Appendix a Derivation of LQI

Appendix A Derivation of LQI

Abstract The Life Quality Index (LQI) is derived from economic principles. The LQI can be used to estimate the societal capacity to commit resources 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, derived 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 principles 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 balance 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 decisions 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 effectiveness 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 established much earlier by Pigou (1920) and Hicks (1939). These ideas continue 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 production 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) required 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 provides 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 consistently on a scale of preference according to the degree of satisfaction (utility) 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 115 117 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 subjective 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 absolute 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 quantity 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 preference 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 function on its dependent variable. For example, given a value of x and a function 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. A.2 with respect to x and y can be obtained in a similar manner:

η = η = UX a and UY b . [A.4]

The exponents of the utility function in Eq. A.2 are the elasticity coeffi- cients of sub-utility functions. Appendix A Derivation of LQI 117 119 A.3.2 Indifference Curve

The assumption of utility maximization is a useful device to explain consumer bahavior. A utility function in this context is only needed to describe an indifference curve in the commodity space, indicating that people are indifferent between various consumption patterns and prefer more to less (Wicksteed 1888). The indifference curve, a concept introduced by Edgeworth, is a locus of points for all those combinations of x and y that produce a constant level of utility function U(x, y) as shown in Fig. A.1, i.e.,

U (x, y) = C = xa yb [A.5] or it can be written in differential form as

∂U (x, y) ∂U (x, y) dU (x, y) = 0 = dx + dy . [A.6] ∂x ∂y

Indifference Curve U(x,y) = C

Y dy

Fig. A.1 The utility indifference curve

For the specific power utility function (Eq. A.2), it can be simplified as 118120 The Engineering Decisions for Life Quality dx dy a + b = 0 . [A.7] x y

The indifference curve is convex to the origin, reflecting the law of diminishing marginal utility. Given a utility function, a scale of preference and, consequently, an indifference curve can be deduced. But it is not possible to deduce a unique utility function from a given indifference equation. There are in general an infinite number of utility functions which can generate a given indifference curve. For example, if we construct one utility function, we can get another by squaring this function or another by taking the logarithm of this function. In fact any monotonic transformation of utility function leaves the indifference curve unchanged. In this sense the utility function is indeterminate.

A.3.3 Marginal Rate of Substitution

The marginal rate of substitution (MRS) of y for x is defined as the amount of y that the person is willing to give up in order to gain an additional unit of x and still remain on the same indifference curve, i.e., maintain the constant level of utility. Mathematically, it is given as the ratio of marginal utility of x to y:

∂U (x, y)/ ∂x MRS − = . [A.8] y x ∂U (x, y)/ ∂y

From Eqs. A.6 and A.7, it can be shown that

dy a y MRS − = − = . [A.9] y x dx b x

Note that MRSy-x decreases with increase in x, which is consistent with law of diminishing marginal utility. It is also shown by the convexity of the utility indifference curve in x–y coordinates. An important point is that MRSy-x is invariant with respect to a monotonic transformation of the utility function. Appendix A Derivation of LQI 119 121 A.3.4 Elasticity of Substitution

Along an indifference curve, the degree of ease with which y can be substituted for x is quantified in terms of the elasticity of substitution σ, defined as the percentage change in the ratio y/x divided by the percentage change in the marginal rate of substitution of y for x (Hicks 1939). It is expressed as

d(y / x) /(y / x) σ = . [A.10] d(MRS y−x ) /(MRS y−x )

For the utility function under consideration, Eq. A.3, both the numerator and denominator turn out to be equal to a/b. It means that the elasticity of substitution is unity, σ = 1 irrespective of the values of exponents a and b.

A.3.5 Remarks

The properties of LQI in the light of utility theory can be summarized as follows: 1. The LQI is analogous to an ordinal utility function through which the society can rank its preferences for quality of life. 2. The LQI consists of two sub-utility functions: Gq and E with constant elasticity. 3. The indifference curve is convex to origin, and the elasticity of substitution is unity. For this particular class of utility functions, the indifference curve and the utility function are directly related to each other. The original derivation of LQI began with the differential equation of indifference curve under the assumption of constant elasticity (Nathwani et al. 1997), whereas the alternate started with an ordinal utility function of the form Eq. A.2 (Pandey and Nathwani 2003a, 2005). In both cases as expected, the marginal rate of substitution of income for longevity is identical. In this sense, the two formulations are identical. Comparing Eqs. A.1 and A.5, the condition to evaluate the societal capacity to commit resources (SCCR) implies that the society’s preference remains on the same utility indifference curve. The fact that the indifference curve may shift over time is not accounted for in the LQI formula- tion. A key parameter in the determination of societal capacity to commit resources is the marginal rate of substitution of income G for lifetime E. 120122 The Engineering Decisions for Life Quality A.4 The Role of GDP in LQI

A.4.1 Definition of GDP

A modern interdependent economy consists of a vast interlocking network of transactions and exchange of goods and services. The economy at a macro level resembles a circular flow of production → income → purchases → production, as shown in Fig. A.2. Production creates income, income creates expenditures, and spending calls forth production. The production is carried out by business firms for sale to consumers. Input to production primarily includes labor, capital, and “know-how” or productive utilization of capital and labor inputs through technology. Production requires the payment of wages and salaries to workers, and interest and dividend to investors. In turn, this income is used by workers and investors to purchase goods and services for consumption and re-investment. Production gener- ates income and income in turn is used to purchase the output (Beckerman 1968). The most comprehensive measure of national output is the gross domestic product (GDP) that captures the monetary value of all goods and services produced annually in a country.

Fig. A.2 The circular flow of economic activity Appendix A Derivation of LQI 123121 A.4.2 Measurement of GDP

In the system of national accounts, only productive activities that contrib- ute to flow of goods and services are included in the national product. Non-productive activities that involve redistribution among the community members of the goods and services produced in the economy are excluded. The calculation of GDP carefully excludes the double counting of inter- mediate goods and includes only the value-added activities of the production process. Transfer payments are excluded. There are two methods of measuring the GDP: the expenditure and the income methods.

A.4.3 Expenditure Method

GDP is the value of total expenditures incurred in the consumption of the final product, i.e., equivalent to aggregate demand, while all other inter- mediate inputs are ignored. GDP is the sum of three major components: private consumption, public consumption, and investment expenditure. The private consumption comprises mainly the consumption of households, i.e., food, clothes, domestic heating, etc. The public consumption is the expenditure incurred by public authorities, e.g., government expenditures on health care, education, policing, and national defense. Approxi- mately 80% of the GDP in OECD countries is spent on private and public consumption. Investment is that part of final output which takes the form of addition to, or replacement, of real productive assets, such as plants, equipment, and infrastructure. The investment in broad terms is seen as the means of adding to the economy’s wealth, i.e., its capacity to produce future incomes. It includes the domestic capital formation, changes in stocks (i.e., inventories of production goods and raw materials) and exports minus imports).

A.4.4 Income Method

GDP is also the total of earnings or income accruing to the basic factors of the production: labor receiving wages and benefits, and capital receiving the remainder (e.g. profit, economic rent). Labor comprises all the human contributions to the output. The capital is the stock of all productive assets, i.e., land, plants, machines, housing, trucks, ships, and so on. The circular flow diagram in Fig. A.2 shows that the flow of GDP is matched by a flow 122124 The Engineering Decisions for Life Quality of gross national income (GNI) in the lower half of the loop. GNI is the sum of all of the income (wages, profits, rent, and interest) earned in the production of GDP. By definition, GNI is equal to GDP. In summary, the two methods of measuring the GDP are merely different points at which the flow of money is measured. If measured at the point money is flowing into the productive sector then it is the expenditure method. If it is measured at the receiving end of household sector, it is the income method. The national accounts estimates are formally reconciled using the two methods to provide a consistent and reliable estimate.

A.4.5 GDP as an Indicator of Welfare

Welfare economists consider social income of individuals, comprising the totality of goods and services valued in terms of money, as a measure of welfare. By this definition, the GDP per person is synonymous with the social income (Hicks 1940). Hicks (1958) stated that the concept of social income is philosophically related to the assumption of integrated wants. If the person’s wants, which are to be satisfied by a set of real goods, are included in the income, then the goods purchased through income are input, and the outflow is a single abstract quantity called the utility. Consumption of different goods con- verges in the production of one single product: the utility. The measurement of social income in terms of production is therefore logically an indicator of welfare. This issue is further discussed in the first Human Development Report (UNDP 1990), which considered human development as the ability of people to live the kind of life they have reason to value. Real GDP per capita provides a good approximation of the relative power to buy commodities and to gain command over resources needed for decent living. In the accounting and assessment of development, the modern concept views individuals as the principal means (or contributors) to development as well as the ends. For example, the productivity of an individual contrib- utes directly to the aggregate wealth creation of a society. The income so generated increases the capacity of society to provide the necessary means, namely infrastructure (hospitals, schools, clean water, safe roads, and structures) that in turn benefits the individual via access to quality health and environment, length of life, and adequate means for cultural expres- sion. The LQI as a tool enhances our decision-making capacity and brings into sharp focus the choices and trade-offs we have to make between extension of life and creation of productive wealth. Appendix A Derivation of LQI 125123 A.5 Production of GDP

A.5.1 Production Function

The production function, Y, is a relationship between the factors of production (input) and the production of goods and services (i.e., GDP) in a period, usually of one year. Labor (W) and capital (K) constitute the factors of production. Labor refers to the effort (in person hours at work) required for production. Capital is the stock of goods used in the production process. The production function proposed by Cobb and Douglas (Samuelson 1970) is as follows:

Y = AK αW β . [A.11]

Here A represents the technological knowledge factor, and α and β are constants that are independent of K and W. Typically, Eq. A.11 is written in terms of index numbers, which are obtained by dividing the current values by those of some base year. For example

α β Y (t) = ⎛ K(t) ⎞ ⎛ L(t) ⎞ . [A.12] ⎜ ⎟ ⎜ ⎟ Yo ⎝ Ko ⎠ ⎝ Wo ⎠

The parameters in Eq. A.12 can be estimated from the analysis of economic production data.

A.5.2 Key Assumptions of Production Economics

A.5.2.1 Positive Marginal Products

The marginal product of capital and labor are both positive, i.e.,

∂Y ∂Y > 0 and > 0 . [A.13] ∂W ∂K

Thus an increase in either capital or labor will always increase the flow of output (production). The condition is satisfied when α > 0 and β > 0. 124126 The Engineering Decisions for Life Quality A.5.2.2 The Law of Diminishing Returns

Although an increase in labor (or capital) increases the output, successive increments in labor (or capital) produce diminishing increments in the flow of output. In mathematical terms, the rate of change of the marginal products of both labor and capital is negative:

∂ 2Y ∂ 2Y < 0 and < 0 . [A.14] ∂W 2 ∂K 2

This condition is satisfied when α < 1 and β < 1.

A.5.2.3 Profit Maximization Principle

In economics a firm’s output, wage rate, and investment are determined from the profit-maximizing decisions. Profit (Φ) is defined as the total revenue generated by the economic value of production minus the total cost of production, given as

Φ = Y −[sW + rK] [A.15] where s is the wage rate ($/hour), and r is the interest rate (or rent of capital) and the production is valued using standardized prices. The first-order condition for maximizing the profit with respect to the factors of production can be written as

∂Φ ∂Y ∂Y = − s = 0 or = s . [A.16] ∂W ∂W ∂W

It can be interpreted that a profit-maximizing firm will hire labor until its marginal value of product is equal to the real wage, s. Similarly, the marginal productivity of capital is equal to the rent, as shown below:

∂Φ ∂Y ∂Y = − r = 0 or = r . [A.17] ∂K ∂K ∂K

The marginal product of labor can be obtained from Eq. A.11 as

∂Y β − α Y = βW 1 (AK ) = β . [A.18] ∂W W Appendix A Derivation of LQI 125 127 Comparing Eqs. A.18 and A.16, we can show that β is a ratio of the total labor wages to the GDP:

(sW ) Wages β = = . [A.19] Y GDP

It is concluded that under the profit-maximizing condition, β is equivalent to the share of labor in the GDP. Similarly, the coefficient α is equivalent to the share of capital in the production. In the Cobb–Douglas production model, the wages are thus a fixed proportion of GDP. In traditional economics, labor productivity, pw, is defined as the ratio of the GDP (output) to the total labor input (work hours), i.e.,

Y s s p = = ⇒ β = . [A.20] W β W pW

For a constant β, an increase in productivity would accompany an increase in wage rates, and is a common empirical observation verified across many countries.

A.5.2.4 The “Return to Scale” Property

A change in the capital and labor will also change the production output. From Eq. A.11, the differential of the production function can be mathematically written as dY dK dW = α + β . [A.21] Y K W

Suppose both capital and labor are changed by a fixed proportion, i.e., dK/K = dW/W = λ, the change in the output is given as dY = λ(α + β ). [A.22] Y

Since α + β = 1, the production increases in the same fixed proportion as the proportion (λ) by which all the factors of production are increased. This is referred to as the constant return to scale property of the production function. If, however, α + β > 1, then Y increases proportionately more than the factors K and W; this is called increasing return to scale, and 126128 The Engineering Decisions for Life Quality its use is not common in production economics. The case of α + β < 1 sig- nifies the decreasing return to scale, i.e., inefficient process of production. Using Eqs. A.11, A.16, and A.17, it can be shown that

(α + β )Y = sW + rK . [A.23]

An interesting implication of the constant return to scale property (α + β =1) is that if capital and labor are fully paid their marginal product in maximizing the profit, the total product will be exhausted without any sur- plus or deficit (Wicksell 1893).

A.5.3 Economics of Labor–Leisure Trade-off

The labor supply decision by consumers is also related to the utility maximization proposition (Samuelson 1970). It is assumed that a rational consumer attempts to maximize consumption, but minimizes the amount of work required to earn income to fulfill the desire for consumption. We here denote consumption as C, work time as W, work-free or leisure time R, the price of consumption as p and labor (wages) as s per unit. Then the consumer faces a budget constraint that the cost of consumption should not exceed the income earned, pC ≤ sW. The maximum amount of labor time available (= T) to a person is fixed, e.g., 24 hours in a day, i.e. T = W + R. The budget constraint can be modified as pC + sT ≤ sW + sT, or pC + sR ≤ sT. The consumer maximizes a utility function, U(C, R) and the corresponding conditions are:

1 ∂U 1 ∂U ∂U / ∂R s = ⇔ = = MRS − . [A.24] p ∂C s ∂R ∂U / ∂C p C R

In summary, the consumer will choose between consumption and leisure so that the ratio of marginal utilities (or marginal rate of substitution of consumption for leisure) is equal to the real wage (s/p). The wage rates by themselves are considered as indeterminate (Samuel- son 1970). Increases in the real wage can have two effects. The first is the substitution effect, meaning that the greater the real wage, the costlier the leisure becomes relative to forgone income (and consumption). Thus the consumer will supply more labor and increase the work time. The second effect is the income effect. It implies that the greater the wage, the less labor time required to support the same level of consumption as before, thereby resulting in an increase in leisure time. In general the substitution ef- Appendix A Derivation of LQI 127 129 fect dominates at low wages, whereas the income effect dominates at higher wages. Empirical evidence over the past century has shown a decrease in the labor time and increase in wage levels, suggesting strong support for the view that the income effect dominates as economies develop (Samuelson 1970).

