Sample chapter for the economic theory of annuities_4

64 • Chapter 8 Now let pF1(z)r1(z) +(1 − p)F2(z)r2(z) pF1(z) +(1 − p)F2(z) = δr1(z) +(1 −δ)r2(z), M ≤ z ≤ T, (8.25) where δ = δ(z) = pF1(z) +(1 − p)F2(z), 0 < δ(z) < 1. (8.26) The future instantaneous rate of return at any age z ≥ M on long-term annuities held at age M is a weighted average of the risk-class rates of return, the weights being the fraction of each risk class in the population.5 Inserting (8.25) into (8.24), the latter becomes Z T Z T p F1(z)(w(z) −c1)dz +(1 − p) F2(z)(w(z) −c2)dz M M Z M + F(z)(w(z) −c)dz = 0. (8.27) 0 From (8.27) it is now straightforward to draw the following conclu-sion: The unique solution to (8.22) and (8.23) that satisfies (8.1), with r(z) given by (8.25), is c = c1 = c2 = c∗ and R∗ = R∗ = R∗, where (c∗, R∗) is the First-Best solution (8.3) and (8.4). A separating competitive equilibrium with long-term annuities sup-ports the first-best allocation. Individuals are able to insure themselves against uncertainty with respect to their future risk class by purchasing long-term annuities early in life. In equilibrium these annuities yield at every age a rate of return equal to the population average of risk class rates of return. The returns from these annuities provide an individual with a consumption level that is independent of risk-class realization. The transfers across states of nature necessary for the first-best allocation are obtained through the revaluation of long-term annuities. The stochastically dominant risk class obtains a windfall because the annuities held by individuals in this class are worth more because of the 5 The change in r(z) is r(z) = δr1 +(1 −δ)r2 f1(z) +δr1(+(1 −δ)r2 f2(z). The sign of this expression can be negative or positive. The change in the hazard 0 rate, F(z), is equal to F(z) f(z) + F(z) . A nondecreasing hazard rate implies that 0 0 − f(z) ≤ F(z) but does not sign f(z) (for the exponential function, f0(z) < 0, and this inequality becomes an equality). Uncertain Future Survival Functions • 65 higher life expectancy of the owners. The other risk class experiences a loss for the opposite reason. Another important implication of the fact that in equilibrium con-sumption is independent of the state of nature is the following. From (8.20) it is seen that when ci = c∗, i = 1,2, the solution to (8.20) has ai(z) = ai(z) = 0, M ≤ z ≤ T. Thus: The market for risk class annuities after age M (sometimes called “the residual market”) is inactive. Under full information, the competitive equilibrium yields zero trading in annuities after age M. As argued above and seen from (8.21), the interpretation of this result is that the flow of returns from annuities held at age M can be matched, using the relevant risk-class survival function of the holder of the annuities, to finance a constant flow of consumption: ∗ R R∗ Fi(z)w(z)dz +a∗(M)RT Fi(z)r(z)dz T Fi(z)dz where a∗(M) is the optimum level of annuities at age M: Z M a∗(M) = F(M) 0 F(z)(w(z) −c∗)dz. (8.28) 8.5 Example: Exponential Survival Functions Let F(z) = e−αz, 0 ≤ z ≤ M, and Fi(z) = e−αMe−αi(z−M), M ≤ z ≤ ∞, i = 1,2. Assume further that wages are constant; w(z) = w. With a constant level of consumption, c, before age M, the level of annuities held at age M is a(M) = F(M)Z0MF(z)(w −c)dz = wα c eαM −1. For the above survival function, the risk-class rates of return at age z ≥ M are αi, i = 1,2. We assume that risk class 1 stochastically dominates risk class 2, α1 < α2. Annuities yield a rate of return, r(z), that is a weighted average of these returns: r(z) = δ(z)α1 +(1 −δ(z))α2, where −α1(z−M) δ(z) = pe−α1(z−M) +(1 − p)e−α2(z−M) . (8.29) The weight δ(z) is the fraction of risk class 1 in the population. It increases from p to 1 as z increases from M to ∞. Accordingly, r(z) decreases with z from pα1 + (1 − p)α2 at z = M, approaching α1 as z → ∞ (figure 8.2). 66 • Chapter 8 Figure 8.2. The rate of return on long-term annuities. Consumption after age M for a risk-class-i individual is con-stant, ci, and the budget constraint is w RFi(z)dz − ci T Fi(z) + aM MFi(z)r(z)dz = 0. For our case it is equal to −αM 1 −e−αi(R −M) − e−αM i i +w −c 1 −e−αMZ Te−αi(z−M)r(z)dz = 0, i = 1,2. (8.30) M Multiplying (8.13) by p for i = 1 and by 1 − p for i = 2, and adding, it can be seen that the unique solution to (8.30) is c∗ = c1 = c2 and R∗ = R = R , where ∗ wα(eαM −1) + p (1 −e−α1(R∗−M)) + 1 − p(1 −e−α2(R∗−M)) α(eαM −1) + p + 1 − p (8.31) C H A P T E R 7 Moral Hazard 7.1 Introduction The holding of annuities may lead individuals to devote additional resources to life extension or, more generally, to increasing survival probabilities. We shall show that such actions by an individual in a competitive annuity market lead to inefficient resource allocation. Specifi-cally, this behavior, which is called moral hazard, leads to overinvestment in raising survival probabilities. The reason for this inefficiency is that individuals disregard the effect of their actions on the equilibrium rate of return on annuities. The impact of individuals disregarding their actions on the terms of insurance contracts is common in insurance markets. Perhaps moral hazard plays a relatively small role in annuity markets, as Finkelstein and Poterba (2004) speculate, but it is important to understand the potential direction of its effect. Following the discussion in chapter 6, assume that survival functions depend on a parameter α, F(z,α). A decrease in α increases survival probabilities at all ages: ∂F(z,α)/∂α ≤ 0. Individuals can affect the level of α by investing resources, whose level is denoted by m(α), such as med-ical care and healthy nutrition. Increasing survival requires additional resources, m(α) < 0, with increasing marginal costs, m0(α) > 0. 7.2 Comparison of First Best and Competitive Equilibrium Let us first examine the first-best allocation. With consumption constant at all ages, the resource constraint is now Z T Z R c F(z,α)dz − F(z,α) w(z)dz +m(α) = 0. (7.1) 0 0 Maximizing expected utility, (4.1), with respect to c, R, and α yields the familiar first-order condition u0(c)w(R) −e(R) = 0 (7.2) 52 • Chapter 7 Figure 7.1. Investment in raising survival probabilities. and the additional condition (u(c) −u0(c)c)Z T ∂F(z,α) dz +Z R ∂F(z,α)(u(c)w(z) 0 0 −e(z))dz −u0(c)m(α) = 0, where, from (7.1), R R F(z,α) w(z)dz −m(α) 0 F(z,α)dz (7.3) (7.4) Conditions (7.1)–(7.3) jointly determine the efficient allocation (c∗, R∗,α∗). Denote the left hand sides of (7.1) and (7.3) by ϕ(c,α, R) and ψ(c,α, R), respectively. We assume that second-order conditions are satisfied and relegate the technical analysis to the appendix. Figure 7.1 holds the optimum retirement age R∗ constant and describes the condi-tions ϕ(c,α, R∗) = 0 and ψ(c,α, R∗) = 0. Under competition, it is assumed that the level of expenditures on longevity, m(α), is private information. Hence, annuity-issuing firms cannot condition the rate of return on annuities on the level of these expenditures by annuitants. Let the rate of return faced by individuals at ... - tailieumienphi.vn