Sample chapter for the economic theory of annuities_3

80 • Chapter 10 particular, the individual typically has some influence on the outcome. Thus, the probability q, which was taken as given, may be regarded, to some extent at least, as influenced by individual decisions that involve costs and efforts. The potential conflict that this type of moral hazard raises between social welfare and individual interests is very clear in this context. Since V∗ < V∗, an increase in q decreases the first-best expected utility. On the other hand, in a competitive equilibrium, V > V , and hence an increase in q may be desirable. C H A P T E R 9 Pooling Equilibrium and Adverse Selection 9.1 Introduction For a competitive annuity market with long-term annuities to be efficient, it must be assumed that individuals can be identified by their risk classes. We now wish to explore the existence of an equilibrium in which the individuals’ risk classes are unknown and cannot be revealed by their actions. This is called a pooling equilibrium. Annuities are offered in a pooling equilibrium at the same price to all individuals (assuming that nonlinear prices, which require exclusivity, as in Rothschild and Stiglitz (1979), are not feasible). Consequently, the equilibrium price of annuities is equal to the average longevity of the annuitants, weighted by the equilibrium amounts purchased by different risk classes. This result has two important implications. One, the amount of annuities purchased by individuals with high longevity is larger than in a separating, efficient equilibrium, and the opposite holds for individuals with low longevities. This is termed adverse selection. Two, adverse selection causes the prices of annuities to exceed the present values of expected average actuarial payouts. The empirical importance of adverse selection is widely debated (see, for example, Chiapori and Salanie (2000), though its presence is visible. For example, from the data in Brown et al. (2001), one can derive survival rates for males and females born in 1935, distinguish-ing between the overall population average rates and the rates appli-cable to annuitants, that is, those who purchase private annuities. As figures 9.1(a) and (b) clearly display, at all ages annuitants, whether males or females, have higher survival rates than the population average rates (table 9A.1 in the appendix provides the underlying data). Adverse selection seems somewhat smaller among females, perhaps because of the smaller variance in female survival rates across different occupations and educational groups. Adverse selection may be reflected not only in the amounts of annuities purchased by different risk classes but also in the selection of different insurance instruments, such as different types of annuities. We explore this important issue in chapter 11. (a) Z Figure 9.1(a). Male survival functions (1935 cohort). (Source: Brown et al. 2001, table 1.1). (b) Z Figure 9.1(b). Female survival functions (1935 cohort). (Source: Brown et al. 2001, table 1.1). Pooling Equilibrium • 69 9.2 General Model We continue to denote the flow of returns on long-term annuities purchased prior to age M by r(z), M ≤ z ≤ T. The dynamic budget constraint of a risk-class-i individual, i = 1,2, is now ˙i(z) = rp(z)ai(z) +w(z) −ci(z) +r(z)a(M), M ≤ z ≤ T, (9.1) where ai(z) are annuities purchased or sold (with ai(M) = 0) and rp(z) is the rate of return in the (pooled) annuity market for age-z individuals, M ≤ z ≤ T. For any consumption path, the demand for annuities is, by (9.1), Z z Z z Z x ai(z) = exp rp(x)dx exp − rp(h)dh M M M ×(w(x) −ci(x) +r(x)a(M))dx , i = 1,2. (9.2) Maximization of expected utility, Z T Fi(z)u(ci(z))dz, i = 1,2, (9.3) M subject to (9.1) yields optimum consumption, denoted ci(z), Z z ci(z) = ci(M)exp (rp(x) −ri(x))dx , M ≤ z ≤ T, i = 1,2 M (9.4) (where σ is evaluated at ci(x)). It is seen that ci(z) increases or decreases with age depending on the sign of rp(z) − ri(z). Optimum consumption at age M, ci(M), is found from (9.2), setting ai(T) = 0, Z T Z x exp − rp(h)dh (w(x) −ci(x) +r(x)a(M))dx = 0, i = 1,2. M M (9.5) Substituting for ci(x), from (9.4), RM exp −RMrp(h)dh (w(x) +r(x)a(M))dx T exp x σ ((1 −σ)rp(h) −ri(h))dh dx i = 1,2. (9.6) 70 • Chapter 9 Since r1(z) < r2(z) for all z, M ≤ z ≤ T, it follows from (9.6) that ˆ1(M) < c2(M). Inserting optimum consumption ci(x) into (9.2), we obtain the optimum demand for annuities, ai(z). Since ai(M) = 0, it is seen from (9.1) that a1(M) > a2(M). In fact, it can be shown (see appendix) that a1(z) > a2(z) for all M < z < T. This is to be expected: At all ages, the stochastically dominant risk class, having higher longevity, holds more annuities compared to the risk class with lower longevity. We wish to examine whether there exists an equilibrium pooling rate of return, rp(z), that satisfies the aggregate resource constraint (zero expected profits). Multiplying (9.1) by Fi(z) and integrating by parts, we obtain Z T Fi(z)(rp(z) −ri(z))ai(z)dz M Z T Z T = Fi(z)(w(z) −ci(z))dz +aM r(z)dz, i = 1,2. (9.7) M M Multiplying (9.7) by p for i = 1 and by (1 − p) for i = 2, and adding, Z T [(pF1(z)a1(z) +(1 − p)F2(z)a2(z))rp(z) M −(pF1(z)a1(z)r1(z) +(1 − p)F2(z)a2(z)r2(z))]dz Z T Z M = p F1(z)(w(z) −c1(z))dz +(1 − p) F2(z)(w(z) −c2(z))dz M T T + a(M) (pF1(z) +(1 − p)F2(z))r(z)dz. (9.8) M Recall that pF1(z)r1(z) +(1 − p)F2(z)r2(z) pF1(z) +(1 − p)F2(z) is the rate of return on annuities purchased prior to age M. Hence the last term on the right hand side of (9.8) is equal to F(M)a(M) = 0 F(z)(w(z) − c)dz. Thus, the no-arbitrage condition in the pooled ... - tailieumienphi.vn