A.6 Derivation of the LQI

The general idea in welfare economics is that a person’s enjoyment of life- quality or utility in an economic sense arises from a continuous stream of resources available for consumption over the entire life. Therefore, income required to support consumption and the time to enjoy it are two determi- nants of the life quality. The potential lifetime utility of a person can be interpreted as the social income (G $/person/year) utility over the work-free (leisure) lifetime (tR). An ordinal utility function can then be defined as

⎛ 1 ⎞ = = q . [A.25] LU U (G)tL ⎜ G ⎟tR ⎝ q ⎠

The function, U(G), is the utility per unit time, since G is a rate quantity. This function used in economic analysis, exhibits a constant relative risk aversion equal to GU"/U' = (1 – q), where U' and U" denote the first and second derivatives of the utility function U. The coefficient of risk aversion also determines the person’s willingness to shift consumption between different periods (Romer 2001). In the present context, the utility function of income U(G) = Gq/q serves to illustrate the important point that there are diminishing returns in transforming income into human capabilities. Considering G as a constant and the remaining lifetime as a random variable, the expected utility can be derived as (Pandey and Nathwani 2003a)

⎛ G q ⎞ = ⎜ ⎟ . [A.26] LQ ⎜ ⎟eR ⎝ q ⎠ where eR is the work-free life expectancy. The derivation is based on LQI described in terms of work time and then using the labor–leisure trade-off. Suppose productive work time per person is w years/year and the number of persons in the society is N. The total labor input is therefore W= wN years/year and capital stock invest- 128130 The Engineering Decisions for Life Quality ment as K = kN $/year. Substituted in the production function [A.11] this gives

α β Y = A(kN) (wN) [A.27] that can be rearranged as

Y α β = G = A(k) (w) . [A.28] N

Now substituting for G from Eq. A.28 and eR = (1 - w)E into Eq. A.26 gives

1 α β L = [Ak ]q (w )q (1− w)E . [A.29] Q q

We assume that the capital investment per person (k) and technological factors (A) are independent of the work-time fraction (w). Using the labor– leisure trade-off, the first-order optimality condition is expressed as dL 1 w Q = 0 ⇒ q = . [A.30] dw β (1− w)

The assumption of k being independent of w is examined in the next section. The use of the production function in the LQI derivation was first presented by Pandey (2005). Substituting for G, q, and eR in Eq. A.26 gives

β − = ⎛ (1 w) ⎞ q − = q LQ ⎜ ⎟G (1 w)E CG E , [A.31] ⎝ w ⎠ where C = β (1 − w)2 / w is an arbitrary constant.

Finally, setting K = 1/q, and setting C = 1, and making the monotonic K transformation LQI = (LQ) , yields the equivalent form of the Life Quality Index used in this book,

LQI = EKG. [A.32]

As economies develop over time, an optimum balance between the productive labor and leisure time is maintained. Changes in the value of w are Appendix A Derivation of LQI 129 131 generally modest in the short term. Rackwitz (2004) analyzed economic and a public opinion data and showed that in developed economies the balance between productive labor and leisure time is close to a stable optimum. In Appendix C we summarize economic data from the Organization for Economic Cooperation and Development (OECD) related to GDP, population and work time that are relevant to the calibration of LQI. This permits accurate calibration of the exponent K.

A.7 Summary

The LQI is derived using the concepts of economics sciences and a lifetime utility function as

LQI = EKG [A.33] where K is a parameter. K depends on the economic structure of a society in terms of the coefficient of labor share in the GDP and the annual work- time fraction. Using OECD economic data, the value of K is determined in Appendix C. K is approximately equal to 5.0. There is a unified basis of deriving the LQI in the context of ordinal utility theory. Appendix C illustrates the process of deriving all LQI parameters specific to a country through economic data analysis.

References

Allen RGD (1935) A note on the determinateness of the utility function. Rev Econ Stud 2(2):155–158 Beckerman W (1968) An introduction to national income analysis. Weidenfeld and Nicolson. London, U.K Hicks JR (1939) The foundation of welfare economics. Econ J 49(196):696–712 Hicks JR (1940) Valuation of social income. Economica, No. 26 Hicks JR (1958) The measurement of real income. Oxford Econ Pap Hicks JR (1975) The scope and status of welfare economics. Oxford Economics Paper 27(3):307–326 Kubler O, Faber MH (2005) LQI: On the correlation between life expectancy and the gross domestic product per capita. Proc 9th International Conference on Structural Safety and Reliability, ICOSSAR 1–8, Rome Lind NC (2002) Social and economic criteria of acceptable risk. Reliab Eng Sys Saf 78(1):21–26 Lind NC (2007) Turning life into life expectancy: The efficiency of life-saving interventions. Int J Risk Assess Manag 7(6/7):884-894 Lind NC, Nathwani JS, Siddal E (1992) Managing risks in the public interest. Institute of Risk Research, University of Waterloo, Waterloo, ON 130132 The Engineering Decisions for Life Quality Maes MA, Pandey MD, Nathwani JS (2003) Harmonizing structural safety levels with life quality objectives. Can J Civ Eng 30(3):500–510 Nathwani JS , Lind NC, Pandey MD (1997) Affordable safety by choice: The life quality method. Institute for Risk Research, University of Waterloo, Waterloo, ON Organization of Economic Development and Cooperation (OECD). http://www.oecd.org/statistics. Accessed Nov 28, 2007 Pandey MD (2005) A discussion of derivation and calibration of the Life-Quality Index. Proc. 9th International Conference on Structural Safety and Reliability, ICOSSAR June 22–25, 2005, Rome Pandey MD, Nathwani JS (2007) Foundational principles of welfare economics underlying the life quality index for efficient risk management. Int J Risk Assess Management 17(6/7):862–882 Pandey MD, Nathwani JS (2003a) Life-Quality Index for the estimation of societal willingness- to-pay for safety. J Struct Saf 26(2):181–199 Pandey MD, Nathwani JS (2003b) Canada Wide Standard for Particulate Matter and Ozone: Cost-benefit analysis using a life-quality index. Int J Risk Anal 23(1):55–67 Pandey MD, Nathwani JS (2003c) A conceptual approach to the estimation of societal willingness-to-pay for nuclear safety programs. Int J Nucl Eng Design 224(1):65–77 Pigou AC (1920) Economics of welfare. Macmillan, London Rackwitz R (2002) Optimization and risk acceptability based on the life quality index. J Struct Saf 24:297–331 Rackwitz R (2003) Optimal and acceptable technical facilities involving risks. Int J Risk Anal 24:146–158 Rackwitz R (2004) The philosophy behind Life Quality Index and empirical verification. Memo- randum to Joint Committee on Structural Safety Romer D (2001) Advanced macroeconomics. McGraw-Hill, New York Samuelson PA (1970) Economics. 8th Edition. McGraw-Hill, New York Stigler GJ (1950) The development of utility theory: Part 1 and 2. J Polit Econ 58:307– 327 and 373–396 United Nations Development Program (1990) The human development report. Oxford Univer- sity Press, UK Wicksell K (1893) Value, capital and rent. Reprints of Economic Classics, 1954. Augustus M. Kelley Publishers, New York Wicksteed PH (1888) The alphabet of economic sciences: Elements of the theory of value or worth. Reprints of Economic Classics, 1970, Augustus M. Kelley Publishers, New York

Appendix B Discounting

Abstract In the first five sections we present the philosophies, recommendations, and directives by selected government agencies and firms. Risk assessment often involves considerations of risks in the distant future, considered in Sect. B.6.

B.1 Introduction

Cost–benefit analysis (CBA) is a structured approach to account for the multiple consequences of a project, a plant or a proposed scheme. Its origin has been traced to the introduction of the USA Flood Control Act in 1936, which stated the simple principle that a flood control project should be deemed reasonable if “the benefits to whomsoever they may accrue are in excess of the estimated costs” (Pearce and Nash 1981). This is perhaps the first introduction of welfare economics in public decision-making without recognizing the limitation of the definition of just whose benefits should count. Cost–benefit analysis is a way to make rational comparisons between alternative investments to assess whether they are worth undertaking. Since these investments have benefit streams that extend over long periods of time, it is necessary to calculate their present value by taking into account the time value of money. This rate of return, conceptually similar to an interest rate, is referred to as the discount rate. The reason for discounting is to represent the generally acceptable proposition that a dollar in a future year is worth less than a dollar in the current year. In other words, people prefer to consume a given amount of resources now than in the future. Different discount rates are used to evaluate private and public investments in the economy (Spiro 2008): • Businesses use return on equity (ROE) or weighted average cost of capital (WACC) after tax rates to discount investment costs and private returns accruing to them on an after-tax basis in unregulated markets. • Regulatory agencies allow companies to earn a specified rate of return on capital depending on the company’s deemed conditions of capital structure and risk. 132136 The Engineering Decisions for Life Quality • Households postpone some consumption in favor of savings, depending on interest rates on savings accounts or other personal savings in- struments. • Governments undertake (or mandate) projects of infrastructural, environmental, or health and safety enhancement in the wider public interest, and merit of projects is assessed in terms of the long-term return to current and future generations of society as whole. The discount rate for public financing of projects in the public interest rate is referred to as a social discount rate (SDR). The SDR is normally applied to projects whose outcomes include benefits, costs, and forgone opportunities that endure into the long term and affect future generations. The benefits of such projects extend beyond the specific services sold to specific customers, but are dispersed widely as societal benefits. For private corporations making such calculations, the discount rate is a relatively straightforward calculation of the actual cost of funds, being a weighted average of return on equity and interest on debt. However, in the case of government projects, the use of the actual borrowing rate as the discount rate can lead to misleading conclusions. The government is able to borrow large sums of money at low interest rates, but this interest rate may not necessarily reflect the opportunity cost of capital. Unlike a corporation, the government’s credit rating does not derive from its balance sheet. Governments are able to borrow money primarily due to their power to raise revenue through taxation. If it is used as the discount rate for evaluating investment projects, it may lead to inefficient use of the government’s borrowing capacity. The social discount rate (also known as the economic cost of capital) seeks to mimic the rate of return that would be earned on private sector investments. Inefficiencies in the government’s use of capital are minimized by requiring government investments to meet a rate of return hurdle similar to that of the private sector. The following four sections of this appendix summarize discount rate policies adopted by government and agencies in Canada, the USA, the UK, and New Zealand. The last section considers discounting of risks, costs, and benefits in the far future.

B.2 The Canadian Cost–Benefit Analysis Guide (2007)

For each option under consideration, the stream of costs and benefits will usually not occur in the same year but be spread over several years. Dis- Appendix B Discounting 137133 counting allows for the systematic comparison of costs and benefits that occur in different time periods by allowing one to calculate the net present value of the intervention. If the costs and benefits are expressed in current prices or nominal dollars, they should be deflated to reflect real prices or prices expressed in terms of the price level of a specific year. In this way, the changes in the reported values of benefits and costs over time that are due purely to inflation are removed. The discounted present value of net benefits is the algebraic sum of the present values of the expected incremental net benefits of the policy option over and above the baseline scenario during the policy’s anticipated impact time period. If the net present value (NPV) is greater than or equal to zero, then the policy is expected to generate more benefits than costs and should be recommended for implementation. However, if the NPV is less than zero, the policy should not be recommended for implementation on efficiency grounds.

B.2.1 Rational Approaches to Discount Rates

Choosing a discount rate has been one of the most controversial aspects of the cost–benefit analysis of regulatory policies. The term discount rate refers to the time value of the costs and benefits from the viewpoint of society. It is similar to the concept of the private opportunity cost of capital used to discount a stream of net cash flows of an investment project, but the implications can be more complex. With costs and benefits expressed in real values, people prefer to make payments later and receive benefits sooner. This is due to the fact there is a time preference for current consumption over future consumption. Simi- larly, there is an opportunity cost of the resources invested in any given activity, as they could have been invested elsewhere if they had not been spent on the activity being evaluated. One approach to discounting is based on the fact that present consumption is valued differently from future consumption. Following this approach, all benefits and costs are first converted into quantities of consumption equivalents before being discounted. In this case, the discount rate is the rate of time preference at which individuals are willing to exchange consumption over time. Another approach considers what society forgoes in terms of pre-tax returns of displaced investment in the country. Using this approach, no account is made for time preference of present versus future consumption. The discount rate is based purely on the opportunity cost of forgone investments. An approach that captures the essential features of both these 134138 The Engineering Decisions for Life Quality two alternatives uses a weighted average of the economic rate of return on private investment and the time preference rate for consumption (Sandmo and Dreze 1971, Arnold 1972). Many professionals have chosen to use a discount rate that follows this weighted average opportunity cost of funds concept. A natural place to look for the relative weights to place on the rate of time preference and the gross rate of return on investment is the re- sponse of the capital market to extractions or injections of funds. On the cost side, the marginal source of funds for both the public and private sec- tors is usually from borrowing either domestically or from abroad. Like- wise, if benefits arise that create income, it will be in the first instance de- posited in financial institutions, where it is available to finance other activities. While this approach is not without its restrictions, these pale in comparison to the practical problems that arise if the rate of time preference is used as the rate of discount for such interventions (Sjaastad and Wisecarver 1977). Other questions have been raised as to whether a lower rate should be used for intergenerational discounting because many of the people affected by some policy or regulation may no longer be alive in the distant future. However, there is little consensus in the literature on discounting for intergenerational policies. Applying one discount rate to the streams of costs and another to the streams of benefits can be tricky and empirically difficult for each policy because of the requirements for converting all the streams of costs into consumption equivalents in a consistent manner. In Appendix B.6 we describe an approach to the discounting of risk in the distant future.

B.2.2 Discount Rates

When a program requires funds that are extracted from the capital markets, the funds are drawn from three sources. First, funds that would have been invested in other investment activities have now been displaced by expenditures required by the policy action. The cost of these funds is the return that would have been earned on the alternative investments. Second, funds come from different categories of savers in the country who postpone their consumption in the expectation of getting a return on their savings. The cost of this part of the funds is reflected in the interest rate that the savers earn net of personal income tax. Third, some funds may come from abroad, that is, from foreign savers. The cost of these funds would be the marginal cost of foreign borrowing. At the margin, the cost associated with incremental foreign borrowing is measured by the interest expense on the incremental borrowings plus the marginal change in the cost of foreign Appendix B Discounting 135 139 borrowing times the quantity of the stock of foreign debt negotiated at variable interest rates. The discount rate will be a weighted average of the costs of funds from the three sources outlined above: the rate of return on postponed investment, the rate of interest (net of tax) on domestic savings, and the marginal cost of additional foreign capital inflows. The weights are equal to the proportion of funds sourced from domestic private-sector investors, domestic private-sector savers, and foreign savers. Based on the above approach, the discount rate for Canada was re- estimated recently by Jenkins and Kuo (2007) and found to be a real rate of approximately 8%. This rate is lower than the real rate of discount of 10% recommended by the Treasury Board of Canada Secretariat in 1998 but is higher than the 7% real rate proposed by Burgess in 1981 and the 7.3% real rate recommended by Brean et al. (2005). This rate of 8% is consistent with the 10% estimated earlier and used in the Treasury Board guidelines of 1976 and 1998 (Jenkins 1972). The effective rate of corporate income tax in Canada has been steadily decreasing over the last two decades. Furthermore, the introduction of the goods and services tax has removed much of the burden of the sales tax system from the value added of capital. Both these policy changes will tend to lower the required gross of tax rate of return on capital. The Treas- ury Board of Canada has recommended that a real rate of 8% be used as the discount rate for the evaluation of regulatory interventions in Canada. In certain circumstances where consumer consumption is involved and there are no or minimal resources involving opportunity costs (such as certain human health and environmental goods and services), some federal departments, governments, and international organizations have taken into consideration factors other than the economic opportunity cost of funds when developing their recommendations for the value of the discount rate. The social discount rates based on these considerations are lower than the 8% recommended by the Treasury Board. One approach is to estimate the social time preference rate, which is based on the rate at which individuals discount future consumption and projected growth rate in consumption. For Canada, the social-time preference rate has been estimated to be around 3%. In these circumstances, the net present value of the results of the analysis can also be carried out using a social discount rate of 3% accompanied by the use of a shadow price of investment that is applied to all the costs of the intervention that results in a postponement or reduction of investment activity. However, there is still controversy in the literature on the use of these social discount rates and further guidance will be needed in the future. 136140 The Engineering Decisions for Life Quality The Canadian government in 2007 established the Center of Regulatory Expertise that for a period of 5 years will help departments and agencies adjust to the new approach to regulating, including cost–benefit analysis, instrument choice, and performance measurement. This assistance will include the provision of specialist analytical services. Departments and agencies are expected to discuss their approach to cost–benefit analysis with their Treasury Board of Canada Secretariat analyst, including the need for and approach to discounting any longer-term costs and benefits associated with proposals involving, for example, health and environmental regulation.

B.3 Benefit–Cost Analysis Guide, Treasury Board of Canada

Unlike most individuals and organizations, governments frequently take two different points of view in assessing investments – the fiscal point of view (is the project a good one from the government's narrow fiscal perspective?) and the social point of view (is the project a good one for the country?). The discount rates can be quite different from these two perspectives.

