Sample chapter for the economic theory of annuities_5

34 • Chapter 5 Figure 5.1. Optimum nonannuitized assets. course, γ = 1.R0 F(z)e−ρz dz. The dynamic budget constraint is b(z) = γa +ρb(z) −c(z), (5.15) with solution Z z b(z) = eρz e−ρx(γa −c(x))dx+ W−a . (5.16) 0 The amount of b(z) changes with age, depending on the consumption path. The only constraint is that b(z) ≥ 0 for all z, 0 ≤ z ≤ T. Hence, W−a ≥ 0. For simplicity, consider the special case σ = 1 (u(c) = lnc), δ = 0, T = ∞, and F(z) = e−αz. For this case, γ = α + ρ. Maximization of expected utility subject to (5.15) yields optimum consumption c∗(z) = c∗(0)e(ρ−α)z. Assume that ρ − α > 0, implying that consumption rises with age. Solve for c∗(0) from (5.16), setting limz→∞ b(z)e−ρz = 0. Since b(0) ≥ 0, it is optimum to set b(0) = 0 and a = W, or c∗(0) = α((ρ +α)/ρ)W. Substituting in (5.16) we obtain the optimum path, b∗(z). It is now seen from (5.15) that b∗(0) = ((ρ −α)/ρ)(ρ+α)W > 0. Nonannuitized assets accumulate and then decumulate to support the optimum consumption trajectory (figure 5.1). Comparative Statics • 35 5.5 Partial Annuitization: Low Returns on Annuities Cannon and Tonks (2005) observe that the issuers of annuities (insurance firms) invest their assets, for reasons of liquidity and risk, mainly in bonds that yield a lower return than equities. While the reasons given for this policy are rather weak (and as annuity markets grow, insurance firms are expected to hold more balanced portfolios), this may be another explanation why individuals annuitize only later in life, holding nonannuitized assets at early ages. To see this, let annuities have a rate of return of ρ0 + r(z), while nonannuitized assets yield a return of ρ, ρ > ρ0. The budget constraint (5.13) now becomes a(z) = (ρ0 +r)a(z) +ρb(z) +w(z) −c(z) −b(z). (5.17) Multiplying both sides of (5.17) by e−ρ0zF(z) and integrating by parts yields Z T Z T e−ρ0zF(z)(w(z)−c(z))dz− [r(z)−(ρ−ρ0)]e−ρ0zF(z)b(z)dz. (5.18) 0 0 Recall that b(z) ≥ 0, 0 ≤ z ≤ T. If the hazard rate, r(z), increases with age so that r(z) −(ρ −ρ0) Q 0 as z Q zc, (5.19) then the individual’s optimum policy is to invest all assets in b up to age zc, switching to annuities afterward. 5.6 Length of Life and Retirement We have seen, in (5.7), that under reasonable conditions for the age profile of changes in longevity, optimum retirement increases with longevity. Recent increases in longevity have largely been concentrated in very old ages (see Cutler, 2004). It is therefore of interest to examine how optimum retirement responds to a steady increase in the length of life. It is simplest to consider a particular case, (3.7), with no uncertainty and a finite lifetime. With a positive time preference and rate of interest, optimum consumption is given by (5.12), and c∗(0) is determined by condition (5.11) with F(z) = 1, 0 ≤ z ≤ T: c∗(0)Z TexpZ z (1 −σ)ρ − δ dxdz −Z R∗e−ρzw(z)dz = 0. 0 0 0 (5.20) 36 • Chapter 5 Jointly with the condition for optimum retirement, u0 c∗(0)expZ R∗ (ρ −δ) dxw(R∗) = e(R∗), (5.21) 0 equations (5.15) and (5.21) determine the optimum (c∗(0), R∗), which depend on the length of life, T. We are particularly interested in the dependence of R∗ on T as it becomes very large. For simplicity, assume that σ = σ(c∗(x)) is constant. Differentiating (5.21) totally with respect to T and inserting the proper expressions from (5.20), we obtain  ! 