Appendix A Derivation of LQI Abstract The Life Quality Index (LQI) is derived from economic princi- ples. The LQI can be used to estimate the societal capacity to commit re- sources to risk reduction, SCCR. The LQI, in a format simple for practical application, can be written as EKG, where G is the real GDP/person/year, E is the life expectancy and K is a constant. The societal capacity SCCR, de- rived from the LQI, equals KG/E and is described in Chap. 2. The purpose of this appendix is to describe the derivation of LQI from economic prin- ciples and to prepare for the calibration of the parameter K using economic data in Appendix C. A.1 Introduction Efficient management of risks to life safety involves the search for a bal- ance between the overall potential for harm and good outcomes. In recent years several acceptability criteria with quantitative rationales have been derived from compound social indicators (Lind et al. 1992, Nathwani et al. 1997, Rackwitz 2002, Lind 2002) to support evaluation of broad program outcomes. Lind et al. (1992) proposed the use of two key social indicators, real GDP per capita and life expectancy at birth (both also used in the UN Hu- man Development Project Index, HDI) for judging the effectiveness of de- cisions about risk and life safety. The concept was expanded by Nathwani et al. (1997) who further developed the Life Quality Index (LQI) to estab- lish a test of efficiency for programs and regulations to manage risks. The LQI is simpler than the HDI and based on well-defined component indices weighted to reflect peoples’ revealed preference for the work/non-work- time ratio and productivity. It allows an explicit valuation of a project’s ef- fectiveness for life extension. Moreover, it is derived rigorously from first principles of human welfare as shown in the following. While the use of social indicators to track progress of nations is a fairly recent development, and their use to evaluate risk mitigation is even more recent, the philosophical foundations of welfare economics were estab- lished much earlier by Pigou (1920) and Hicks (1939). These ideas con- tinue to influence the development of social and economic policies cen- tered around the concept of human welfare. The LQI builds on the concept of a social indicator that is a function of mortality and economic produc- tion and places an implicit value on reduction of life risk. The implied 114116 The Engineering Decisions for Life Quality value is the increase in wealth production (i.e., real GDP per person) re- quired to neutralize a small unit increase in mortality (Lind et al. 1992, Nathwani et al. 1997, Lind 2002, Rackwitz 2002, 2003, 2004). Use of the LQI based on key social indicators offers a great advantage in that it pro- vides a criterion of acceptable risk that harmonizes with national social and development objectives as reflected in the social indicator (Kubler and Fa- ber 2005, Lind 2007, Maes et al. 2003). The purpose of this appendix is to present an analytical approach to the derivation and calibration that is consistent with established principles of economic sciences. The derivation is based on the idea that the LQI can be interpreted as a lifetime utility function of an individual. The proposed LQI calibration is based on the concepts of production economics utilizing available economic data, thereby removing a simplifying assumption used in the original study (Nathwani et al. 1997). The notation in this appendix is different from the notation in the main body of the book, in order to facilitate the derivation of LQI in the macro- economic context. We may write one form of the LQI as q LQ = CG E, [A.1] where C is an arbitrary constant and q is a parameter to be determined. A.2 The Utility Function The notion of “utility” was introduced in the economic theory of value to describe the satisfaction experienced by a person through the consumption or use of a commodity. Individual demand behavior is then mathematically modeled as the process of maximizing utility under given constraints (Stig- ler 1950). As utility is not inherent in any commodity, but depends on the value a consumer places on it, it is essentially subjective and influenced heavily by psychological factors (Wicksell 1893). The measurement and comparison of utility in an absolute sense is of little merit. Nevertheless, a fundamental assumption of the demand theory is that the people are capa- ble of rank-ordering different commodities and their combinations consis- tently on a scale of preference according to the degree of satisfaction (util- ity) derived from them. The utility function attaches a utility number to each object, but the number itself has no physical meaning. The only thing that matters is the direction of change of the number for various objects. Mathematically, the utility is determined only up to a monotonic increasing transformation. In this sense, the utility is regarded as an ordinal concept. If a person prefers Appendix A Derivation of LQI 115117 an object A to B to C, then U(A) > U(B) > U(C) where U(X) denotes the utility associated with object X. However, nothing can be said about whether the difference between the utilities provided by A and B is greater or less than the difference between the utilities provided by B and C. Utili- ties can be compared but differences between utilities cannot; this is the essence of the ordinal concept. A much larger question is whether, and under what conditions, ordinal utility can be taken as measurable. The assumption that individuals can distinguish increments of utility (or intensity of preferences) and they can order these increments in the same way as the preferences themselves brings about the notion of cardinal utility. The cardinal utility function provides consistent ordering of utility increments. It means that if U(A) > U(B) > U(C), then (U(A) - U(B)) > (U(B) - U(C)) can also be arranged in the order of preference. Mathematically, the cardinal utility is said to be determinate only up to a linear transformation (Allen 1935). Classical economists such as Marshall, Walras, and Edgeworth con- ceived that the utility is a measurable quantity in theory, provided that enough facts can be collected. Pareto abandoned this idea and postulated the ordinal concept of utility as a scale of preference. The ordinal concept of utility has far-reaching consequences, as it has transformed the subjec- tive theory of value into a general logic of choice in the realm of economic theory (Hicks 1939, 1975). Hicks (1939) demonstrated that the theory of value requires only the marginal rate of substitution; it does not require measurability of the abso- lute utility. If an individual possesses two goods X and Y, the marginal rate of substitution of any good Y for any other good X is defined as the quan- tity of goods Y which would just compensate him for the loss of a marginal unit of X. If the individual gets less than this quantity of Y, he would be worse off than before the substitution took place. If an individual prefer- ence is to be in equilibrium with respect to a system of market prices, his marginal rate of substitution between any two goods must equal the ratio of prices. Otherwise he would find an advantage in substituting some quantity of one for an equal value (at the market rate) of the other. In summary, the theory of equilibrium in a market depends on the directions of indifference, and does not involve anything more. This paved the way for an interpretation of utility as a value function (or scale of preference) that describes consumer behavior. The utility function is just a function, and real empirical significance resided in its objective properties and restrictions placed on it, such as diminishing marginal rate of substitution. In summary, ordinal utility theory is sufficient to derive the economic theories of value, consumer demand, prices, and wages, which has ren- 116118 The Engineering Decisions for Life Quality dered the cardinal utility concept rather inconsequential in economics (Al- len 1935). A.3 Technical Concepts and Definitions For illustrative purposes, consider a utility function U(x, y) of the form U (x, y) = V (x)H (y) = x a y b [A.2] where V(x) = xa and H(y) = yb denote the sub-utility functions with respect to the attributes x and y, respectively. The utility function is taken as sepa- rable function in x and y similar to the LQI expressions in Eq. A.1. The marginal utility of x is defined as the partial deriva- tive (∂U (x, y)/ ∂x) . It should be positive to imply that the utility increases with an increase in consumption of x. The law of diminishing marginal utility, a fundamental notion in economics, implies that the rate of change of marginal utility of x decreases with increase in its values. It imposes a condition that a ≤ 1 and b ≤ 1. A.3.1 Elasticity of a Function The elasticity of a function is a measure of the responsiveness of the func- tion on its dependent variable. For example, given a value of x and a func- tion V(x), the elasticity of V(x), ηx, is defined as the ratio of percentage change in V(x) divided by the percentage change in x. In particular, for V(x) = xa, we have dV (x)/V (x) x dV (x) η = = = a . [A.3] x dx / x V (x) dx The elasticity of the utility function U(x, y) in Eq.