpj Ệ 0 j h j 1 2 . Thus with the given total resources an increase in one individual s survival probability decreases his or her optimum consumption but the positive effect of higher survival probability on expected utility dominates. The effect on the welfare of other individuals facing only resource redistribution depends on the shape of the social welfare function. Competitive Annuity Market with Full Information In a competitive market with full information on the survival probabilities of individuals and a zero rate of interest the price of a unit of second-period consumption c2h is equal to the survival probability of each annuitant. Individuals maximize expected utility subject to a budget constraint C1h phC2h yh h 1 . H where yh is the given income of individual h. Demands for first- and second-period consumption annuities C1h and c2h are given by C1h C2h Ch yh 1 ph . The first-best allocation can be supported by a competitive annuity market accompanied by an optimum income allocation. Equating consumption levels under competition Ch