B.3.1 The Fiscal Discount Rate

The fiscal discount rate is the government's cost of borrowing. It is appropriate to use the actual cost of borrowing when the analysis is from the narrow fiscal point of view of the government and the marginal funds for the investment come from borrowing rather than from increased taxes. The fiscal discount rate tends to be low because governments are generally preferred borrowers (taxation is in the background as a guarantee of repayment). The use of the fiscal point of view and thus of the fiscal discount rate is only appropriate when the proposed investment has few, if any, social implications. Examples are decisions to purchase computers or lease minor accommodation. If the project is large enough to matter to the general economy or if it has aspects that are of interest to the public, then the narrow fiscal point of view is probably inappropriate. Appendix B Discounting 137 141 B.3.2 The Social Discount Rate

The social discount rate is roughly equal to the opportunity cost of capital, weighted according to the source of investment capital. For the Government of Canada, this is foreign borrowing, forgone investment in the private sector, or forgone consumption. If you know what the government's investment is displacing and what the rates of return would have been for the displaced uses, then you can calculate the opportunity cost. Essentially, the argument is that the government must achieve a return on investment at least equivalent to what the money would earn if left in the private sector to justify taxing the private economy to undertake public-sector investments. If the government cannot achieve this it would be better for Canada that the money be left untaxed in the private sector. Since 1976 Treasury Board has required that benefit–cost analysts use a social discount rate of 10% “real” per annum – that is, a 10% discount rate applied to real dollars (constant, inflation-adjusted dollars). This rate is a stable one because it reflects an opportunity cost in the private sector where the average rate of return to investment (over the whole economy) changes very slowly over the years, if at all. The government’s estimate of the social discount rate has been robust, despite some challenges over the years. Social discount rates as low as 7.5% real and as high as 12% real have been proposed and supported by various economists (Treasury Board, 1998, 2007). Estimates by the Department of Finance, however, have consistently supported the 10% real estimate of the social discount rate. Currently, the only serious challenge to the 10% social discount rate is from those who argue that high discount rates unfairly devalue benefits to future generations, who have as much right to such basics as clean water and clean air as the current generation does. This argument for low discount rates in the public sector is not well based, however. A project with a high rate of return when all its costs and benefits are counted is better for the present generation and, through reinvestment, better for future generations as well. Only when benefits are non-renewable and consumed rather than reinvested is there conflict across generations, with one generation paying and another benefiting. Manipulating the discount rate does not lessen this conflict. It has to be addressed directly by intergenerational consumption analysis. 138142 The Engineering Decisions for Life Quality B.3.3 The Rate-of-Time Preference for Consumption

Considerable confusion in benefit–cost analysis has been caused by analysts using different numeraires (the units of value). To avoid confusion, one should generally use a “dollar of investment” as the numeraire and 10% per annum real as the social discount rate. This common approach to investment and rates of return is familiar to economists and non-economists alike. On the other hand, it is possible (and perhaps theoretically more precise) to use a dollar of consumption as the numeraire. After all, investment is not a final value in the way consumption is. The social rate-of-time preference for consumption is normally taken to be about 4%. This is obviously a much lower discount rate than 10% and on the surface may seem more attractive to those who think that benefits in the distant future (say, general environmental benefits) should not be discounted too heavily. However, if you use a dollar of consumption as the numeraire and a social rate-of-time preference for consumption as the discount rate, then (to make the analysis fair and consistent) you must calculate shadow prices for the investment funds in terms of a stream of consumption forgone. Economists who have calculated the shadow price of a dollar of investment funds in Canada and the United States have found that it is about $2.50 in “consumption dollars”. The important point is that the rate used in the first approach (10% discounting of costs and benefits, expressed in an investment-dollar numeraire) and the rate used in the second approach (4% discounting of costs and benefits, expressed in a consumption-dollar numeraire, with a dollar of investment funds shadow priced at $2.50) give the same result when properly applied. Because the outcome of either approach, properly done, is the same, it makes sense to stay with the more easily understood concept of an investment-dollar numeraire and a 10% discount rate (on which everyday thinking about investment and rate of return is based). What is not acceptable is to confuse the two approaches. To use a 4% discount rate without shadow-pricing the investment funds is incorrect.

B.3.4 Strategic Effects of High and Low Discount Rates

The choice of a discount rate is important. It has a strong (although hidden) influence on the choices that an organization will make with respect to investment in specific projects and policies. A low discount rate is favorable for the following: Appendix B Discounting 139 143 • An active investment program, because capital seems inexpensive • Outright purchase of assets • Many and larger projects and programs • Projects whose benefits may be long-term A high discount rate is favorable for the following: • A cautious capital investment program, because capital seems expensive • Leasing and other kinds of deferred-payment options • Short-term, flexible planning • Labor-intensive rather than capital-intensive solutions

B.3.5 The Discount Rate as a Risk Variable

The 1976 Treasury Board Benefit–cost Analysis Guide recommended a social discount rate of 10% real, and 5% to 15% real per annum in sensitivity analysis. Experience has shown, however, that this range was too broad. Most projects look good at a 5% discount rate and poor at a 15% discount rate. A credible and more useful range for the social discount rate is normally about 8–12% real per annum (for risk analysis), with a most likely value of 10% real per annum. Because there is some uncertainty about the correct value of the discount rate, it is important to note that the discount rate should be included as a risk variable in the parameter table of a benefit–cost and risk analysis using simulation. This makes it less important to fix on a precise value of the discount rate and places more emphasis on identifying the likely range of values of the discount rate and on interpreting the results of the financial simulation.

B.3.6 Best Practice – Inflation Adjustments and Discounting

• To ensure that changes in relative prices are properly recognized, tables of costs and benefits should be first constructed in nominal dollars, and cash flows should be set out for each period to the investment (financing) horizon. Conversions to constant dollars or to present value dollars should wait until all costs and benefits over time are worked out in nominal dollars. • Adjusting for inflation is not the same thing as discounting to present values, so each should be done independently. 140144 The Engineering Decisions for Life Quality • The appropriate discount rate depends on the point of view of the analysis and also on the choice of numeraire. • The Government of Canada uses a fiscal discount rate (based on a narrow “internal” point of view that is appropriate mostly for small projects) and a social discount rate (based on a nation-wide point of view). With the normal dollar of investment as the numeraire, the appropriate social discount rate (as measured by the Department of Finance and Treasury Board of Canada Secretariat) is about 10% real per annum. The plausible range for risk analysis is 8–12%.

B.4 US Office of Management and Budget (OMB) A94: Guidelines for Benefit–Cost Analysis of Federal Programs

B.4.1 General Principles

Benefit–cost analysis is recommended as the technique to use in a formal economic analysis of government programs or projects. Cost-effectiveness analysis is a less comprehensive technique, but it can be appropriate when the benefits from competing alternatives are the same or where a policy decision has been made that the benefits must be provided. The standard criterion for deciding whether a government program can be justified on economic principles is net present value – the discounted monetized value of expected net benefits (i.e., benefits minus costs). Net present value is computed by assigning monetary values to benefits and costs, discounting future benefits and costs using an appropriate discount rate, and subtracting the sum total of discounted costs from the sum total of discounted benefits. Discounting benefits and costs transforms gains and losses occurring in different time periods to a common unit of measurement. Programs with positive net present value increase social resources and are generally preferred. Programs with negative net present value should generally be avoided. Although net present value is not always computable (and it does not usually reflect effects on income distribution), efforts to measure it can produce useful insights even when the monetary values of some benefits or costs cannot be determined. Appendix B Discounting 145141 B.4.2 Discount Rate Policy

In order to compute net present value, it is necessary to discount future benefits and costs. This discounting reflects the time value of money. Ben- efits and costs are worth more if they are experienced sooner. All future benefits and costs, including non-monetized benefits and costs, should be discounted. The higher the discount rate, the lower is the present value of future cash flows. For typical investments, with costs concentrated in early periods and benefits following in later periods, raising the discount rate tends to reduce the net present value.

B.4.3 Real Versus Nominal Discount Rates

The proper discount rate to use depends on whether the benefits and costs are measured in real or nominal terms. 1. A real discount rate that has been adjusted to eliminate the effect of expected inflation should be used to discount constant-dollar or real benefits and costs. A real discount rate can be approximated by subtracting expected inflation from a nominal interest rate. 2. A nominal discount rate that reflects expected inflation should be used to discount nominal benefits and costs. Market interest rates are nominal interest rates in this sense.

B.4.4 Public Investment and Regulatory Analyses

The guidance in this section applies to benefit–cost analyses of public investments and regulatory programs that provide benefits and costs to the general public. In general, public investments and regulations displace both private investment and consumption. To account for this displacement and to promote efficient investment and regulatory policies, the following guidance should be observed. 1. Base-case analysis: Constant-dollar benefit–cost analyses of proposed investments and regulations should report net present value and other outcomes determined using a real discount rate of 7%. This rate ap- proximates the marginal pre-tax rate of return on an average investment in the private sector in recent years. Changes in this rate are reflected in updates on an ongoing basis. 142146 The Engineering Decisions for Life Quality 2. Other discount rates: Analyses should show the sensitivity of the discounted net present value and other outcomes to variations in the discount rate. The importance of these alternative calculations will depend on the specific economic characteristics of the program under analysis. For example, in analyzing a regulatory proposal whose main cost is to reduce business investment, net present value should also be calculated using a higher discount rate than 7%. 3. Analyses may include among the reported outcomes the internal rate of return implied by the stream of benefits and costs. The internal rate of return is the discount rate that sets the net present value of the program or project to zero. While the internal rate of return does not generally provide an acceptable decision criterion, it does provide useful information, particularly when budgets are constrained or there is uncertainty about the appropriate discount rate. 4. Using the shadow price of capital to value benefits and costs is the analytically preferred means of capturing the effects of government projects on resource allocation in the private sector. To use this method accurately, the analyst must be able to compute how the benefits and costs of a program or project affect the allocation of private consumption and investment. OMB concurrence is required if this method is used in place of the base case discount rate.

B.5 The UK Treasury Green Book

B.5.1 Introduction

This section shows how the discount rate of 3.5% real is derived and the circumstances in which it should be applied.

B.5.2 Social Time Preference Rate

Social time preference is defined as the value society attaches to present, as opposed to future, consumption. The social time preference rate (STPR) is a rate used for discounting future benefits and costs, and is based on comparisons of utility across different points in time or different generations. This guidance recommends that the STPR be used as the standard real discount rate. Appendix B Discounting 147143 The STPR has two components: 1. The rate at which individuals discount future consumption over present consumption, on the assumption that no change in per capita consumption is expected, represented by ρ 2. An additional element, if per capita consumption is expected to grow over time, reflecting the fact that these circumstances imply future consumption will be plentiful relative to the current position and thus have lower marginal utility. This effect is represented by the product of the annual growth in per capita consumption (g) and the elasticity of marginal utility of consumption (μ) with respect to utility. The STPR, represented by r, is the sum of these two components, i.e., r = ρ + μg. Each element of STPR is examined in turn below. Estimates of ρ comprise two elements, namely catastrophe risk (L); and pure time preference (δ). The first component, catastrophe risk, is the like- lihood that there will be some event so devastating that all returns from policies, programs or projects are eliminated, or at least radically and un- predictably altered. Examples are technological advancements that lead to premature obsolescence, or natural disasters, major wars, etc. The scale of this risk is, by its nature, hard to quantify (Newbery 1992, Kula 1987, Pearce and Ulph 1995, OXERA 2002). The second component, pure time preference, reflects individuals’ preference for consumption now, rather than later, with an unchanging level of consumption per capita over time (Scott 1989). The evidence suggests that these two components indicate a value for ρ of around 1.5% a year for the near future (Scott 1977, 1989, OXERA 2002). Estimates of μ are based on the available evidence that suggest the elasticity of the marginal utility of consumption is around 1.4. This implies that a marginal increment in consumption to a generation that has twice the consumption of the current generation will reduce the utility by half. Estimates of g are also based on study of past economic growth. Maddi- son (2001) shows growth per capita in UK to be 2.1% over the period 1950 to 1998. Surveying the evidence, OXERA (2002) also suggests a figure of 2.1% for output growth to be reasonable. The annual rate of g is therefore put at 2% per year. A representative value of STPR can thus be calculated using the data g = 2%, ρ = 1.5%, and μ = ,, which turns out to be STPR (= 1.5 + (1×2)) = 3.5%. 144148 The Engineering Decisions for Life Quality B.5.3 Long-Term Discount Rates

Where the appraisal of a proposal depends materially upon the discounting of effects in the very long term, the received view is that a lower discount rate for the longer term (beyond 30 years) should be used (OXERA 2002). The main rationale for declining long-term discount rates results from uncertainty about the future. This uncertainty can be shown to cause declining discount rates over time (Weitzman 2001, Gollier 2002). In light of this evidence, it is recommended that for costs and benefits accruing more than 30 years into the future, appraisers use the schedule of discount rates provided in Table B.1 below.

Table B.1 The declining long-term discount rate

Period of years 0–30 31–75 76–125 126–200 201–300 300+ Discount rate 3.5% 3.0% 2.5% 2.0% 1.5% 1.0%

Table B.2 Summary of social discount rates recommended by selected agencies

Social discount rate Agency (SDR) Remarks Treasury Board of Canada Real 8% As low as Different approaches to SDR (p. 35) Secretariat, Canadian Cost–Benefit 3% Social rate of time preference (RTP), Analysis Guide 2007 e.g. interest after tax on personal savings; Social opportunity cost (SOC) of capital before tax Where personal consumption is involved, use RTP ~ 3% for Canada (pp. 37–38) Costs and benefits to be discounted at same rate (p. 38) Uncertain costs, benefits treated in Monte Carlo rather than adjusting discount rates (p. 36) Canada Gazette, Part II, Vol. 140, Real 7% Real For funding conservation No.23, Nov 15, 06 5% –10% Ontario Ministry of Public Infra- Real 5% Real SDR based on social opportunity cost structure Renewal, and formerly at (SOC) of capital Ontario Ministry of Finance, Jan 3% – 7% Jan 2007 paper calculates SDR 5% real 2007, update Mar 2008 (P. Spiro) based on cost of capital before Ont. tax, after fed. tax Appendix B Discounting 145 149

Mar 2008 paper calculates SDR 5% real rate after tax as= (real after-tax ROE=7.7% * nonfinancial corp equity/assets = 50%) + (Ontario Govern- ment real long bond yield=2.2% × D/E=50%) Acknowledges that its after-tax view conflicts with most Canadian studies. Neglects recycling of fed. tax revenues to Ontario residents, since 2005 recycling was only 15% States that discrate not to be adjusted for project-specific uncertainties; costs themselves adjusted States that government pooling of risk justifies lower discrate than implied by corporate bonds States that Infrastructure investment could have a positive impact on private sector productivity Assumes gov’t borrowing does not crowd-out private investment Implied WACC before tax (OPA calculation) = 7.0% = 0.50 × 7.7%/(1 - 34.5%) + 0.50 × 2.2% DDS Management Consultants, Real 5% Real 3% Results robust over wide range of al- “Replacing Ontario’s Coal-Fired – 10% ternative values for discount rate Electricity Generation,” prepared Used Monte Carlo and sensitivity anal- for the Ontario Ministry of Energy, ysis Apr 2005 H. M. Treasury, Green Book, Real 3.5% Note 5.49 Appraisal and Evaluation in Central Social rate of time preference (RTP), Government, Jan 16, 2003 calculated as after-tax rate of return on money lent or borrowed US Office of Management & Real 3%, Budget, Circular No. A-94, App. C, Revised Jan 2007 UK Department for Business, Real 2.2% Enterprise & Regulatory Reform, (p.62) Meeting the Energy Challenge: A White Paper on Nuclear Power, Jan 2008 C.D. Howe Institute, Real 2% Real Main conceptual bases of SDR: Building the Future – Issues in Pub- 2% – SOC = marg. rate of return on low-risk lic Infrastructure in Canada, Policy 6.6% private investment before tax (pp. 91– Study 34, “The Social Discount 96) Rate in Canada” by M.A. Moore, Value based on Industrial equities and A.E. Boardman, D.H. Greenberg, long bonds ~ real 8.8% (Footnote May 2001 p. 94) 146150 The Engineering Decisions for Life Quality

