FINITE LIFE EXPECTANCY AND THE AGE-DEPENDENT VALUE OF A STATISTICAL LIFE

In this short paper, we investigate the behavior of the age-dependent value of a statistical life (VSL) within a lifecycle framework with a finite maximal possible lifespan. Some existing results, obtained under the unrealistic assumption of an infinite life expectancy, are reversed. In particular, we show that when the market interest rate is equal to (or less than) the sum of age-specific mortality rate and the discounting rate in time preference at any age over the remaining lifetime, then VSL declines. We also show that an inverted-U shape of VSL profile over the life cycle emerges under realistically plausible circumstances. An innovation is that we characterize the changes in optimal consumption and instantaneous utility with age, showing that such changes are proportionate to the difference between the sum of age-specific mortality rate and the discounting rate in time preference and the market interest rate, which may prove to be useful in addressing other issues related to VSL.Value of life; life expectancy; interest rates; time preference; mortality.

Simultaneously continuous retraction and Bishop-Phelps-Bollob\'as type theorem

We study the existence of a retraction from the dual space $X^*$ of a (real or complex) Banach space $X$ onto its unit ball $B_{X^*}$ which is uniformly continuous in norm topology and continuous in weak-$*$ topology. Such a retraction is called a uniformly simultaneously continuous retraction. It is shown that if $X$ has a normalized unconditional Schauder basis with unconditional basis constant 1 and $X^*$ is uniformly monotone, then a uniformly simultaneously continuous retraction from $X^*$ onto $B_{X^*}$ exists. It is also shown that if $\{X_i\}$ is a family of separable Banach spaces whose duals are uniformly convex with moduli of convexity $\delta_i(\varepsilon)$ such that $\inf_i \delta_i(\varepsilon)>0$ and $X= \left[\bigoplus X_i\right]_{c_0}$ or $X=\left[\bigoplus X_i\right]_{\ell_p}$ for $1\le p<\infty$, then a uniformly simultaneously continuous retraction exists from $X^*$ onto $B_{X^*}$. The relation between the existence of a uniformly simultaneously continuous retraction and the Bishsop-Phelps-Bollob\'as property for operators is investigated and it is proved that the existence of a uniformly simultaneously continuous retraction from $X^*$ onto its unit ball implies that a pair $(X, C_0(K))$ has the Bishop-Phelps-Bollob\'as property for every locally compact Hausdorff spaces $K$. As a corollary, we prove that $(C_0(S), C_0(K))$ has the Bishop-Phelps-Bollob\'as property if $C_0(S)$ and $C_0(K)$ are the spaces of all real-valued continuous functions vanishing at infinity on locally compact metric space $S$ and locally compact Hausdorff space $K$ respectively.Comment: 15 page