1 (1 −σ)ρ −δ dR∗ A 1 −exp −1 ((1 −σ)ρ −δ)T dT  σ AT where (1 −σ)ρ −δ = 0, (1 −σ)ρ −δ = 0, (5.22) e−ρR∗w(R∗) w0(R∗) e0(R∗) R∗e−ρzw(z)dz w(R∗) e(R∗) (5.23) Expression A is positive by the second-order condition for the opti-mum R∗. Hence, dR∗/dT > 0. Assume that limT→∞A is finite, say, A. Then, from (5.22),  ∗  ((1 −σ)ρ −δ), lim dT =  0, (1 −σ)ρ −δ > 0, (5.24) (1 −σ)ρ −δ ≤ 0. Thus, when σ ≤ 1, optimum retirement age may increase indefinitely as life expectancy rises, provided the rate of time preference is small. When this condition is not satisfied, then optimum retirement approaches a finite age. This is seen most clearly when wages and labor disutility are assumed constant, w(z) = w, e(z) = e, and ρ = δ > 0. From (5.20) and (5.21), R∗ is then determined by the condition 0 w(1 −e−ρR∗) 1 −e−ρT (5.25) Comparative Statics • 37 Figure 5.2. Optimum retirement age and length of life (R is defined by u0 (w(1 −e−ρR))w = e). (assuming that the parameters w and e yield an interior solution, R∗ < T). On the other hand, when ρ = δ = 0, (5.21) becomes u0 wR∗ w = e. (5.26) With positive discounting, as T becomes large, optimum retirement approaches a finite age, while with no discounting R∗/T remains constant (figure 5.2). The reason for the difference in the pattern of optimum retirement is straightforward. Without discounting, the importance of a marginal increase in the length of life does not diminish even at high levels of longevity and, accordingly, the individual adjusts retirement to maintain consumption intact. With discounting, the importance of a marginal increase in the length of life diminishes as this change is more distant. Accordingly, the responses of optimum consumption and retirement become negligible and eventually vanish. Subsequently, we shall continue to assume that ρ = δ = 0. The discussion above, concerning different patterns of optimum retire-ment response to increasing longevity, is of great practical importance. Many countries have recently raised the normal retirement age (NRA) 38 • Chapter 5 for receiving social security benefits: In the United States the NRA will reach 67 in 2011, up from 65. Other countries, such as France, Germany, and Israel have also raised their SS retirement ages to 67. In all these cases, postponement of eligibility for “normal” SS benefits seems to be primarily motivated by the long-term solvency needs of the SS systems rather than by consumer welfare considerations. The above analysis points out that in designing future retirement ages for SS systems, consumer preference considerations may provide widely different outcomes. In particular, when the rise in optimum retirement age tapers off as life expectancy rises, this will exacerbate the financial constraints of SS systems, requiring a combination of a reduction of benefits and an increase in contributions. 5.7 Optimum Without Annuities Suppose that there is no market for annuities but that individuals can save in other assets and use accumulated savings for consumption. Denote the level of these assets at age z by b(z). These assets yield no return. Precluding individuals from dying with debt implies that they cannot incur debt at any age; that is, b(z) ≥ 0 for all 0 ≤ z ≤ T. The dynamics of the budget constraint are thus b(z) = w(z) −c(z), (5.27) where b(z) is current savings, positive or negative. The non-negativity constraint on b(z) is written Z z b(z) = (w(x) −c(x))dx ≥ 0, 0 ≤ z ≤ T. (5.28) 0 (Again, it is understood that w(z) = 0 for z ≥ R). Having no bequest motive, the individual plans not to leave any assets at age T3: Z T b(T) = (w(z) −c(z))dz = 0. (5.29) 0 Assuming that assets (at the optimum) are strictly positive at all ages (and hence (5.28) is nonbinding), maximization of (4.1) subject to (5.29) yields the first-order condition F(z)u0(c(z)) −λ = 0, (5.30) 3 Death at any earlier age, z < T, may leave a positive amount of unintended bequests, b(z) > 0. By assumption, this has no value to the individual, but for aggregate analysis this has to be taken into account. ... - tailieumienphi.vn