Low Risk: captures systematic (market- aggregate) risk, but not unsystematic (project-or firm-specific) risk; since lat- ter represents transfers within society and can be diversified away Assumes: project financed entirely by borrowing from residents; investment responsive to changes in interest rates; savings not responsive to changes in interest rates Implies: project crowds out mostly private investment and some private consumption Contra-indications: project may borrow from nonresidents; project may be financed by taxes; crowding out is di- minished if the economic resources are not fully utilized (e.g. if there is significant unemployment); project could en- courage private invest. due to infrastructure improvement RTP = marg. rate of time preference (pp. 96–98) Willingness to exchange current consumption for future consumption Value based on real yield on long government bonds after tax ~ 0% – 4% Assumes: project financed entirely by taxes which affect mostly consumption, no borrowing from nonresidents Contra-indications: savers earn less on incremental saving than borrowers do on decremental borrowing; ignores effects on future generations Range of recommended real SDR values Environmental/health effects for long- term: use 2.0% Long-term projects w. intergenerational effects: use 3.2% No intergenerational effects: Tax-financed projects use 3.3% Debt-financed projects use 6.6% University College London Real 2.4% Real 2% Evidence for UK Centre for Social and Economic – 4% Research on Global Environment

New Zealand Treasury, Working RTP reflects social preferences in addi- Paper 02/21, Sept 2002 tion to financial sector considerations (p. 5) Shadow price of capital complicated to calculate, not directly observable in the Appendix B Discounting 147 151 marketplace (p. 7)

Saha International, Discount Rate SDR as Wt. Avg. of RTP and SOC, to for Application in Grid Investment balance social and commercial prefer- Test, Australian Electricity ences (p. 45) Commission, Nov 30, 2004 Discount rate may be different for NPV of plan and for setting revenue requirements (p. 24) SDR ensures analysis for resource utilization of to maximize utility of society as a whole (p. 25) SDR not concerned with income or wealth transfer between agents within society (p. 26) From a societal cost–benefit viewpoint, there is widespread acceptance that RTP (or some other rate less than WACC) is most appropriate to reflect social preferences (p.26 and 28). This does not focus on what’s best for investors, but maximizing net benefits to society as a whole (p. 36) RTP usually estimated in range 3%– 5% (p. 42) Not necessary to consider unsystematic risks (i.e. risks independent of general aggregate market risks), since investors can diversify their portfolios (p. 28) To maximize economic efficiency or reflect interests of electricity customers, use RTP. To effect a reliability objective, use a relatively low SDR. For greatest transparency, use WACC, since RTP difficult to calculate (pp. 41– 43) 148152 The Engineering Decisions for Life Quality Table B.3 Illustrative discount rates used by specific firms

Agency Discount Remarks rate Ontario Energy Board, Canada Guidance to Ontario electricity distributors “Guidelines For Electricity Distributor undertaking conservation and demand man- Conservation And Demand Management” agement programs EB-2008-0037, Mar 28, 2008 With regard to the Total Resource Cost test, Report (p. 33) states, “For the purpose of calculating the net present value, a distributor should use a discount rate equal to the incremental after-tax cost of capital, based on the prospective capital mix, debt and preference share cost rates, and the latest approved rate of return on common equity” Ontario Energy Board, Canada Rulings mandate the after-tax nominal cost “Cost of Capital Parameter Updates for of debt and equity to Ontario Distribution 2008 Cost of Service Applications,” Companies for rate-making. Mar 7, 2008 Allowed nominal after-tax ROE=8.57%, (based on Government of Canada Long Bond nominal yield at 4.46%) Allowed nominal long-term Debt Rate 6.10% Implied WACC Before Tax WACC = 6.76% = 0.40 × {(1 + (8.57% / (1- 34.5%)))/ (1 + 2%) - 1} + 0.60 × {(1 + 6.1%)/(1 + 2%)) - 1} Ontario Energy Board, Canada Document deals with the nominal after-tax Report of the Board on Cost of Capital and cost of debt and equity to Ontario distribu- 2nd Generation Incentive Regulation for tion companies for the purposes of rate- Ontario’s Electricity Distributors, making. The ROE will be based on Gov- Dec 20, 2006 ernment of Canada long bond yield plus and equity risk premium Ontario Energy Board, Canada Rulings mandate the after-tax nominal cost “Revenue Requirement and Charge De- of debt and equity to Hydro One for rate- terminant Order Arising from the EB- making 2006-0501 Decision with Reasons” Allowed nominal after-tax ROE = 8.35% Allowed nominal debt rate = 5.74% (based on weighted average of allowed costs of third party debt at 5.85%, short-term debt at 4.14%, deemed long-term debt at 6.08%) Deemed capital structure = 60% debt and 40% equity Corporate income tax rate = 34.5% for 2008 Hydro One Inc., Canada Hydro One estimate of nominal after tax WACC for transmission in 2008 is 5.63% Implied real WACC before tax WACC = 6.43% ={(1+(5.63%/(1- 34.5%)))/(1+2%)} - 1 Appendix B Discounting 149 153

Ontario Power Generation Review Com- Real Report cites nominal rates. Inflation rate = mittee 7.8% 2%, from OPG benefit–cost analysis 2004 “Transforming Ontario’s Power Genera- 10% nominal is corporate discount rate and tion Company,” Mar 15, 2004 “corporate weighted average cost of capital” (pp. 51–53) 15% nominal discount rate represents higher risk that might be associated with nuclear re- furbishment projects British Columbia Utilities Commission Real BCUC decided that “BC Hydro borrows at (BCUC) Decision, May 11, 2007 2.5% rates that reflect the Provincial Govern- Regarding: 2006 Integrated Electricity ment’s credit rating and current nominal in- Plan, Submitted by: BC Hydro, Mar 29, terest rate on 20 to 30-year debt for BC Hy- 2006 dro, and thus its ratepayers, is approximately 4.60 percent per annum. The Commission Panel concludes this is the appropriate discount rate for BC Hydro to use to evaluate resource options under the current assumption of 100 percent debt financing.” (p. 203) In the absence of any submissions or interventions to the contrary, BCUC “finds no justification for the use of different discount rates for the economic analysis and the ratepayer impact analysis.” (p. 204) BC Hydro submitted that the economic analysis of its resource acquisition plan be done at a discount rate equal to its risk-laden before-tax WACC of 6% real (since taxes are a transfer within society and not a cost to the project for evaluation purposes), and that its ratepayer impact analysis be done at a discount rate equal to its embedded debt interest rate (which exceeds its incremental 4.6% nominal cost of debt) BCUC “accepts BC Hydro’s argument that two tests may be considered for use in project evaluation. The first, and the more important, is an economic analysis of a project, which should only use the incremental cash flows disbursed by BC Hydro as its key input. The second, and less material test is a ratepayer impact analysis which examines how BC Hydro will recover a project’s costs from its ratepayers and which may include items typically not found in a conventional economic analysis such as sunk costs…” (pp. 200–201) Regarding uncertainties and sensitivity analysis, BCUC “considers the issue of risk to be dealt with adequately through the sensitivity and scenario analysis. However, the 150154 The Engineering Decisions for Life Quality

Commission Panel does continue to see value in sensitivity analyses around a single discount rate.” (p. 204). BCUC believes that the discount rate should be the same for all projects, with project-specific uncertainties to be reflected in the cashflows themselves

Distinguishing BC Hydro from private investors, BCUC decided that project evaluation “should not seek to artificially compensate for real differences in project impacts, including possible differences in the cost of capital between BC Hydro and other devel- opers. With respect to the cost of capital, BC Hydro projects will clearly have an advantage as a result of 100 percent debt financing and access the Province’s high credit rating.” (p. 205) Manitoba Hydro Real Sections 6.1.3 and 6.5 Submission to Manitoba Clean Environ- 6.08% Discount rate, although cited as WACC, is ment Commission on the Wuskwatim Pro- treated as a hurdle for a project’s IRR. ject, (Sec. 6), Apr 2003 For low-risk projects, 6.08% discount rate Higher rates, up to 15% real, used for higher risk projects Canadian Energy Research Institute Real Projects public financed: “Levelised Unit Electricity Cost Compari- 8% Real discount rate 8% son of Alternate Technologies for Basel- Alternatives: Real 6% –12% oad Generation in Ontario,” Sep 2004 Projects Merchant Financed: Real ROE 12%; real interest rate 8%; Debt/equity 50/50 Alternatives: ROE 12%–20%, interest 6%- 8%; D/E 70/30 World Nuclear Association Estimates made using 5% and 10% discount “The Economics of Nuclear Power,” rates May 2008 International Energy Agency, Estimates made using 5% and 10% discount OECD’s Nuclear Energy Agency rates “Projected Costs of Generating Electric- ity,” 2005 Source: OPA, IPSP, June 2008

B.6 Discounting Risks in the Far Future

B.6.1 Introduction

Risks to life and health in the future must be discounted in quantitative risk analysis. Yet, risks in the distant future become trivialized if any reason- Appendix B Discounting 155151 able constant interest rate is used. Our responsibility toward future generations rules out such drastic discounting. A solution to this problem is proposed here, resting on the ethical principle that our duty with respect to saving lives is the same to all generations, whether in the near or far future. It is shown that when a choice between prospects involving different risks has a financing horizon T, then ordinary principles of discounting apply up to this time T, while no further discounting is justifiable after T. The principle implies that risk events beyond the financing horizon should be valued as if they occurred at the financing horizon. It is often necessary to choose between alternatives that present different risks to life in a distant future. For example, engineers design bridges, tunnels, waste repositories, or dams with design lives in the order of hundreds of years; the safety margins against failure under extreme loadings such as earthquake or flood must be selected to conserve life to a prescribed or optimal degree. The disposal of highly radioactive nuclear waste is an extreme case. For example, in Canada it is required not to harm humans for at least 10,000 years. When it is necessary to spend resources to reduce a later risk to life or health, the question of what discount rate to apply to the funding (or, equivalently, to the risk) is crucial and cannot be avoided. Considerations similar to those in the following apply to environmental risks as well. The assessment of some environmental risks, e.g., climate change, can be very sensitive to discounting; the discount rate has a major impact on the optimal climate strategy (Toth 1995). Interest and discount rates in the following are net of inflation and taxes. The simplest approach would be to not discount at all. At first sight, this would seem supported by the ethical sentiment that “A life is simply worth saving with the same effort now and in the future,” as Rackwitz (2003) has expressed it. But this conclusion would be a fallacy because, if effort is measured by what it costs, an effort costs more now than in the future. Many studies have shown that not to discount future risk leads to self- contradiction; costs of life-saving must be discounted at the same rate as other investments to avoid inconsistency (Weinstein and Stason 1977, Paté-Cornell 1984, Rackwitz et al. 2005). The reasoning is that if the amount A, invested in a life-saving intervention now, will reduce a risk at a future time t by B units (e.g., QALYs = life years, quality-adjusted for state of health), then it could alternatively be invested, expected to increase to an amount greater than A for some time, and therefore used to deliver more than B units at time t. Indeed, even more so in view of the general progress in life-saving technology that is to be expected. This standard reasoning that risks must be discounted like finances is correct and compelling as far as it goes. However, it is neither necessary nor credible that the interest rate should be constant over time. It is 152156 The Engineering Decisions for Life Quality unlikely that the requisite alternative investment would remain intact for centuries. The reasoning breaks down once the periods of construction and financing is over. The duty remains to reduce risks imposed on future generations in some equitable manner. The literature on interest rates in risk analysis is extensive; Rackwitz (2003) presents a summary from the perspective of life risk management. He gives a thorough account of related work, considering the classical Ramsay model, rate of time preference of consumption, the rate of economic growth, etc. A project or policy involves and affects various parties such as its designer, owner, users, bystanders, and the public. All have different economic interests and exposure to the associated risks. A different interest rate and a different discount rate for risk would apply to each party. Interest rates are related to the rate of benefit obtained. Rackwitz et al. (2005) have shown that if the rate of public benefit is constant, then the public interest rate, if constant, should be positive and should not exceed the rate of public benefit. The public makes financial profit only by its economic growth, so long-term public interest rates should be close to the long-term rate of economic growth per capita – an average of one or two percent or less. The owner would apply financial market rates. The designer (risk analyst) owes it to the owner to apply financial market rates of discounting, but owes it to the public to apply the public rates. As explained in the following this dilemma can be resolved when, as is common, the undertaking has a finite financing horizon. The essence of the problem is that even an extremely small constant financial discount rate effectively trivializes risks to life and health in the distant future to the point of absurdity. To illustrate: At the modest interest rate of 1% per year, the present value of a quantity 10,000 years from now is discounted by a factor of 1043. Even at a rate as low as 0.05% per year, the present value of a quantity 10,000 years from now is discounted by a factor of 150. There is a choice between being inconsistent, being arbitrary, and being unethical. It is useful at the outset to consider why interest is charged. One reason is pragmatic: It is a prerequisite for obtaining financing. A psychological reason is that we prefer consumption now to consumption later. In part this may perhaps be called “short-sightedness,” or “impatience,” but it is also partly because of foresight. We may not be around to enjoy the later consumption – gross mortality is in the order of 1% per year. The interest rate for financing is dictated by the market and rooted in the rate of time preference of consumption. Both lose their relevance after a few generations. In the assessment of risks to life and health, long-term in- Appendix B Discounting 157153 terest rates must be smaller than the rates usually used in financing over in- termediate periods (very roughly up to 30 years); otherwise the risks become negligible. The problem remains, however: Precisely what value should the very long term present value (discount) factor, D(t), have for very large t? The need to discount derives from principles of financing and consistency, but discounting beyond the relatively short term must be moderated, because we do have responsibilities to future generations. It thus seems necessary to consider the problem in terms that can be ar- gued on a moral basis: The human rights of future generations. The present paper suggests a quantitative argument that derives from inherently ethical considerations. Briefly, the argument that dictates the applicable interest rate rests on a principle of indifference: Is our duty to one generation in the far future greater or less than our duty to any other generation? If not, then it is reasonable to assert that they are the same.

B.6.2 Financing Horizon

The quantitative model presented in the following rests on the notion that every prospect (i.e., project, regulation, intervention, etc.) involving life risks in the distant future has (or should have) a financing horizon. The financing horizon T is the upper limit of the time interval (0,T) during which the prospect is to be financed. The financing horizon is specific to the prospect. It is almost always well defined for properly planned undertak- ings in public health or engineering. For infrastructure facilities the financing horizon lies typically at the end of the useful service life or design life, in the order of 50–100 years. For large-scale public works financed by tolls (e.g., bridges, toll roads, tunnels) the financing horizon is the end of the amortization period. Maintenance or demolition is entirely up to future generations. Facilities for the disposal of highly radioactive nuclear waste must protect hundreds of generations and may take centuries for comple- tion; the financing horizon would be approximately the time at which the last spent fuel was removed from a reactor: T = 100–200 years. From the perspective of an individual whose life expectancy is e years the financing horizon T is no greater than e, since you cannot lend or borrow longer than you live. Likewise, many public prospects ought to have a financing horizon roughly equal to the mean life expectancy of the population (roughly 40 years), since financing beyond this limit shifts the burden forward one generation or more. This requirement is ethical rather than financial. In- deed, there are projects for which it is difficult to define a financing horizon, for example when they are financed by government out of taxes at the 154158 The Engineering Decisions for Life Quality time of construction. When choosing a life-saving option the designer may reason that financing is completed when the construction is paid for. Still, the designer may equally reason that any funds not allocated to saving future lives remain available for later use – as long as the regime is intact and not bankrupt. The corresponding financing horizon is then in the unfore- seeable future and ill-defined. Such reasoning runs into our generation’s moral duty not to enjoy benefit by imposing risk on future generations. There is thus the obligation to choose a financing horizon commensurate with the duration of the benefit.

B.6.3 Equivalence Principle

It seems reasonable to postulate the following simple equivalence principle, paraphrased from Rackwitz’s statement (2003) cited in the introduction: All persons, now and at any time in the future, are equally worthy of risk reduction. This is an ethical principle. It is merely a symmetry argument that follows from the golden rule or Kant’s categorical imperative. Deviations from this principle would need to be justified by appeal to necessity or to some higher principle.

