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Library Introduction to The Economic Theory_5

Các chức năng sống còn trong chương 2, độ tuổi được đưa đến là một biến liên tục, ký hiệu là z, có phạm vi từ 0 đến tối đa suốt đời, ký hiệu là T. chính thức, nó có thể cho phép T = ∞. Khi xem xét quyết định cá nhân, tuổi 0 nên được hiểu là độ tuổi sớm nhất mà tại đó các quyết định được thực hiện. Không chắc chắn về tuổi thọ | 34 Chapter 5 Figure 5.1. Optimum nonannuitized assets. course Y 1 f0F z e pz dz. The dynamic budget constraint is b z Ya pb z - c z 5.15 with solution b z ep z i -px ya - c x dx W - a 0 5.16 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 Ơ 1 u c ln c 8 0 T X and F z e-az. For this case Y a p. Maximization of expected utility subject to 5.15 yields optimum consumption c z c 0 e p-a z. Assume that p - a 0 implying that consumption rises with age. Solve for c 0 from 5.16 setting limz . b z e-pz 0. Since b 0 0 it is optimum to set b 0 0 and a W or c 0 a p a p W. Substituting in 5.16 we obtain the optimum path b z . It is now seen from 5.15 that b 0 p - a p p a 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 p0 r z while nonannuitized assets yield a return of p p p0. The budget constraint 5.13 now becomes a. z p0 r a z pb z w z - c z - b z . 5.17 Multiplying both sides of 5.17 by e-p0zF z and integrating by parts yields e-p0zF z w z -c z dz-Ị r z - p - p0 e-p0zF z b z dz. 5.18 Recall that b z 0 0 z T. If the hazard rate r z increases with age so that r z - p - p0 I 0 as z I 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 .