Cost and benefit including value of life and limb measured in time units

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1 Secial Worksho on Risk Accetance and Risk Communication March 26-27, 2007, Stanford University Cost and benefit including value of life and limb measured in time units Ove Ditlevsen and Peter Friis-Hansen Technical University of Denmark Nils Koels Allé, Building 403, DK 2800 Kgs. Lyngby, Denmark, Abstract: Key elements of the authors ork on money equivalent time allocation to costs and benefits in risk analysis are ut together as an entity. This includes the data suorted dimensionless analysis of an equilibrium relation beteen total oulation ork time and gross domestic roduct leading to the definition of the life quality time allocation index (LQTAI). A ostulate of invariance of the LQTAI allocates societal value in terms of time to avoid life shortening fatalities as ell as serious injuries that shorten the life in good health. The observed large difference beteen the actual costs used by the oner for economically otimizing an activity motivates a simle risk accet criterion suited to be imosed on the oner by the ublic. The aer ends ith an examle concerning allocation of economical means for mitigation of loss of life and limb on a ferry in fire. 1. Introduction An engineering activity is lanned to be realized ithin a statistically homogeneous economical region. By dividing all costs and benefits in the exected net cost equation by the average age er time unit over the oulation of the region the equation gets hysical dimension as time, and the equation becomes indeendent of local inflation an urchase oer. The equation may contain terms that reresent loss of life and limb, and ossibly also environmental damages. Such terms have in the ast been considered as intangibles causing them to be excluded from the cost-benefit analysis. Hoever, ideas based on macro economical reasoning have in the last decade oened ossibilities of making rational evaluations (e.g. by use of the life quality index (LQI) defined in (Nathani, Lind, & Pandey 1997, Pandey, Nathani, & Lind 2006) and extensively studied and alied in (Rackitz 2002, Rackitz, Lentz, & Faber 2005), or, for environmental damage, the Nature reservation illingness index defined in (Friis-Hansen & Ditlevsen 2003)). In the time formulation the life and limb losses may at a first glance simly be ritten as the increment of the exected life time in good health caused by the loss giving accident. Hoever, this ould be an oversimlification because a art of the loss is ork time of larger societal value than the free time. The correction can be made by use of the criterion of invariance of the LQI, or, directly in time units, by use of invariance of the life quality time allocation index (LQTAI) defined in (Ditlevsen & Friis-Hansen 2005, Ditlevsen & Friis-Hansen 2007, Ditlevsen 2007). The results may be slightly different because the LQTAI is an extended more general version of the LQI. This aer gives a short recaitulation of the authors thoughts behind the LQTAI including its emirical suort. To the authors surrise these thoughts and the suorting emirical findings have turned out to be controversial. For large rojects the decision making by the oner is restricted by ublic requirements. The oner is rimarily focused on maximizing the rofit only including the direct costs to be sent on insurance remiums, uncovered losses and damage comensations. These costs are often much less than the societal value of life and limb as obtained from the invariance of the LQI or the LQTAI. Therefore the society must consider the ossibility of this larger loss and rotect itself under the consideration that the society has a ositive 1

2 interest in the realization of the roject. Rational reasoning leads to a ublic accet criterion formulated in (Friis-Hansen & Ditlevsen 2003, Ditlevsen 2003) and is recaitulated herein. Finally the aer gives an examle of using the LQTAI to assess the exected societal time value loss of life and limb due to a fire on a ferry. 2. Standard exected loss function examle The standard examle of technical decision roblem is related to an oeration (shi transort, factory, structure, etc.) that in the mean roduces a net gain g er time unit facing the fact that n different indeendent categories of cost generating accidents occur in time at random time oints that for the ith category is in accordance ith a homogeneous Poisson rocess of intensity λ i, i = 1,..., n. Denoting the mean of the random cost of an accident in category i by µ i, the exected loss function related to the oeration becomes asymtotically as t, (Ditlevsen 2003) L(λ 1,..., λ n, µ 1,..., µ n ) = c(λ 1,..., λ n, µ 1,..., µ n ) + 1 γ n λ i µ i g γ i=1 (1) here c(λ 1,..., λ n, µ 1,..., µ n ) is the initial cost invested at time zero and γ is the interest rate. Of dimensional reasons it follos that the initial cost function must be linear in µ 1,..., µ n. The unit of (1) may be the currency of the relevant country, and as such the amounts are time deendent on inflation and current urchase oer of the currency in the actual country. Thus the µ-values must be corrected to equivalent values at time zero. This correction is made automatic if the current values are divided by the average current age (salary) er time unit over the entire oulation of the country. Thus the mean losses µ i and the gain g become exressed as the time it takes to earn the amounts by the average age. To the first aroximation these amounts measured in time units become indeendent of time and to some degree of the same size from country to country. The oner of the oeration then acts otimally if the oeration is designed so that λ 1,..., λ n, µ 1,..., µ n get values that minimize the loss function (1). Even though some of the µ-values may reresent values of life and limb these values are not intangibles from the oners oint of vie. These values are simly those that are imlied by common insurance remiums and damage comensation ractice. These oner s values are usually considerably less than the societal values obtained from life quality index considerations (LQI) as first advocated by Nathani, Lind, and Pandey a decade ago (Nathani, Lind, & Pandey 1997), and since then further develoed by the same authors (Pandey, Nathani, & Lind 2006) and by Rackitz (Rackitz 2002, Rackitz, Lentz, & Faber 2005). The observation that the societal loss value is much larger than the oners loss value related to some accident causes the society to set u certain restrictions on the oner s activity to ensure that the societal loss is fully comensated by the benefit the society obtains from the oner s activity. A rational ublic accetance rule based on the concet of oner s value versus societal value is formulated in (Ditlevsen 2003, Friis-Hansen & Ditlevsen 2003), and is recaitulated in Section 6. Having a ish to better understand the first rinciles behind the LQI formulation, and to see to hich degree the LQI may be emirically justifiable and to hich degree it is based on a ostulate of consistent behavior, the authors came to a slightly different and more general societal life quality model based on dimensionless formulation of reference equilibrium beteen ork time and free time (leisure time). The formulation of this model is in short form recaitulated in the next section based on the authors aers (Ditlevsen & Friis-Hansen 2005, Ditlevsen & Friis-Hansen 2007, Ditlevsen 2007). 2