B.6.4 Application

In the literature various units are used to express the life and health benefit of a life-saving intervention, such as “life years saved” or “the value of a statistical life.” The US Public Health Service Panel on Cost-Effectiveness in Health and Medicine has recommended using quality-adjusted life years (QALYs) to measure life and health benefit. To illustrate the implications of the equivalence principle, suppose some project extends over just two accounting periods (T = 2), at the end of which it is to be fully paid for. Interest on the net cash flow is 10% payable at the end of each period. Suppose that life risks amounting to 4, 6, 3, and 2 QALYs are assigned to times t = 0, 1, 2, 3 respectively. For proper comparison the first three risks are calculated at their net simultaneous value, which can be either at t = 0, 1, or 2. The first risk may be counted at its present value 4 or equivalently as 4(1.10) at t = 1 or 4(1.21) at t = 2. The second risk can be counted either as 6/1.10, 6, or 6(1.10) at t = 0, 1, or 2 Appendix B Discounting 155 159 respectively. The problem concerns the last risk assigned to t = 3; its mitigation must also be paid for at any time before the time horizon. To treat it equal to the others it must be reduced by the factors 1, 1/1.10, and 1/1.21 if it its value is counted at t = 2, 1, or 0, respectively. The net present value of the risk totals 4 + 6/1.1 + 3/1.21 + 2/1.21 = 13.59 QALY. It is seen that the principle in effect requires that mitigation of any risk expected after the end of financing period must be counted as occurring at the time horizon.

Fig. B.1 a Present value factor D(t) and b effective interest rate r*(t)

All amounts and interest rates in the following are net of inflation. Let D(t) denote the present value factor, i.e., the present value of a unit of cash flow or risk flow at time t. Before and at the financing horizon the present value factor equals 156160 The Engineering Decisions for Life Quality

t − τ D(t) = ∫ e r( )dτ , t ∈ (0,T ) [B.1] 0 in which r(t) is the instantaneous rate of interest at time t in the financing scheme for the prospect. Let A denote one unit of the measure of life risk, e.g., QALY. When deciding about a life-saving option that has a financing horizon T years hence, the decision maker would consider the present value of A just after time T to be equal to the present value just before time T for reasons of continuity, and thus equal to AD(T). The present value of A at any time after time T would also have to equal AD(T) by the equivalence principle. With QALYs as the measure of risk reduction, in particular, it follows that for saving a year of life in good health in any future generation beyond the financing horizon one should be willing to spend the same amount: D(t) = const. for all t > T. Setting exp[-r*(t)] = D(t) for all t > 0 yields the apparent or effective time-independent interest rate r*(t) at which a unit of cash flow or risk flow at time t is discounted (see Fig. B.1). If in particular the interest rate is constant over the financing period, then the effective interest rate decreases as a hyperbolic function of t after the financing horizon. Tables B.4 and B.5 give an idea of the discounting. For illustration consider a hypothetical repository for high-level nuclear waste deep underground in igneous rock (perhaps similar to the Canadian concept (Canadian Environmental Assessment Agency 1998)). The repository is being financed from the time the first power was produced, and completed some 200 years later when the repository is closed. The project is to be financed entirely by a surcharge for the power delivered from time t = 0 to the time the last reactor is shut down, t = T = 150 years. The electric utilities are financed by bonds at a true interest rate equivalent to 3% per year. In particular, consider the collective radiation risk R to persons up to 10,000 years later, assumed to eat fish from a lake or stream that has become contaminated from the waste. This risk is diluted by radioactive decay. By appropriate design of multiple barriers the risk can be delayed and attenuated. The risk to each individual is small, but it extends over a long time (a few isotopes have half lives greater than 100,000 years) and will potentially affect many persons. To allocate funds rationally to the mitigation, the risk is assigned a net present value at t = 0 of R/(1.03)150 = R/84. At first sight it may seem draconian to reduce a future risk value by a factor of 84, but this is not different to the treatment of risks just before closure of the facility to, for example, a construction worker. Unlike the individual risk, the population risk may be a significant part of the total Appendix B Discounting 157 161 risk from nuclear power, so the discounting has significant and justifiable implications for nuclear waste disposal.

Table B.4 Present value factor D(r,T,t), Eq. 1, for t > T

T, years: r 10 20 50 100 0.5% 0.951 0.905 0.779 0.607 1% 0.905 0.819 0.607 0.368 2% 0.819 0.670 0.368 0.135 3% 0.741 0.549 0.223 0.0498 4% 0.670 0.449 0.135 0.0183 5% 0.607 0.368 0.082 0.0067

Table B.5 Effective interest rates r∗(r,t) for financing horizon T = 50 years

t = 50 100 200 500 1,000 2,000 5,000 10,000 years years years years years years years years 0.00% 0.50% 0.25% 0.10% 0.05% 0.03% .001% 0.005% 2.00% 0.00% 0.50% 0.20% 0.10% 0.05% 0.02% 0.010% 3.00% 0.50% 0.75% 0.30% 0.15% 0.08% 0.03% 0.015% 4.00% 0.00% 0.00% 0.40% 0.20% 0.10% 0.04% 0.020% 5.00% 2.50% 1.25% 0.50% 0.25% 0.13% 0.05% 0.025%

B.6.5 Discussion

It is important to notice that “discounting” a future risk does not mean that it is considered less worthy of risk mitigation. It means simply that $1 today can be invested – or borrowing it can be deferred – growing to a greater value in the future. In the foregoing example it would be expected to grow to $84 in today’s money (perhaps thousands of dollars in contem- porary currency). This deferred amount could be spent towards protective gear for the construction worker or some more bentonite clay to help protect future generations. Like so many earlier studies, this analysis shows the importance of financing in the valuation of risks in the far future. Table B.1 shows that rate of interest is important, but also that the financing horizon is more important yet. For example, a unit of risk 50 years in the future, when financed within T = 20 years at 5%, has a present value of 0.368, the same value as if financed within T = 50 years at 2%. 158162 The Engineering Decisions for Life Quality If the interest rate is constant, the effective interest rate is exponential up to the financing horizon and hyperbolic thereafter. A hyperbolic discounting function was also proposed Pandey and Nathwani (2003) (based on people’s apparent preferences of consumption), by Ainslie (1975), and by Lowenstein and Prelec (1992). The present study fixes the discounting function for risks in the far future of a prospect in relation to its financing.

B.6.6 Conclusions

While risks to life and health that occur in the far future must be discounted, the discounting rate should not extend forever and need not be constant. Most if not all prospects have a financing horizon, i.e., the time when the prospect’s financing is to be completed. For such prospects the well-known argument that risks must be discounted for self-consistency does not apply beyond the financing horizon. The governing principle remains that all persons in the future, near or far, are equally worthy of risk reduction. This implies that risks in the far future are to be counted as if they occurred at the financing horizon. When the egalitarian principle enunciated by Rackwitz (2003) is interpreted as put forth here, the implications are that the long-term average rates decrease hyperbolically in a manner similar to that proposed by Pan- dey and Nathwani (2003), and that long-term effective discount rates become limited to “one or two percent or less,” as is suggested by Rackwitz (2003).

References

Ainslie G (1975) Specious rewards: A behavioral theory of impulsiveness and impulse control. Psych Bull 82:463–509 Arnold C (1972) On measuring the social opportunity cost of public funds. In: Harberger AC (ed) Project evaluation—collected papers. University of Chicago Press, Chicago Brean D, Burgess D, Hirshhorn D, et al (Mar 2005) Treatment of private and public charges for capital in a “full-cost accounting” of transportation, Final Report Burgess DF (1981) The social discount rate for Canada: Theory and evidence, Canadian Public Policy Canadian Environmental Assessment Agency (1998) Nuclear fuel waste management and disposal concept. Report of the Nuclear Fuel Waste Management and Disposal Concept. Environ- mental Assessment Panel Gollier C (2002) Time horizon and the discount rate. IDEI, University of Toulouse, mimeo HM Treasury (2003) The Green Book, appraisal and evaluation in central government. London. http://www.hm-treasury.gov.uk/d/2(4).pdf. Accessed August 25, 2008 Appendix B Discounting 159 163 Jenkins GP (1972) In: Niskanen WA et al (eds) Measurement of rates of return and taxation from private capital in Canada, Benefit–costs Analysis. Chicago Jenkins GP (1981) The public-sector discount rate for Canada: Some further observations, Cana- dian Public Policy Jenkins G, Kuo C-Y (2007) The economic opportunity cost of capital for Canada—An empirical update, QED Working Paper Number 1133. Department of Economics, Queen’s University, Kingston Kula E (1987) Social interest rate for public sector appraisal in the United Kingdom, United States and Canada. Proj Apprais 2(3):169–174 Lowenstein G, Prelec D (1992) Anomalies in intertemporal choice: Evidence and interpretation. Q J Econ 57:573–598 Maddison A (2001) The world economy: a millennial perspective. OECD Paris Newbery D (1992) Long term discount rates for the Fores Enterprise. Department of Applied Economics, Cambridge University, for the UK Forestry Commission, Edinburgh OXERA (2002) A social time preference rate for use in long-term discounting. Report for ODPM, DfT and Defra Pandey MD, Nathwani JS (2003) Discounting models and the life-quality index for the estimation of societal willingness-to-pay for safety. In: Maes MA (ed) Proc 11th IFIP WP 7.5 Working Conference on Structural Reliability and Optimization of Structural Systems, Banff, Netherlands, Rotterdam, AA Balkema Paté-Cornell ME (1984) Discounting in risk analysis: Capital versus human safety. In: Grigoriu M. (ed) Proc Symp Struct Tech and Risk. University of Waterloo, Waterloo Pearce DW, Nash CA (1981) The social appraisal of projects. McMillan Press, London Pearce D, Ulph D (1995) A social discount rate for the United Kingdom. CSERGE Working Pa- per 95-01. School of Environmental Studies, University of East Anglia, Norwich Rackwitz R (2003) Discounting for optimal and acceptable technical facilities involving risks. Proc. 9th ICASP Conf Berkeley Rackwitz R Lentz A, Faber M (2005) Economically sustainable civil engineering infrastructures by optimization. J Struct Saf 27:187–229 Sandmo A, Dreze JH (1971) Discount rates for public investment in closed and open economies Scott MFG (1977) The test rate of discount and changes in base level income in the United Kingdom. Econ J 219–241 Scott MFG (1989) A new view of economic growth. Clarendon Paperbacks Sjaastad LA, Wisecarver DL (1977) The social cost of public finance, J Polit Econ 85(3):3 Spiro PS (2008) The Social Discount Rate for Ontario Government Investment Projects. Policy Paper Toth FL (1995) Discounting in integrated assessment of climate change. Energy Policy 1995 23(4/5):403–409 Treasury Board of Canada (1998) The social discount rate, benefit/cost analysis guide, section 5.5.2. http://www.tbs-sct.gc.ca/fin/sigs/revolving_funds/bcag/bca2_e.asp. Accessed August 23, 2007 Treasury Board of Canada Secretariat (2007) Canadian cost–benefit analysis guide, catalogue BT58–5. http://www.regulation.gc.ca/documents/gl-ld/analys/analyseng.pdf. Accessed March 21, 2008 US OMB (1992) Guidelines and discount rates for benefit–cost analysis of federal programs, circular A-94:22. http://www.whitehouse.gov/omb/circulars/a094/a094.html. Accessed August 28, 2007 Weinstein MC, Stason WB (1977) Foundations of cost-effectiveness analysis for health and medical practices. New Engl J Med 296(31):716–721 Weitzman M (2001) Gamma discounting. Am Econ Rev 91(1):124–149 Appendix C Calibration of the LQI

Abstract We calibrate the LQI = EKG by determining the value of K that reflects the relative value of discretionary time and the rate of production of wealth. Calibration can be specific for a country or a group of countries. The value of K also defines the labor-demographic constant D = K/E and the societal capacity (SCCR) C = KG/E = DG. The calibration employs six time series of national statistics: the GDP per person; its contribution from labor; the employment; the number of hours worked per worker; the population; and the life expectancy. These define the time series K(t) from which a characteristic value of the parameter K is selected. OECD data ag- gregated for 27 countries for the years 1976–2004 yield K = 5.0.

C.1 Calibration

To calibrate the LQI = EKG is to determine the value of K that reflects the relative value generally placed in a society upon the importance of discretionary time, for which E is taken as a measure, and the rate of production of wealth as reflected in G. This relative value in the margin is found by considering infinitesimal variations dE and dG and requiring that the first variation of the LQI be positive (Pandey et al. 2006), giving an optimality condition for a quantified risk encompassing life, limb, health, wealth, and income. Calibration can be specific for a country (or province, etc.), as when the objective is to assess a prospective project or regulation, or for a group of countries (e.g., the EU) when warranted by circumstances such as international comparison or regulations. The value of K defines the labor- demographic constant D = K/E and the societal capacity (SCCR) C = KG/E = DG. As described in detail by Pandey et al. (2006) and illustrated in Fig. C.1, the calibration employs six time series of national statistics. Given that many economists, regulators, and decision-makers are more familiar with concepts of economics and utility theory, this rigorous derivation should promote communication to a wider audience. The time series are for each country i: the gross domestic product per person G = G(t); its contribution from labor (wages and salaries, etc.) Gw(t); the employment M(t); the average number of hours worked per worker h(t); the population P(t); and the life expectancy at birth E = E(t). From these, three other national time series are calculated: the labor exponent β(t), the work-time fraction w(t), 162168 The Engineering Decisions for Life Quality and from these (see Appendix A) the time series K(t). We assume that the capital investment per person and technological factors are independent of the work time fraction w. Using the labor–leisure trade-off, the first order optimality condition (see Eq.A.30) is expressed as

K = 1/q = β (1/w – 1). [C.1]

GDP per person G Labor Exponent LQI β Labor component of G LQI Exponent GDP w

K Capacity Employment M SCCR

C h w Average hours worked

Population P D= K/E

E Life Expectancy at birth

Fig. C.1 Flow diagram of the calculations of the LQI exponent K, the labor-demographic factor D, and the SCCR C. G is the GDP, Gw is its labor component, M is the employment, h is hours worked, P is the population, E is life expectancy at birth,β is labor exponent, and w is work-time fraction

The analysis is based on statistical data for the period 1976-2004 for 27 countries (OECD 2007) i = {Australia, Austria, . . . , USA} (see Table C.1). First, the population average of the work-time/total-time ratio w(i,t) is determined from the population P = P(i,t), the employment M = M(i,t) and the average hours worked, h = h(i,t), as Appendix C Calibration of the LQI 163 169 w = hM/P. [C.2]

The work-time fraction for workers is calculated from the average number of hours worked per employed person/year, estimated from labor market surveys. According to Statistics Canada’s definition, hours worked means the total number of hours that a person spends working, which includes regular and overtime hours, breaks, travel time, training in the workplace, and time lost in brief work stoppages where workers remain at their posts. On the other hand, time lost due to strikes, lockouts, annual va- cation, public holidays, sick leave, maternity leave or leave for personal needs are not included in total hours worked. The work-time fraction at the population level is obtained as the total number of work hours divided by the national population. Note that the total number of hours worked per year is the product of the number of employed people and the annual number of work hours per worker. Fig. C.2 is a graph of w(i,t) for selected countries.

w 0.12

0.11

0.1

0.09

0.08

0.07 Australia Denmark United Kingdom Canada Italy United States 0.06 1975 1980 1985 1990 1995 2000 2005 2010

Fig. C.2 Work-time fraction for selected countries. The straight line is the trend of the simple average for 27 countries (OECD 2007)

Next, the labor exponent β(i,t) is determined. According to Cobb– Douglas production theory, the ratio of wage to GDP is a measure of the share of labor coefficient,β, in the production function. This ratio is plotted in Fig. C.3. The wages include all payments in cash or in kind made by the domestic producers to workers for services rendered. It includes total pay- 164170 The Engineering Decisions for Life Quality roll of paid workers and an imputed income of self-employed workers. The time series β(i,t) can be determined in two ways using the Cobb–Douglas production model. Let Y be the product, J the capital, W the labor and A a factor reflecting the technology. Then the Cobb–Douglas relationship is

Y = A Jα Wβ. [C.3]

The simplest approach is to apply the profit-maximizing assumption that a firm will hire labor until the marginal value of the product is equal to the real wage. As shown by Pandey et al. (2006) (see Eq.A.19) this gives

β = wages/GDP, [C.4] where wages is the total compensation, properly including all benefits accruing to labor. The other approach makes no assumption of profit maximization, but uses the return-to-scale condition α + β = 1. Taking logarithms of both sides of Equation C.2 and normalizing with respect to a particular year (we used year 2000) gives ln(Y/Y0) = (1 - β) ln (J/J0) + β ln(W/W0), [C.5] from which

β = ln(Y/Y0) - ln(J/J0)]/[ ln(W/W0) - ln(J/J0)]. [C.6]

The two formulas [C.4] and [C.6] give values of w(i,t) that differ by one percent on average and at most by +20% for the 27 countries. The results by Eq.C.6 are shown in Fig. C.3. The labor productivity has greatly increased in the study period (1976- 2004). The increasing level of capital investment in production technology is a primary contributor to productivity growth. The growth in productivity has not caused any major variation in the work time. The data do not show dependence between the capital per worker and the work time. The data supports the assumption made in the calibration of LQI. The wage to GDP ratio fluctuates in very narrow ranges, and it can be assumed as a constant for practical purposes. It also represents the factor β used in the production equation and referred to as share of labor. In this respect, the data satisfies the assumption of the Cobb–Douglas production function that α and β are independent of the factors of production. These observations suggest that the assumptions related to the calibration (Eq.C.1) are empirically justified.

