شاخص تخصیص زمان کیفیت زندگی - نسخه سازگار اقتصاد تعادلی شاخص کیفیت زندگی کنونی

The definition the life quality index for a country as originally suggested by Nathwani, Lind and Pandey is based on the gross domestic product (GDP), the expected life in good health at birth, and the fraction of life time the anonymous citizen of the country is occupied with money making work. The LQI is invented to serve as a mean to evaluate how much money that reasonably can be allocated to safety improving investments by simply requiring constancy of the LQI. By choosing that the importance of increments in the two first variables should be measured relative to the current values of the variables themselves, the relative increment of the LQI becomes defined as a convex combination of the two relative increments. The combination parameter is obtained by an optimality argument about the anonymous citizen’s distribution of his or her time between free time and work time. In the original definition this equilibrium economy principle is applied under the assumption that the GDP is directly proportional to the work time fraction. This direct proportionality has been relaxed by the first author in two earlier papers with an essential effect on the combination parameter. The present paper presents a further development casting the definition into dimensionless quantities that make the index get a pure unit of time and not the somewhat obscure unit as a power product of a money unit and a time unit. To avoid confusion, this new variant of the LQI is called the life quality time allocation index (LQTAI). Moreover, the Danish data from the period from 1948 to 2003 show good agreement with the relation between the productivity and the work time as obtained from the optimality argument. The data fitting leads to an estimate of the combination coefficient of c = 0.092 together with a reduction factor of r = 0.92 to be applied to the total life expectation at birth to obtain the expected life in good health. Among other infinitely many choices of (c, r) there are (0.085, 1.0) and (0.1, 0.85).

In [1] the first author presented an alternative and more general definition of the life quality index (LQI) as compared to the definition in [2], originally given by Nathwani, Lind, and Pandey in [3] on the basis of quite reasonable principles. The LQI is intended as a social indicator that reflects the expected length of “good” life, in particular the enhancement of the quality of life by good health and wealth. The definition is attached to the concept of a societal economy, a concept invented as a terminology in [1]. A societal economy has members. The members are all human beings that live and for a part of their life make productive work within a geographical region in which there is statistical homogeneity of wealth and expected life at birth. A societal economy can be thought of as a part of a country, an entire country or a suitably selected group of countries of similar standard of living of their populations. When talking about “average” it relates to average over a considered social economy. In short, the life quality index differential dQ is defined such that the average importance of the increment dQ is dQ/Q and this importance is simply set to a convex linear combination of the importance of dG and the importance of d[(1 − w)E], where G is the gross domestic product per person, E is the life expectancy at birth, and w is the fraction of time spent with money making work. Thus where 0 < c < 1 is a suitably chosen combination coefficient that must be a constant under variation of the variables G, E, and w, of course. The differential form is an exact differential for the function The question in [1] is then how to choose the coefficient c for the convex combination. In [1] it is assumed that G is proportional to some differentiable function g of w − w0, where w0 is the fraction of time with unpaid work, here assuming that w contains both the money making work w − w0 and the unpaid work w0 needed to stay in healthy and clean condition to be fit for work. Simplifying relative to [1] by neglecting the controversial contribution w0 (that is, setting w0 = 0), the assumption in [1] is that G = pg(w), where p is a variable productivity factor independent of w and g(w) is a suitable differentiable function. Then where g′ is the derivative of g. It follows from (3) that dQ/Q is insensitive to variations of w at the value of w for which the coefficient to dw is zero, that is, when in which w is now the actual value we for the considered societal economy. Thus, it is postulated that the actual value of the work time ratio corresponds to a quasi static societal economy in which there is no gain or loss of life quality by a moderate change of the work time ratio. It is pointed out in [1] that a societal economy that develops according to the function G = pg(w) for some g can only correspond to such an equilibrium economy for some specific value we of the work ratio. If g is known, the value we should then be used to define c by (4). In the period of development of a societal economy towards the equilibrium state in which the LQI is insensitive to variations of w in the neighborhood of we, the coefficient to dw in (3) is different from zero. Keeping to the principle that the same value of c should be used for all societal economies, the question is raised in [1] of how it can be decided on a world wide basis what the optimal value of we is. The suggestion in [1] is to start out by setting c = 0.3, say, and then calculate the value of Q = GcE1 − c for all major societal economies of the world. The societal economy that comes out with the largest value of Q is next used to determine a new value of c corresponding to the value of we for the chosen societal economy. Next a new calculation of all Q values are made and the largest is selected. If that value corresponds to the same societal economy as before, the value of c is set. Otherwise a new calculation of c is made, and the calculation of all the values of Q is made again. In case of convergence, the value of c is set. In the unlikely case of divergence an average value of the sequence of repeated winners is used for c. This procedure will define the common c for all societal economies and all other societal economies than the appointed optimal can adjust according to (3) by increasing or decreasing w. This is still a valid proposal that may be supplemented by what was overlooked in [1]. Since c is a constant it follows that (4) is a differential equation that determines the function g uniquely except for an arbitrary factor. The solution is where K is an arbitrary positive constant. This means that a societal economy that develops in equilibrium and with constant LQI must develop according to the function (5). Thus, the postulate of optimal allocation of time for work and for work free time in the sense of having insensitivity of the LQI to small work time changes under the development of the societal economy has important consequences. These consequences are investigated in the following, where a more general and essentially dimensionless version of the LQI is derived. This new version will be denoted the life quality time allocation index (LQTAI) even though in its essence it is equivalent to the LQI.

A variant of the Life Quality Index denoted as the Life Quality Time Allocation Index is defined by the use of dimensionless quantities such that the LQTAI gets the dimension of time. This is in contrast to the LQI that has the dimension as a power product of a money unit and a time unit. Moreover, a family of relations between the time equivalent productivity p and the work time fraction w is determined such that there is an optimal balance between work time and free time, given an invariant value of the LQTAI. Considering w as a function of p there are two branches, an upper increasing branch and a lower decreasing branch that join together through a vertical tangent at a minimal value of p. A single relation from this family is selected by comparison with Danish statistical data recorded from 1948 to 2003. Surprisingly it is the upper branch that fits the data. Though strongly fluctuating along the curve, the trend is that the time equivalent productivity decreases together with the work time fraction during the years of registration. Moreover, it is surprizing that the comparison with the data necessitates that the combination coefficient c in the differential form of the LQTAI must be set to as small a value as about 0.1. This implies that the Implied Cost of Averting a Fatality (ICAF) is as large as about five times the GDP times the expected life at birth. Besides the unit property difference the substantial difference of the logic of the LQTAI and the LQI is that for the LQTAI the combination coefficient is determined by the entire history of work market data while for the LQI the combination coefficient is chosen arbitrarily as the current work time fraction w. Thus, the LQI for Denmark would have the combination coefficient value about 0.15 in 1948 and about 0.09 in 2003. However, the logics of changing the unit of the LQI correspondingly by time is mysterious and faulty according to common scientific principles.