3 3. Preference equilibrium beteen ork time and free time in good health This section closely follos the text in (Ditlevsen 2007) ith a minor but imortant generalization resent in the earlier aer (Ditlevsen & Friis-Hansen 2005) but not exlicitly addressed in the cited aers on the LQI. The model formulation in Section 2 of (Ditlevsen & Friis-Hansen 2005) starts ith random variable modeling of the relevant quantities for the individual member of the oulation hereuon exectations are taken over the oulation. Herein as in (Ditlevsen & Friis-Hansen 2007, Ditlevsen 2007) the modeling starts directly on the level of mean values over the entire oulation. The slight generalization is that only a fraction r of the time unit reresents time in good health hile the fraction 1 r reresents time ith illness or bad health, that is, time that cannot be used as (moneyroducing) ork time. Thus r here is the fraction of the time unit used for ork, and the ratio /r 1 is the fraction of the time in good health used for ork Time value modeling An increment d has a monetary value directly roortional to the monetary value of the roduction in the time increment, of course. Hoever, hat herein is called exerienced orth is reasonably defined as the dimensionless ratio of this increment value to the value of the roduction in the entire ork time fraction. This definition of exerienced orth of ork time is simly based on the sychological feeling that a small increment of much has less orth than a small increment of little. The definition is consistent ith the emirical Weber-Fechner la that states that there is a logarithmic relation beteen the hysical magnitudes of stimuli and the erceived intensity of the stimuli (source htt://en.ikiedia.org/iki/weber- Fechner la). Thus the exerienced orth of the increment d is measured relatively as d( )/, here is a factor defined as the ratio beteen the total yearly GDP (Gross Domestic Product) G ithin a geograhical region such as a country and the total yearly ork based salary S aid out to the inhabitants of the region. The factor may be interreted as an enhancement factor that increases (or decreases) the societal value of the ork time fraction from to. The corresonding money value is then G = S = (S/), here S/ is the salary er ork time unit. It is seen that aears as an overall (or equivalent ) roductivity that enhances S to G. This enhancement includes all ossible sources to G, that is, also those sources not directly related to the used manoer. Basically G is a measure of the economical ealth of the society and S is a measure of ho much of this economical ealth is distributed directly to the orking oulation. In the folloing ill be denoted as the time equivalent roductivity realizing that the ord roductivity is used differently in roduction economics. More details are discussed in (Ditlevsen 2007). Similarly it is assumed that the exerienced orth of the free time in good health is measured relatively as d(r )/(r ) hich for a constant r is the same as d(1 /r)/(1 /r). Note that d(1 /r)/(1 /r) is negative for a ositive ork fraction increment d. Therefore a balance beteen d()/ = d(/r)/(/r) and d(1 /r)/(1 /r) may exist for hich constant orth is maintained. This consideration leads to the balance equation c d( /r ) + (1 c) d( 1 /r ) = 0 (2) /r 1 /r for some constant c beteen 0 and 1. More correctly, c should be ritten as a function c(r) of r, but for simlicity of riting (r) is suressed herever it is not needed exlicitly in the folloing. For exlanation of (2) it is noted that any relation beteen differentials er definition is linear in the differentials. That the necessarily ositive coefficients sum to one is just the result of a convenient normalization obtained by dividing the equation by the sum of the coefficients. Assume no that the ork market tries to maintain this balance hatever the values of and and assume that locally may be a function of (or, mathematically equivalent, that locally may be a function of ). 3