Appendix C Calibration of the LQI 165 171

Beta

0.6

0.5

0.4

Australia Denmark United Kingdom United States Canada Italy 0.3 1975 1980 1985 1990 1995 2000 2005 2010

Fig. C.3 Labor exponent β for six countries. The straight line is the trend of the simple average for 27 countries (OECD 2007)

K(i,t) by Eq.C.1 is shown in Fig. C.4 for selected countries. Fig. C.4 shows that K(i,t) varies in quite narrow ranges. When estimating the societal capacity to commit resources (SCCR) for practical risk assessment it is therefore reasonable to treat K as a constant for each country. The variation between the countries is noted; 577 values of K(i,t) average 5.23 with an overall coefficient of variation of 19%. The straight line in Fig. C.4 shows the linear trend of the average K(t) = avgi[K(i,t)]. It is noted that the trend (for practical purposes, for the applications) is quite constant over the quarter-century, decreasing at a rate of 0.2%pa. The trend over the latest 10-year period, 1995–2004, is perhaps a more likely indicator of the K-values to be expected in the near future, decreasing more rapidly at the rate of about 2.5%pa. Projection of this linear trend shows an average of K = 5.00 over the period from 1995 to 2015. The value of K for each country is shown in Table C.1. The coefficient of variation over the studied period for each country is adequately modest for applications to risk assessment. K is also fairly close to the simple average over the 27 countries during the period 1976-2004. 166172 The Engineering Decisions for Life Quality

K 6.5

5.5

4.5 Australia Denmark United Kingdom United States Canada* Italy* 4 1975 1980 1985 1990 1995 2000 2005 2010 Fig. C.4 LQI exponent K(i,t) for selected countries. The straight line is the trend of the simple average for 27 countries (OECD 2007)

Kubler and Faber (2005) presented an alternative approach to the LQI calibration by relating K to the correlation between G and E. Their analysis of the 1960–2001 time series of G and E for 208 countries resulted in an estimate of K = 1/0.14. LQI 1.3 1.2 Australia Denmark United Kingdom United States Canada* Italy* 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 1975 1980 1985 1990 1995 2000 2005 2010

Fig. C.5 LQI for 27 countries. The straight line is the trend of the simple average for 27 countries (OECD 2007) Appendix C Calibration of the LQI 167 173 A constant K-value is necessary for comparison amongst countries. Fix- ing the parameter at K = 5.0, the resulting Life Quality Index LQI = E5G for 2004 for the 27 OECD countries is shown in Table C.2. Similar to the Human Development Index, LQI rankings can vary from year to year. Dif- ferences in LQI less than 5% cannot be considered significant given variation in data and measurement uncertainties. Fig. C.5 is a graph of the annual values of LQI(i,t) = E(i,t)K(i,t)G(i,t) in terms of purchasing-power parity, normalized such that LQI(USA,2000)=1.00. The substantial progress for the countries shown is typical of OECD countries. Past values of C(i,t) in purchasing-power-parity constant dollars per year are shown graphically in Fig. C.6. As Table C.2 shows, the coefficient of variation for the individual countries averages some 13% – still modest for practical applications.

SCCR C, $PPP 3000 Australia Denmark United Kingdom United States Canada* Italy* 2500

2000

1500

1000

500 1975 1980 1985 1990 1995 2000 2005 2010 Fig. C.6 Societal capacity (SCCR) C(i,t) in constant PPP$. The straight line is the trend of the simple average for 27 countries (OECD 2007)

The labor-demographic factor D(i,t) varies from 0.040 (Greece) to 0.086 (Germany), averaging 0.064, Table C.1. As Fig. C.7 shows, the relative variation in D(i,t) is for each country fairly constant over time, with coeffi- cients of variation averaging 5.6%. For decision-making practice in any one of the OECD countries this means that the societal capacity SCCR can be determined simply and quite accurately as C = DG. For the purpose of risk assessment it is suggested to use the forecast value of D(i,t) in Table C.4 together with the most recent or forecast values of the GDP per person in local currency. 168174 The Engineering Decisions for Life Quality

D 0.09

0.08

0.07

0.06

0.05 Australia Denmark United Kingdom United States Canada* Italy* 0.04 1975 1980 1985 1990 1995 2000 2005 2010 Fig. C.7 The labor-demographic factor D(i,t) = K(i,t)/E(i,t)

C.2 Summary

This appendix illustrates the process of deriving all parameters for the LQI method specific to a country through economic data analysis. The LQI, derived in Appendix A using the concepts of economics sciences and a lifetime utility function as L = EKG, employs a parameter K that depends on the economic structure of a society as reflected in the labor share in the gross domestic product G and the annual work-time fraction. Using economic data for each of 27 developed countries over the period 1976–2004 the country-specific time series for K are determined. For practical risk analysis we give values of the SCCR and the labor-demographic factor D. An overall average value of K = 5.0 is proposed for international comparisons. Appendix C Calibration of the LQI 169 175 Table C.1 Synopsis of the calibration of K and D.

K(2004) COV of K D(2004) COV of D i 1976–2004 1976–2004 Australia 4.64 5.7% 0.058 8.3% Austria 5.30 0.8% 0.067 1.9% Belgium 6.79 3.2% 0.081 3.0% Canada 4.58 5.0% 0.057 6.4% Czech Republic 3.60 3.8% 0.047 2.9% Denmark 5.39 2.7% 0.069 3.1% Finland 4.86 8.2% 0.062 7.1% France 6.66 6.0% 0.083 3.8% Germany 6.75 3.8% 0.086 2.5% Greece 3.16 2.4% 0.040 3.2% Hungary 4.56 4.3% 0.063 5.5% Iceland 5.81 6.4% 0.072 7.6% Ireland 4.23 13.1% 0.049 15.3% Italy 4.53 3.0% 0.056 5.5% Japan 6.23 4.2% 0.076 5.2% Korea 5.05 7.0% 0.062 4.5% Luxembourg 3.48 11.4% 0.042 12.6% Netherlands 5.89 5.2% 0.075 5.7% New Zealand 4.82 4.4% 0.061 6.7% Norway 4.93 2.8% 0.062 3.3% Poland 4.25 2.3% 0.057 2.5% Portugal 4.60 4.1% 0.058 4.6% Slovak Rep. 4.67 3.1% 0.063 2.6% Spain 5.18 10.0% 0.064 11.7% Sweden 6.05 3.7% 0.075 4.0% UK 5.61 4.4% 0.072 6.0% USA 5.28 5.2% 0.069 6.7% Average: 5.07 5.0% 0.064 5.6% Minimum: 3.16 0.8% 0.040 1.9% Maximum: 6.79 13.1% 0.086 15.3% COV: 0.19 0.182

Note: Values based on interpolation or extrapolation are shown in italics 170176 The Engineering Decisions for Life Quality Table C.2 The Life Quality Index LQI = E5G for the year 2004 normalized with respect to USA for the year 2000

Life GDP per LQI Rank expectancy person (2004) (years) ($PPP) Luxembourg 78 53373 1.67 1 Norway 79.6 37646 1.30 2 Iceland 81.2 30215 1.15 3 USA 77.4 35185 1.06 4 Australia 80.3 29242 1.06 5 Canada 79.9 29413 1.04 6 Japan 81.8 25894 1.03 7 Ireland 78.3 32154 1.02 8 Sweden 80.2 28225 1.01 9 Austria 78.8 29178 0.96 10 Netherlands 78.6 28511 0.93 11 Italy 79.7 26146 0.91 12 France 79.4 26403 0.90 13 Belgium 78.8 27209 0.90 14 Finland 78.4 27512 0.88 15 Denmark 77.5 28991 0.88 16 United Kingdom 78.5 27134 0.88 17 Average of 27 0.86 Germany 78.6 25724 0.84 18 Spain 80.3 22183 0.80 19 Greece 78.9 23600 0.78 20 New Zealand 79.2 22394 0.76 21 Korea 77.3 18383 0.55 22 Portugal 77.3 18366 0.55 23 Czech Republic 75.4 16155 0.43 24 Hungary 72.6 14076 0.31 25 Slovak Republic 73.8 12190 0.29 26 Poland 74.7 11417 0.29 27 Appendix C Calibration of the LQI 177171 Table C.3 Synopsis of the calibration of the SCCR and the LQI

SCCR COV(C) LQI(2004)

C(i,2004) 1976–2004 (USA2000=1.00) i ($PPP) Australia 1709 9.4% 1.093 Austria 1979 4.2% 1.006 Belgium 2273 10.9% 0.931 Canada 1726 28.8% 1.092 Czech Rep. 801 10.9% 0.457 Denmark 2050 14.9% 0.900 Finland 1757 19.6% 0.936 France 2230 16.7% 0.972 Germany 2234 7.8% 0.845 Greece 986 9.9% 0.821 Hungary 927 9.6% 0.327 Iceland 2306 7.0% 1.214 Ireland 1616 18.8% 1.084 Italy 1459 11.8% 0.970 Japan 2011 5.2% 1.074 Korea 1183 40.4% 0.566 Luxembourg 2317 9.5% 1.819 Netherlands 2179 7.2% 0.980 New Zealand 1395 6.2% 0.773 Norway 2401 20.2% 1.371 Poland 681 3.4% 0.309 Portugal 1079 13.7% 0.594 Slovak Rep. 810 9.3% 0.311 Spain 1450 6.9% 0.825 Sweden 2197 12.6% 1.078 UK 2007 43.0% 0.905 USA 2486 11.6% 1.106 Average: 1713 13.7% 0.902 Minimum: 681 3.4% 0.309 Maximum: 2486 43.0% 1.819

COV: 0.33 0.368 Note: Values based on interpolation or extrapolation are shown in italics 172178 The Engineering Decisions for Life Quality Table C.4 Linear regression forecasts of the labor-demographic factor D(i,t), based on D(i,1995)–D(i,2004)

Year i 2010 2015 2020 Australia 0.054 0.052 0.049 Austria 0.064 0.062 0.060 Belgium 0.078 0.075 0.072 Canada 0.053 0.049 0.045 Czech Republic 0.050 0.052 0.054 Denmark 0.066 0.063 0.061 Finland 0.054 0.050 0.045 France 0.082 0.082 0.082 Germany 0.088 0.090 0.091 Greece 0.038 0.036 0.034 Hungary 0.054 0.049 0.044 Iceland 0.065 0.060 0.055 Ireland 0.037 0.027 0.017 Italy 0.051 0.048 0.044 Japan 0.068 0.062 0.056 Korea 0.058 0.055 0.052 Luxembourg 0.034 0.027 0.020 Netherlands 0.069 0.064 0.059 New Zealand 0.060 0.060 0.061 Norway 0.062 0.061 0.060 Poland 0.055 0.053 0.050 Portugal 0.056 0.054 0.052 Slovak Republic 0.068 0.070 0.073 Spain 0.045 0.031 0.017 Sweden 0.074 0.073 0.073 United Kingdom 0.071 0.071 0.070 United States 0.070 0.070 0.071

References

Kubler O, Faber MH (2005) LQI: On the correlation between life expectancy and the gross domestic product per capita. Proc 9th Int Conf Structural Safety and Reliability, ICOSSAR, June 22–25, Rome Organization of Economic Development and Cooperation (OECD) (2007) http://www.oecd.org/statistics. Accessed Jan. 17, 2008 Pandey MD, Nathwani JS, Lind NC (2006) The derivation and calibration of the life-quality index from economic principles. J Struct Saf 28 (1–2):341-360 Appendix D The Life Table and Its Construction

Abstract The parts and construction of a life table is explained, and its use in life risk assessment is illustrated.

D.1 Introduction

The life table, primarily a demographic tool of actuarial science, is a valuable analytical tool in demography, epidemiology, biology, reliability engineering, and now in risk management. The life tables for all countries are widely available and updated by the World Health Organization and national statistical agencies. The Max Planck Institute’s Human Life Table Database is also a valuable resource. We first explain the construction of the conventional life table and show next how it is used in risk management to calculate the gain or loss in life expectancy and quality-adjusted life expectancy. The current life table gives a cross-sectional view of the mortality and survival experience of a population during a current year depending on the age-specific death rates prevailing in the year for which it is constructed. Such a table projects the life span of each individual in a hypothetical population (of 100,000 people) on the basis of actual death rates in a given population. The current life table is a reflection of the mortality experience of a real population during a calendar year, and is widely known to provide a sound basis for statistical inference about the population under consideration. The life expectancy at birth in the current year means the expected value of the length of life that would be obtained if an infant born in the current year were subjected to prevailing age-specific death rates through- out his or her life. Life tables may be either complete or abridged. In a complete life table the functions are computed for each year of life; an abridged life table generally deals with age intervals of 4–5 years, except the first year of life. A common set of intervals is 0–1, 1–4, 5–9, 10–14, . . . , 85, as in Table D.1. The set of all persons in an age interval is called a cohort. 7 Engineering Decisions forLife Quality 174

Table D.1 Abridged life table of Canada (both sexes 1985) from Keyfitz and Flieger (1990), p. 338

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] Mid Observed Observed Probability Numbers of Number of Probability Number Total Number of Total number Life Age Population Deaths Death of death years lived in Survivors of Survival of Deaths Years Lived in of Years LivedExpectancy (thousands) (thousands) Rate interval age interval beyond age x (years)

P D Mx qx ax lx px dx Lx Tx ex 0 373.7 2.982 0.00798 0.00797 0.0835 100000 0.9920 797 99269 7644205 76.44 1 1478.3 0.633 0.00043 0.00171 1.5 99203 0.9983 170 396386 7544936 76.06 5 1784.1 0.403 0.00023 0.00113 2.5 99033 0.9989 112 494884 7148550 72.18 10 1813.7 0.469 0.00026 0.00129 2.8695 98921 0.9987 128 494333 6653666 67.26 15 1975.6 1.433 0.00073 0.00362 2.658 98793 0.9964 358 493128 6159333 62.35 20 2396.3 2.151 0.00090 0.00448 2.5225 98436 0.9955 441 491086 5666205 57.56 25 2341.2 1.987 0.00085 0.00423 2.547 97995 0.9958 415 488956 5175119 52.81 30 2159.3 2.137 0.00099 0.00494 2.5845 97580 0.9951 482 486736 4686163 48.02 35 1978.8 2.348 0.00119 0.00591 2.673 97098 0.9941 574 484155 4199427 43.25 40 1558.3 3.015 0.00193 0.00962 2.711 96524 0.9904 929 480494 3715273 38.49 45 1293.1 4.089 0.00316 0.01568 2.709 95595 0.9843 1499 474542 3234779 33.84 50 1241.7 6.608 0.00532 0.02623 2.6955 94096 0.9738 2468 464794 2760237 29.33 55 1207.3 10.362 0.00858 0.04195 2.675 91628 0.9580 3844 449204 2295443 25.05 60 1120.6 14.935 0.01333 0.06435 2.666 87784 0.9356 5649 425736 1846239 21.03 65 880.2 18.659 0.02120 0.10037 2.6425 82135 0.8996 8244 391241 1420503 17.29 70 729.7 23.897 0.03275 0.15081 2.618 73891 0.8492 11144 342912 1029262 13.93 75 500.8 25.491 0.05090 0.22501 2.575 62747 0.7750 14119 279499 686350 10.94 80 303.6 24.141 0.07952 0.33049 2.553 48629 0.6695 16071 203817 406851 8.37 85 221.9 35.583 0.16036 1 6.0075 32558 0 32558 203034 203034 6.24 Appendix D The Life Table and Its Construction 175 D.2 Explanation of Life Table Variables

