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The 'older the more valuable'
 It is generally believed that life is more valuable for the young as they expect to live longer. It is also generally believed that time is more precious for the middle-aged and the old as they have higher wage-rates andlor have not much time left, making the marginal utility of time high. In this paper, it is shown that results opposite to the above common-sense beliefs may be true. In particular, the marginal utility of time may decrease dramatically with age despite the fact that the wage-rate and the dollar value of time both increase with age, and despite the absence of depreciation in the ability to enjoy life. Moreover, the time value of life (the number of current years one's life is worth) may also increase with age.Most people believe that life is more valuable for the young as they expect to live longer. Even if a year at middle age is worth more than a year when young, life should still be more valuable for the young who will lose, if death strikes, not only the middle-age and older years but also the younger years. It is also generally believed that time is more precious for the middle-aged and the old as they have higher wage-rates andlor have not much time left, making the marginal utility of time high, We show in this paper that results opposite to the above common-sense beliefs may be true. In fact, this is likely to be true provided that the rate of interest is high. In a study of the value of life, Ng (1992) obtained an apparently counterintuitive but explicable result that the monetary (or dollar) value of the remaining life may increase dramatically as one ages (by 11 times in his example), though the utility value of life decreases monotonically with age. The reason lies in that the marginal utility of income declines remarkably as one gets older if the real interest rate is reasonably high. For his purpose, it is sufficient to take utility as a function of consumption only, ignoring the role of leisure. We extend the analysis to address the value of time by explicitly considering the worwleisure choice. In particular, we show that the marginal utility of time may decrease dramatically with age despite the fact that the wage-rate and the dollar value of time both increase with age. Despite their low wage-rates, the young may lose more than the elderly in utility terms by wasting time. Moreover, not only the dollar value of life may increase as one ages due to the decrease in the marginal utility of income, but even the time value of life (number of current years one's life is worth) may also increase with age.
Other analyses of the value of life show decreasing value even in dollar value. The economics approach to the valuation of life is based on the (maximum) willingness to pay to avoid a certain risk of death or the minimum amount of compensation required for a marginal increase in the probability of death. The dollar value of life is obtained by multiplying the compensating variation or equivalent variation for a marginal change in the probability of death by the inverse of this probability change. For a marginal change, the compensating equals the equivalent variation and either measure may be used. For example, if an individual is willing to pay a maximum of $1,000 to avoid a probability of death equal to 0.05%, her life is valued at $$1,000/0.05 = $2,000,000. This measure of the value of life is that to the individual and by the individual, ignoring the external effects to someone else. However, for the value of life, the external effects should on average be already reflected in the 'internal' value captured above. This is so because of the following consideration. If there are many persons with strong external benefitdcosts on others, then the values (either in dollar or in utility) of lives of most individuals will already be made highedlower by these external effects.
The estimation of the willingness to pay may be made either by: (1) actually asking the individual directly, (2) inferring from her revealed preference (e.g. fiom choices between different airlines of different safety records), or (3) calculating from her utility function. If the individual is a rational expected utility maximizer and in the absence of significant mistakes, the three measures should be approximately equal to each other.
The utility value of life is simply the utility expected to be enjoyed in the rest of one's life. If the realization of future utility is uncertain, an uncertainty discount should be rationally applied. Nevertheless, this uncertainty discount is similar in effect to the rate of depreciation in one's capacity to enjoy income allowed in our model. Thus, that rate may be taken to be the combined rate of depreciation (which may be negative) and rate of uncertainty discount. Note that this uncertainty discount is on future utility, not future (real) income or consumption which should be discounted at the (real) rate of interest.
The utility value of time or the marginal utility of time (at any age) is the increase in utility if the available time for that particular age (i.e. once-only rather than a permanent change) can be hypothetically increased by a marginal unit. (Alternatively, it is the decrease in utility if the available time is wasted by a marginal unit.) As is well known, in the standard labor supply models the dollar value of leisure time at the margin is equal to the wage rate. It is certainly true that one cannot enjoy a time endowment of more than 24 hours a day. However, one needs to go beyond the wage rate and hypothetically consider the marginal utility of time in order to measure the value of disposable time at different ages. To see the difference between the wage rate (which equals the marginal rate of substitution between leisure and consumption when utility is maximized) and the actual increase/decrease in utility of a hypothetical increase/waste of time.
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