Sample chapter for the economic theory of annuities_1

Utilitarian Pricing of Annuities • 111 Figure 13.1. First-best allocation of utilities. while V∗ = (1 + ph) u ! Ph=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 productivity.2 The first best allocation pro-vides higher (expected) utility to those with a higher capacity to produce utility. In the appendix to this chapter it is shown that 0 > 1 c∗pn ∂ph > −1, (13.7) while ∂c∗/∂pj R 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 (13.7) imply ∗ ∗ ∂ph = u(c∗) +(1 + pn)u0(cn) ∂pn > 0, (13.8) while ∂V∗/∂pj R 0, j = h, j = 1,2,..., H.3 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. 13.2 Competitive Annuity Market with Full Information In a competitive market with full information on the survival proba-bilities 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, 2,..., H, (13.9) where yh is the given income of individual h. Demands for first- and second-period consumption (annuities), c1h and c2h, are given by c1h = b2h = ch = yh/(1 + ph). The first-best allocation can be supported by a competitive annuity market accompanied by an optimum income allocation. Equating con-sumption levels under competition, ch, to the optimum levels, c∗(p), yields unique income levels, yh = (1+ ph)ch(p), that support the first-best allocation. In particular, with an additive W, all individuals consume the same amount: c∗ = Ph=1(1 + ph), hence 1 + ph h h=1(1 + ph) (13.10) 3 In the extreme case when W = min[V ,V ,...,VH], optimum expected utilities, V∗ = (1 + ph)u(c∗), are equal, and hence optimum consumption, c∗, strictly decreases with ph (and increases with pj, j = h). Utilitarian Pricing of Annuities • 113 13.3 Second-best Optimum Pricing of Annuities Governments do not engage, for well-known reasons, in unconstrained lump-sum redistributions of incomes. In contrast, most annuities are supplied directly by government-run social security systems and taxes/subsidies can, if so desired, be applied to the prices of annuities offered by private pension funds. These prices can be used by govern-ments to improve social welfare. Although deviations from actuarially fair prices entail distortions (i.e., efficiency losses), distributional im-provements may outweigh the costs.4 Suppose that individual h purchases annuities at a price of qh. With an income yh, his or her budget constraint is c1h +qhc2h = yh, h = 1,2..., H. (13.11) Maximization of (13.2) subject to (13.11) yields demands cih = cih(qh, ph, yh), i = 1,2, and h = 1,2,..., H. Maximized expected utility, h, is V (qh, ph, yh) = u(c1h) + phu(c2h). Assume that no outside resources are available for the annuity market, hence total subsidies/taxes must equal zero, H (qh − ph)c2h = 0. (13.12) h=1 Maximization of W(V ,V ,...,VH) with respect to prices (q1,...,qH) subject to (13.12) yields the first-order condition ∂W ∂qh +λc2h +(qh − ph) dqh = 0, h = 1,2,..., H, (13.13) where λ > 0 is the shadow price of constraint (13.12). In elasticity form, using Roy’s identity (∂V /∂qh = −(∂V /∂yh)c2h), (13.13) can be written qh − ph θh qh εh (13.14) where εh = −(qh/c2h)(∂c2h/dqh) is the price elasticity of second-period consumption of individual h and θh = 1 − λ ∂W ∂yh 4 In practice, of course, prices do not vary individually. Rather, individuals with similar survival probabilities are grouped into risk classes, and annuity prices and taxes/subsidies vary across these classes. 114 • Chapter 13 is the net social value of a marginal transfer to individual h through the optimum pricing scheme. Equation (13.14) is a variant of the well-known inverse elasticity optimum tax formula, which combines equity (θh) and efficiency (1/εh) considerations. The implication of (13.14) for the optimum pricing of annuities depends on the welfare function, W, and on the joint distribution of incomes, (y1,..., yH), and probabilities, (p1,..., pH). To obtain some concrete examples, let W be the sum of expected utilities. Then ∂W/∂V = 1, h = 1,2,..., H. Assume further that V = lnc1h + ph lnc2h. In this case, demands are b1h = 1 + ph , c2h = 1 + ph qh , (13.15) and bh = (1 + ph)ln 1 + ph + ph ln qh . (13.16) Conditions (13.14) and (13.12) now yield the solution ! qh = φ Pβh , (13.17) h=1 h where H φ = ph > 0 and h=1 βh = 1 + ph > 0. Consider two special cases of (13.17): (a) Equal incomes: (yh = y = R/H; h = 1,2,..., H) Condition (13.17) now becomes qh = φ(ph/(1 + ph)), where PH φ = h=1 h (>1). (13.18) H h h=1 1 + ph It is seen (figure 13.2) that optimum pricing involves subsidization (taxation) of individuals with high (low) survival probabilities.5 5 In figure 13.2, it can be shown that φ/2 < 1. Utilitarian Pricing of Annuities • 115 Figure 13.2. Optimum annuity pricing in a full-information equilibrium. (b) yh = y(1 + ph) This, one recalls, is the first-best utilitarian income distribution, and since all price elasticities are equal to unity, we see from (13.17), as expected, that qh = ph; that is efficiency prices are optimal. More generally, it is seen from (13.17) that a higher correlation between incomes, yh, and survival probabilities, ph, decreases—and possibly eliminates—the subsidization of high-survival individuals. In contrast, a negative correlation between incomes and survival probabilities (as, presumably, in the female/male case) leads to subsidies for high- survival individuals, possibly to the commonly observed uniform pricing rule. ... - tailieumienphi.vn