In this section, construction of a life table is explained, using for illustration the Abridged Life Table of Canada (Table D.1) for the year 1985 (Keyfitz and Flieger 1990, Chian 1984). The meaning of various columns of Table D.1 are discussed below. Column [1] Age of the cohort, x. Column [2] Mid-year population in a particular cohort. For example, population in age interval 30–34, P30 , is 2,159.3 thousands. Column [3] The number of deaths actually noted in a particular age group (D). For example, the number of deaths observed between age 30 and 34,

D30 , is 2137. Column [4] The age specific death rate observed in the population. For example, the age specific death rate between age 30–34 is

M 30 = D30 / P30 = 2.137/2,159.3 = 0.00099. [D.1]

Column [5]: The probability of death of a person exactly x years old dying before reaching age (x+n), i.e., the next age group. This probability (q) is commonly approximated in terms of the age-specific death rate as

nM x . qx = [D.2] 1+ ( ax M x )

For example, the probability of death between age 30–34 is

5x0.00099 q = = 0.00494 . [D.3] 30 1 + (2.584x0.00099)

Note that M30 = 0.00099 and a30 = 2.5854. Column [6] The average number of years lived in an age group, x to x+n, by those who die after ax years. ax is an input parameter for constructing the life table. For example, persons dying between age 30 and 34 would live 2.584 years, on average, after their 30th birthday. Column [7] The number of survivors reaching age x in a cohort of 100,000 people, calculated as 176182 Engineering Decisions for Life Quality lx = l(x-n) × (1 - q(x-n)). [D.4]

For example, the number of survivors at age 30 is obtained as l30 = 97,995 × (1 - 0.00423) = 97,580. [D.5]

Column [8] The probability of survival at age x given as =− px 1 qx . [D.6]

For example, the probability of survival at age 30 is p30 = 1 - 0.00494 = 0.99506. [D.7]

Column [9] The number of deaths between age x and x+n in a cohort of 100,000 people, calculated as

= × d x qx lx . [D.8]

For example,

× d30− 34 = 0.00494 97,580 = 482. [D.9]

Column [10] The number of years lived between age x and (x + n). This quantity is calculated as =−+ Lnx ()lx dadx x x . [D.10]

For example, the number of years lived between age 30 and 34 are calculated as × L30− 34 = 5(97,580 - 482) + (2.584 482) = 486,736 years [D.11]

Column [11] The total number of years lived beyond age x by survivors of the original cohort reaching age x, calculated as the sum of Lx values from age x to the end of the life table. Thus,

w = TLxx∑ [D.12] x Appendix D The Life Table and Its Construction 177 183 where w is the terminal age in the life table. There is an obvious relationship, =+ Tx Lx Tx+n . [D.13]

For example, the number of years lived beyond age 30 are calculated as =+ T30L 30T 35= 486,736 + 4,199,427 = 4,686,163 years [D.14]

Column [12] The expectation of life at age x is given as = ex Tx / lx . [D.15]

For example, the expectation of life at age 30 is calculated as = e30T 30/ l 30 = 4,686,163/97,580 = 48.02 years. [D.16]

D.3 Explanation of the Modified Life Table, Table D.2

The purpose of the modified life table is to illustrate the calculation of the influence of a risk on the life expectancy and to determine the influence of health-related quality adjustment. Column [3A] The risk modifies the probability of death by the factor k(x), specific to each age group x. The chosen example in Table D.2 shows a risk that is increasingly serious as age increases over 40 years, doubling the mortality in the highest age group. The total risk is calculated in the last row of column 4. Column [14A] The age-specific effect of declining health quality, h(x), factors the life years in column 15. All other columns are calculated exactly as in the life table, Table D.1.

D.4 Life Table Data

Life tables for Canada are reproduced here for illustration only. For current life table information for use in risk analysis, see the references. 178184 Engineering Decisions for Life Quality Extended male and female Life Tables for Canada (2000–2002) are presented in Tables D.3 and D.4, respectively (Statistics Canada, 2002). The notations used in these tables are as follows: lx = number of survivors dx = number of deaths px = probability of survival qx = probability of death Lx = number of years lived in an age interval Tx = number of years lived beyond age x ex = life expectancy at age x pedxD h ieTbeadIsCntuto 179 Appendix D The Life Table and ItsConstruction

Table D.2 Modified life table of Canada (both sexes): death rates factored; health-related quality-of-life factored. Death rate factor in column 3A; HRQ factor in column 14A

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] Age Mid population Observed Observed Probability Numbers of Number of Probability of Number of Total num- Total num- Life (thousands) deaths death rate of death years lived in survivors survival deaths ber of years ber of years expectancy (thousands) interval lived in lived beyond (years) age interval age x

P D Mx qx ax lx px dx Lx Tx ex 0 373.7 2.982 0.00798 0.00797 0.0835 100,000 0.992 797 99,269 7,644,205 76.44 1 1478.3 0.633 0.00043 0.00171 1.5 99,203 0.9983 170 396,386 7,544,936 76.06 5 1784.1 0.403 0.00023 0.00113 2.5 99,033 0.9989 112 494,884 7,148,550 72.18 10 1813.7 0.469 0.00026 0.00129 2.8695 98,921 0.9987 128 494,333 6,653,666 67.26 15 1975.6 1.433 0.00073 0.00362 2.658 98,793 0.9964 358 493,128 6,159,333 62.35 20 2396.3 2.151 0.0009 0.00448 2.5225 98,436 0.9955 441 491,086 5,666,205 57.56 25 2341.2 1.987 0.00085 0.00423 2.547 97,995 0.9958 415 488,956 5,175,119 52.81 30 2159.3 2.137 0.00099 0.00494 2.5845 97,580 0.9951 482 486,736 4,686,163 48.02 35 1978.8 2.348 0.00119 0.00591 2.673 97,098 0.9941 574 484,155 4,199,427 43.25 40 1558.3 3.015 0.00193 0.00962 2.711 96,524 0.9904 929 480,494 3,715,273 38.49 45 1293.1 4.089 0.00316 0.01568 2.709 95,595 0.9843 1499 474,542 3,234,779 33.84 50 1241.7 6.608 0.00532 0.02623 2.6955 94,096 0.9738 2468 464,794 2,760,237 29.33 55 1207.3 10.362 0.00858 0.04195 2.675 91,628 0.958 3844 449,204 2,295,443 25.05 60 1120.6 14.935 0.01333 0.06435 2.666 87,784 0.9356 5649 425,736 1,846,239 21.03 65 880.2 18.659 0.0212 0.10037 2.6425 82,135 0.8996 8244 391,241 1,420,503 17.29 70 729.7 23.897 0.03275 0.15081 2.618 73,891 0.8492 11144 342,912 1,029,262 13.93 75 500.8 25.491 0.0509 0.22501 2.575 62,747 0.775 14119 279,499 686,350 10.94 80 303.6 24.141 0.07952 0.33049 2.553 48,629 0.6695 16071 203,817 406,851 8.37 85 221.9 35.583 0.16036 1 6.0075 32,558 0 32558 203,034 203,034 6.24 180 Engineering Decisions for Life Quality

Table D.3 Life table for Canada (male 2000–2002) (Statistics Canada)

Age (years) lx dx px qx Lx Tx ex

0 100000 577 0.99423 0.00577 99486 7691802 76.92 1 99423 35 0.99965 0.00035 99405 7592316 76.36 2 99388 21 0.99979 0.00021 99376 7492911 75.39 3 99367 21 0.99979 0.00021 99356 7393535 74.41 4 99346 20 0.9998 0.0002 99336 7294179 73.42 5 99326 16 0.99983 0.00017 99318 7194843 72.44 6 99310 13 0.99987 0.00013 99303 7095525 71.45 7 99297 9 0.99991 0.00009 99293 6996222 70.46 8 99288 8 0.99992 0.00008 99284 6896929 69.46 9 99280 8 0.99992 0.00008 99276 6797645 68.47 10 99272 10 0.9999 0.0001 99267 6698369 67.48 11 99262 10 0.9999 0.0001 99257 6599102 66.48 12 99252 15 0.99985 0.00015 99244 6499845 65.49 13 99237 23 0.99977 0.00023 99225 6400601 64.5 14 99214 34 0.99966 0.00034 99197 6301376 63.51 15 99180 45 0.99955 0.00046 99158 6202179 62.53 16 99135 56 0.99943 0.00057 99107 6103021 61.56 17 99079 65 0.99934 0.00066 99047 6003914 60.6 18 99014 72 0.99928 0.00072 98978 5904867 59.64 19 98942 76 0.99922 0.00078 98904 5805889 58.68 20 98866 81 0.99918 0.00082 98825 5706985 57.72 21 98785 85 0.99915 0.00085 98742 5608160 56.77 22 98700 85 0.99913 0.00087 98658 5509418 55.82 23 98615 86 0.99913 0.00087 98571 5410760 54.87 24 98529 85 0.99915 0.00085 98487 5312189 53.92 25 98444 81 0.99917 0.00083 98404 5213702 52.96 26 98363 79 0.99919 0.00081 98323 5115298 52 27 98284 79 0.9992 0.0008 98244 5016975 51.05 28 98205 81 0.99918 0.00082 98164 4918731 50.09 29 98124 82 0.99916 0.00084 98083 4820567 49.13 30 98042 86 0.99912 0.00088 97999 4722484 48.17 31 97956 90 0.99909 0.00091 97911 4624485 47.21 32 97866 94 0.99904 0.00096 97819 4526574 46.25 33 97772 97 0.999 0.001 97724 4428755 45.3 34 97675 102 0.99895 0.00105 97623 4331031 44.34 35 97573 108 0.9989 0.0011 97519 4233408 43.39 Appendix D The Life Table and Its Construction 181

Age (years) lx dx px qx Lx Tx ex

36 97,465 113 0.99884 0.00116 97,409 4,135,889 42.43 37 97,352 120 0.99877 0.00123 97,293 4,038,480 41.48 38 97,232 128 0.99868 0.00132 97,168 3,941,187 40.53 39 97,104 137 0.99859 0.00141 97,035 3,844,019 39.59 40 96,967 147 0.99848 0.00152 96,893 3,746,984 38.64 41 96,820 159 0.99836 0.00164 96,740 3,650,091 37.7 42 96,661 173 0.99822 0.00178 96,575 3,553,351 36.76 43 96,488 187 0.99805 0.00195 96,395 3,456,776 35.83 44 96,301 205 0.99787 0.00213 96,198 3,360,381 34.89 45 96,096 224 0.99767 0.00233 95,984 3,264,183 33.97 46 95,872 244 0.99745 0.00255 95,750 3,168,199 33.05 47 95,628 267 0.99721 0.00279 95,495 3,072,449 32.13 48 95,361 290 0.99696 0.00304 95,216 2,976,954 31.22 49 95,071 314 0.99669 0.00331 94,914 2,881,738 30.31 50 94,757 341 0.9964 0.0036 94,587 2,786,824 29.41 51 94,416 372 0.99606 0.00394 94,230 2,692,237 28.51 52 94,044 408 0.99566 0.00434 93,840 2,598,007 27.63 53 93,636 450 0.99519 0.00481 93,411 2,504,167 26.74 54 93,186 497 0.99467 0.00533 92,938 2,410,756 25.87 55 92,689 547 0.9941 0.0059 92,415 2,317,818 25.01 56 92,142 603 0.99346 0.00654 91,841 2,225,403 24.15 57 91,539 665 0.99274 0.00726 91,207 2,133,562 23.31 58 90,874 731 0.99195 0.00805 90,508 2,042,355 22.47 59 90,143 803 0.9911 0.0089 89,742 1,951,847 21.65 60 89,340 877 0.99018 0.00982 88,901 1,862,105 20.84 61 88,463 960 0.98915 0.01085 87,983 1,773,204 20.04 62 87,503 1,048 0.98802 0.01198 86,979 1,685,221 19.26 63 86,455 1,142 0.98679 0.01321 85,884 1,598,242 18.49 64 85,313 1,239 0.98549 0.01451 84,693 1,512,358 17.73 65 84,074 1,339 0.98407 0.01593 83,405 1,427,665 16.98 66 82,735 1,449 0.98248 0.01752 82,010 1,344,260 16.25 67 81,286 1,570 0.9807 0.0193 80,501 1,262,250 15.53 68 79,716 1,693 0.97876 0.02124 78,870 1,181,749 14.82 69 78,023 1,817 0.97671 0.02329 77,115 1,102,879 14.14 70 76,206 1,947 0.97445 0.02555 75,232 1,025,764 13.46 71 74,259 2,086 0.9719 0.0281 73,216 950,532 12.8 72 72,173 2,240 0.96896 0.03104 71,053 877,316 12.16 73 69,933 2,398 0.96571 0.03429 68,734 806,263 11.53 74 67,535 2,552 0.96221 0.03779 66,258 737,529 10.92 182 Engineering Decisions for Life Quality

Age (years) lx dx px qx Lx Tx ex

75 64,983 2,707 0.95835 0.04165 63,629 671,271 10.33 76 62,276 2,864 0.95401 0.04599 60,844 607,642 9.76 77 59,412 3,025 0.94909 0.05091 57,899 546,798 9.2 78 56,387 3,175 0.94369 0.05631 54,799 488,899 8.67 79 53,212 3,305 0.9379 0.0621 51,560 434,100 8.16 80 49,907 3,417 0.93154 0.06846 48,198 382,540 7.67 81 46,490 3,512 0.92445 0.07555 44,734 334,342 7.19 82 42,978 3,590 0.91647 0.08353 41,183 289,608 6.74 83 39,388 3,629 0.90786 0.09214 37,573 248,425 6.31 84 35,759 3,623 0.89871 0.10129 33,948 210,852 5.9 85 32,136 3,578 0.88865 0.11135 30,347 176,904 5.5 86 28,558 3,504 0.87732 0.12268 26,806 146,557 5.13 87 25,054 3,398 0.86434 0.13566 23,355 119,751 4.78 88 21,656 3,250 0.84996 0.15005 20,031 96,396 4.45 89 18,406 3,047 0.83442 0.16558 16,882 76,365 4.15 90 15,359 2,806 0.81736 0.18264 13,956 59,483 3.87 91 12,553 2,530 0.7984 0.2016 11,288 45,527 3.63 92 10,023 2,234 0.77717 0.22283 8,906 34,239 3.42 93 7,789 1,720 0.77914 0.22086 6,930 25,333 3.25 94 6,069 1,448 0.76133 0.23867 5,344 18,403 3.03 95 4,621 1,190 0.74246 0.25754 4,026 13,059 2.83 96 3,431 952 0.72249 0.27751 2,954 9,033 2.63 97 2,479 741 0.70142 0.29858 2,109 6,079 2.45 98 1,738 557 0.67923 0.32077 1,460 3,970 2.28 99 1,181 406 0.65594 0.34406 977 2,510 2.13 100 775 286 0.63154 0.36846 632 1,533 1.98 101 489 193 0.60604 0.39396 393 901 1.84 102 296 124 0.57947 0.42053 234 508 1.71 103 172 77 0.55185 0.44815 133 274 1.6 104 95 45 0.52322 0.47678 72 141 1.48 105 50 26 0.49363 0.50637 38 69 1.38 106 24 13 0.46313 0.53687 17 31 1.29 107 11 6 0.43178 0.56822 9 14 1.2 108 5 3 0.39964 0.60036 3 5 1.11 109 2 1 0.3668 0.6332 1 2 1.04 Appendix D The Life Table and Its Construction 183

Table D.4 Life table for Canada (female 2000–2002) (Statistics Canada)

lx dx px qx Lx Tx ex

0 100,000 467 0.99533 0.00467 99,589 8,203,072 82.03 1 99,533 35 0.99965 0.00035 99,514 8,103,483 81.41 2 99,498 20 0.9998 0.0002 99,486 8,003,969 80.44 3 99,478 15 0.99985 0.00015 99,471 7,904,483 79.46 4 99,463 12 0.99988 0.00012 99,456 7,805,012 78.47 5 99,451 10 0.9999 0.0001 99,445 7,705,556 77.48 6 99,441 9 0.99992 0.00008 99,437 7,606,111 76.49 7 99,432 7 0.99993 0.00007 99,429 7,506,674 75.5 8 99,425 7 0.99993 0.00007 99,421 7,407,245 74.5 9 99,418 7 0.99993 0.00007 99,415 7,307,824 73.51 10 99,411 9 0.99991 0.00009 99,407 7,208,409 72.51 11 99,402 8 0.99991 0.00009 99,398 7,109,002 71.52 12 99,394 13 0.99987 0.00013 99,387 7,009,604 70.52 13 99,381 15 0.99984 0.00016 99,374 6,910,217 69.53 14 99,366 20 0.9998 0.0002 99,356 6,810,843 68.54 15 99,346 24 0.99976 0.00024 99,333 6,711,487 67.56 16 99,322 28 0.99972 0.00028 99,308 6,612,154 66.57 17 99,294 31 0.99969 0.00031 99,278 6,512,846 65.59 18 99,263 33 0.99967 0.00033 99,246 6,413,568 64.61 19 99,230 33 0.99966 0.00034 99,214 6,314,322 63.63 20 99,197 34 0.99966 0.00034 99,180 6,215,108 62.65 21 99,163 33 0.99966 0.00034 99,146 6,115,928 61.68 22 99,130 33 0.99966 0.00034 99,114 6,016,782 60.7 23 99,097 33 0.99967 0.00033 99,080 5,917,668 59.72 24 99,064 33 0.99967 0.00033 99,047 5,818,588 58.74 25 99,031 32 0.99967 0.00033 99,015 5,719,541 57.76 26 98,999 33 0.99967 0.00033 98,982 5,620,526 56.77 27 98,966 33 0.99967 0.00033 98,950 5,521,544 55.79 28 98,933 34 0.99965 0.00035 98,916 5,422,594 54.81 29 98,899 36 0.99963 0.00037 98,881 5,323,678 53.83 30 98,863 39 0.99961 0.00039 98,843 5,224,797 52.85 31 98,824 42 0.99958 0.00042 98,803 5,125,954 51.87 32 98,782 45 0.99954 0.00046 98,760 5,027,151 50.89 33 98,737 50 0.9995 0.0005 98,711 4,928,391 49.91 34 98,687 54 0.99945 0.00055 98,660 4,829,680 48.94 184 Engineering Decisions for Life Quality

lx dx px qx Lx Tx ex

35 98,633 60 0.99939 0.00061 98,603 4,731,020 47.97 36 98,573 66 0.99933 0.00067 98,539 4,632,417 46.99 37 98,507 72 0.99927 0.00073 98,471 4,533,878 46.03 38 98,435 78 0.99921 0.00079 98,397 4,435,407 45.06 39 98,357 83 0.99915 0.00085 98,315 4,337,010 44.09 40 98,274 91 0.99908 0.00092 98,229 4,238,695 43.13 41 98,183 97 0.99901 0.00099 98,134 4,140,466 42.17 42 98,086 107 0.99891 0.00109 98,033 4,042,332 41.21 43 97,979 117 0.9988 0.0012 97,920 3,944,299 40.26 44 97,862 129 0.99868 0.00132 97,797 3,846,379 39.3 45 97,733 142 0.99855 0.00145 97,662 3,748,582 38.36 46 97,591 156 0.9984 0.0016 97,513 3,650,920 37.41 47 97,435 171 0.99824 0.00176 97,350 3,553,407 36.47 48 97,264 187 0.99807 0.00193 97,170 3,456,057 35.53 49 97,077 204 0.9979 0.0021 96,975 3,358,887 34.6 50 96,873 222 0.99771 0.00229 96,761 3,261,912 33.67 51 96,651 243 0.99749 0.00251 96,530 3,165,151 32.75 52 96,408 266 0.99724 0.00276 96,275 3,068,621 31.83 53 96,142 293 0.99695 0.00305 95,996 2,972,346 30.92 54 95,849 323 0.99663 0.00337 95,687 2,876,350 30.01 55 95,526 355 0.99628 0.00372 95,349 2,780,663 29.11 56 95,171 390 0.9959 0.0041 94,976 2,685,314 28.22 57 94,781 427 0.99549 0.00451 94,568 2,590,338 27.33 58 94,354 466 0.99506 0.00494 94,121 2,495,770 26.45 59 93,888 505 0.99462 0.00538 93,636 2,401,649 25.58 60 93,383 548 0.99413 0.00587 93,109 2,308,013 24.72 61 92,835 595 0.99359 0.00641 92,538 2,214,904 23.86 62 92,240 649 0.99296 0.00704 91,915 2,122,366 23.01 63 91,591 709 0.99226 0.00774 91,236 2,030,451 22.17 64 90,882 772 0.9915 0.0085 90,496 1,939,215 21.34 65 90,110 841 0.99067 0.00933 89,689 1,848,719 20.52 66 89,269 915 0.98975 0.01026 88,812 1,759,030 19.7 67 88,354 999 0.98869 0.01131 87,854 1,670,218 18.9 68 87,355 1086 0.98757 0.01243 86,812 1,582,364 18.11 69 86,269 1175 0.98638 0.01362 85,682 1,495,552 17.34 70 85,094 1271 0.98507 0.01493 84,458 1,409,870 16.57 71 83,823 1378 0.98355 0.01645 83,134 1,325,412 15.81 72 82,445 1503 0.98177 0.01823 81,694 1,242,278 15.07 73 80,942 1635 0.97981 0.02019 80,124 1,160,584 14.34 Appendix D The Life Table and Its Construction 185

lx dx px qx Lx Tx ex

74 79,307 1,768 0.9777 0.0223 78,423 1,080,460 13.62 75 77,539 1,913 0.97533 0.02467 76,582 1,002,037 12.92 76 75,626 2,074 0.97258 0.02742 74,589 925,455 12.24 77 73,552 2,255 0.96934 0.03066 72,425 850,866 11.57 78 71,297 2,441 0.96576 0.03424 70,076 778,441 10.92 79 68,856 2,621 0.96193 0.03807 67,546 708,365 10.29 80 66,235 2,809 0.9576 0.0424 64,830 640,819 9.67 81 63,426 3,011 0.95252 0.04748 61,920 575,989 9.08 82 60,415 3,235 0.94646 0.05354 58,798 514,069 8.51 83 57,180 3,470 0.93932 0.06068 55,445 455,271 7.96 84 53,710 3,691 0.93128 0.06872 51,865 399,826 7.44 85 50,019 3,879 0.92245 0.07755 48,080 347,961 6.96 86 46,140 4,015 0.91297 0.08703 44,132 299,881 6.5 87 42,125 4,088 0.90296 0.09704 40,081 255,749 6.07 88 38,037 4,095 0.89233 0.10767 35,990 215,668 5.67 89 33,942 4,039 0.88101 0.11899 31,922 179,678 5.29 90 29,903 3,914 0.86912 0.13088 27,946 147,756 4.94 91 25,989 3,722 0.85678 0.14322 24,128 119,810 4.61 92 22,267 3,471 0.84412 0.15588 20,532 95,682 4.3 93 18,796 3,212 0.82913 0.17087 17,190 75,150 4 94 15,584 2,911 0.8132 0.1868 14,129 57,960 3.72 95 12,673 2,582 0.79624 0.20376 11,382 43,831 3.46 96 10,091 2,238 0.77823 0.22177 8,972 32,449 3.22 97 7,853 1,891 0.75917 0.24083 6,908 23,477 2.99 98 5,962 1,556 0.73906 0.26094 5,184 16,569 2.78 99 4,406 1,243 0.71791 0.28209 3,784 11,385 2.58 100 3,163 962 0.69575 0.30425 2,682 7,601 2.4 101 2,201 721 0.6726 0.3274 1,841 4,919 2.23 102 1,480 520 0.64849 0.35151 1,220 3,078 2.08 103 960 361 0.62349 0.37651 779 1,858 1.94 104 599 241 0.59763 0.40237 478 1,079 1.8 105 358 154 0.57098 0.42902 281 601 1.68 106 204 93 0.54362 0.45638 158 320 1.57 107 111 54 0.51561 0.48439 84 162 1.46 108 57 29 0.48704 0.51296 43 78 1.36 109 28 15 0.458 0.542 20 35 1.27 186 Engineering Decisions for Life Quality D.5 References

Chian CL (1984) The life table and its applications. Robert Krieger Publishing, Malabar Keyfitz N, Flieger W (1990) World population growth and aging. University of Chicago Press, Chicago Max Planck Institute of Demography. Human life-table database. http://www.lifetable.de/cgi-bin/datamap.plx Statistics Canada. Life tables, Canada and Provinces and territories. http://www.statcan.gc.ca/pub/84-537-x/4064441-eng.htm World Health Organization (WHO). Life tables for WHO member states. http://www.who.int/whosis/database/life_tables/life_tables.cfm

Index

Affordability, 2 marginal, 8, 138, 139 Air Quality, 12, 53–59, 66, 68, 69, 113 opportunity, 136–139, 141, 148, 149 ALARA, 30 Cross-Entropy, 98–102, 105 ALARP, 30 Assets, 37, 50, 87–94, 124, 143, 149 Defense, 88, 91, 94, 109 exposed, 87, 90, 94 Demography, 181 Dike, 97, 105 Calibration, 12, 22, 32, 97, 115, 116, 132, Discount Rate, 74, 135, 136, 139–142, 167, 170–172, 175, 177 144–148, 153, 154 CANDU, 71, 81 fiscal, 140, 144 Capacity, 1, 2, 4, 7, 8, 12–14, 27, 28, nominal, 145, 153 30–34, 37–39, 50, 54, 55, 62, 64, 71, 72, real, 139, 145–147, 154 75–77, 79, 90, 91, 102, 109, 110, 112, social, 136, 139–142, 144, 148 115, 122, 124, 125, 136, 167, Discounting, 29, 45–50, 61–69, 74, 75, 168, 171, 173 77, 79, 84, 99, 102, 103, 135–138, societal, 1, 2, 4, 7, 12–14, 28, 31, 33, 140, 142, 144, 145, 147, 148, 154–157, 34, 37, 38, 50, 54, 55, 62, 64, 71, 72, 76, 160–162 77, 79, 109, 112, 115, 122, 167, 171, Distribution, 97, 100, 101 173 monoscopic, 97, 100, 101 sustainable, 28, 31, 50, 54 panscopic, 101 Climate Change, 87, 155 Cobb-Douglas, 33, 128, 169, 170 Elasticity, 74, 118, 121, 147 Cohort, 24, 55, 181–184 Error, 110 Consumption, 20, 38, 45, 61, 62, 73, Exclusion, 109 75, 116, 118, 119, 122, 124, 125, 129, Expenditure, 4, 5, 9, 11, 12, 17, 20, 27, 30, 130, 136–139, 141, 142, 146–148, 150, 38, 49, 50, 53–55, 62–64, 69, 71, 72, 74, 156, 162 75, 79, 82–84, 112, 122–124, 138 Correlation, 22, 172 Exposure, 6, 10, 27, 31, 49, 55, 58, 63, 71, Cost, 3, 8, 10, 12, 26, 28, 37–39, 45, 50, 79–83, 87–95, 113, 156 53–57, 59, 60, 65–69, 75, 76, 78–80, 84, 113, 135–149, 151, 153 Factor, 30–34, 168, 173, 174, 178, 179 benefit, 10, 12, 26, 28, 37–39, 45, 50, Labour-Demographic, 30–34, 168, 173, 53–57, 59, 60, 65–69, 75, 76, 78–80, 84, 174, 178, 179 113, 135–137, 140–148, 151, 153 188 Index

Financing Horizon, 99, 102–104, 144, Parameter, 8, 12, 20, 38, 48, 56, 74, 100, 155–158, 160–162 101, 115, 116, 122, 126, 132, 143, 152, Flood, 12, 87–91, 93–95, 97–105, 113, 167, 173, 174, 178, 183 135, 155 Pollution, 6, 7, 27, 39, 53–63, 66–68, 78, 79, 110 Hazard, 31, 39, Port City, 88, 90, 91 private, 31 Power Plant, 6, 37, 71, 82, 84 public, 31, 39 Preference, 115 revealed, 115 Income, 7, 12, 19, 20, 24, 27, 28, 30, 38, Principle, 10, 11, 31, 111 42–44, 73, 74, 121–125, 129, 130, 138, compensation, 10, 111 139, 145, 151, 152, 167, 170 De minimis, 11 Index, 4, 11, 12, 14, 17–23, 28, 29, 33, 37, Democratic, 31 38, 53, 62, 69, 71, 75, 82, 84, 112, 115, Equivalence Principle, 158, 160 131, 173, 176 Kaldor-Hicks, 10, 111 Human Development, 18, 19, 21, 22, Life Measure, 158 33, 115, 173 Profit maximization Principle, 127 Life Quality, 4, 11, 12, 14, 17, 18, Profit maximization, 127, 156, 170 20–23, 28, 29, 33, 37, 38, 53, 62, 69, 71, Profit, 124, 127, 129, 156, 170 75, 82, 84, 112, 115, 131, 173, 176 Indifference Curve, 119–122 Radiation, 6, 71, 79–83, 113, 160 Inflation, 102, 103, 137, 141, 144, 145, Ranking of nations, 19 153, 155, 159 Rationality, 112 Redundancy, 81, 109 Levee, 97, 100–104, 106, 113 Resilience, 109 Life expectancy, 4–7, 11–13, 17–21, Resources, 109, 110, 112, 113, 115, 122, 23–27, 29–34, 38–42, 44, 48, 49, 55, 61, 130, 135, 137, 139, 145, 150, 155, 171 64, 65, 73–77, 79, 90, 111–113, 130, Returns, 13, 110, 126, 128, 130, 135, 157, 167, 168, 176, 181, 185 137, 147 Life Table, 181, 182 diminishing, 110, 126, 130 abridged, 181, 182 Risk, 2, 3, 5, 6, 9, 14, 17, 25, 26, 28, 38, complete, 181 42, 58, 61, 62, 64, 75, 77–79, 98, 112, 115, 161 cancer, 6, 79 Mortality, 6–8, 17, 20, 24–26, 29, 38–43, mitigation, 2, 3, 9, 75, 98, 115, 161 50, 53, 55–61, 63, 64, 66–69, 79, 84, of death, 5, 25, 26, 38, 58, 61–64, 77, 104, 115, 116, 156, 181, 185 78, 112 infant, 8, 17, 25

Safety first, 110 Nation, 10, 14, 17–19, 21, 22, 24, 25, 30, Satisficing, 110 74, 93, 94, 115, 144 Savings, 136, 138, 139, 148, 150 Ranking, 19, 93, 94 Shadow Price, 139, 142, 146, 151 Nuclear Power, 6, 71, 81, 82, 84, 149, Smoking, 5, 6 154, 161 Social Indicator, 4, 11, 12, 17–20, 22–24, Numeraire, 111, 142, 144 26, 33, 74, 98, 112, 115, 116 Species, 98, 109 OECD, 4, 22, 32, 33, 74, 124, 131, 132, Standard, 19, 26, 30, 46, 48, 56, 57, 154, 167–169, 171–173 59, 60, 62, 67, 69, 77, 88, 101, 144, 147, 155 Index 189

Storm, 87, 90, 101 Value, 2–4, 13, 14, 18, 20–24, 28, 33, Substitution, 117, 120–122, 129 38–40, 44, 48, 49, 54, 55, 57–61, Survival, 8 63–68, 73–80, 83, 90, 94, 98–100, 102, Child, 8 103, 105, 106, 111, 112, 115–118, infant, 8 121–127, 131, 132, 135, 137, 139, SWTP, 49, 76, 77, 80 142–150, 152, 154–161, 167, 170, 171, 173–175, 177, 179, 181, 184 Tradeoff, 130, 131, 168 Vita mensura, 4 Labour-Leisure, 130, 131, 168 VSL, 57–61, 63, 65, 66, 77–79 Trial, 103, 104, 110 Wages, 32, 117, 122, 124, 127–129, 167, Uncertainty, 2, 21, 58, 90, 113, 143, 146, 169, 170 148 Welfare, 1, 2, 17, 20, 27, 33, 115 Utility, 28, 47, 62, 73, 74, 116–119, 121, human, 1, 2, 17, 20, 27, 33, 115 129, 130, 132, 174, 178 Willingness-to-pay (WTP), 61, 63, 73, 75, cardinal, 117, 118 77, 78, 80 function, 28, 47, 62, 73, 74, 116–119, 121, 129, 130, 132, 174, 178 ordinal, 116, 117