4 Then (2) reduces to the simle differential equation ( c /r 1 c ) d(/r) = c d (3) 1 /r beteen /r and. It is seen that d = 0 for /r = c and an elementary analysis of the variation of the sign of d ith /r in the neighborhood of /r = c shos that is minimal for /r = c. The general solution to (3) is = min (1 /r) 1 1/c (/r) 1 (1 c) 1 1/c c 1 (4) here min = min Data The relation (4) is suorted by data from different OECD countries. The numerical data for Denmark are given in full ith detailed exlanations in (Ditlevsen & Friis-Hansen 2005), and for the other OECD countries in diagrammatic form in (Ditlevsen & Friis-Hansen 2007) all based on the data in (OECD 2004) Denmark ADAMS Denmark ADAMS 0.16 Denmark ADAMS; c =0.084; K = = ex[!0.044 (y!1948)] year Germany 0) year Germany Germany; c =0.080; K = )15 0)14-0)080.0)12& ex23!0)0&4 45!1" )13 0)12 0)11 0) )0" ) "50 1"60 1"&0 1"80 1"" ear Canada year Canada Canada; c =0.095; K = = ex[!0.041 (y!1948)] year year Figure 1. To: Data oints for Denmark in the years Recorded ork time fraction and time equivalent roductivity (ratio of salary to GDP er same time unit). The smooth curves are jointly least square fits of (5) (left diagram) and (4) (for r = 1) ith (5) substituted (middle diagram) and are consistent ith the curve in the right diagram for (, ) ( min = 1.81, c = 0.084). Middle: Data for Germany in the years ( min = 1.80, c = 0.080). Bottom: Data for Canada in the years ( min = 1.89, c = 0.095). 4

5 The to first to diagrams in Fig. 1 sho the data for and as functions of the year for Denmark in the eriod from 1948 to The data oints for the air (, ) are shon in the third diagram. The largest value of corresonds to year 1948, and the connecting lines beteen the data oints sho the succession of the oints ith the year. The second and third ro of diagrams sho the corresonding data time series for Germany and Canada only reaching back to 1960, hoever, and therefore giving less strong suort than the Danish data. The smooth curves in the left diagrams are fits of the decreasing exonential function = c + a e b(t 1948) c as t (5) adoting the hyothesis that the asymtotic value as t is equal to the value of at hich () = (/1, 1) takes its minimal value min. The unknon arameters a, b, c, min are estimated by a simultaneous least square fit of (4) and (5) to the to data sets for (t, ) and (t, ). The squared deviations are divided by the average squared deviation for each of the to data sets to bring the squared deviations into the same size scale. In (Ditlevsen 2007) the time ordered Danish data samles of the ork time ratio and the time equivalent roductivity are analyzed don to details that make it ossible to formulate a hite noise driven stochastic rocess model for the joint time develoment of the to dimensionless quantities. It is remarkable that the data residuals ith resect to the redicted relation as function of time sho stationary random behavior in all essentials. Both rocesses are ell described as autoregressive rocesses in discrete time. Simulations ith the model sho samle curves of great similarity to the data samle curves. Thus all significant systematic behavior of the Danish data is catured by the relation (4) for r = 1 and the exonential time decay (5). It is seen in all three ros of diagrams in Fig. 1 that the Danish and the German data have reached a statistically stationary state after about 1975 hile a stationary state is reached before 1960 for the Canadian data. As judged by the eye, the diagrams in (Ditlevsen & Friis-Hansen 2007) for 18 OECD countries reveal that only the data for Iceland, Ireland, Jaan and Ne Zealand seem to be deviate significantly from the redictions of the model The time reduction factor r and the ealth index I r As it is seen from (4), the time reduction factor r simly defines an affinity maing that moves all data oints (, ) to (, /r). The grah of (, ) defined by (4) for r = 1 mas to a curve that deviates from the grah of (4). Hoever, a simle test calculation as done in (Ditlevsen & Friis-Hansen 2007) shos that the deviation in beteen the to grahs for fixed ithin the relevant value domain of is at most about 0.5% for r > 2/3. Since the oint (c, min ) mas to the oint (c/r, min ) it follos that c(r 0 ) = c/r 0 for the actual value r 0 of r. In rincile the value of r 0 can be estimated through a oulation health investigation. A value of about 0.9 may be a reasonable assessment. The deviations beteen the maed curve and the curve defined by (4), both corresonding to r and coinciding in the oint of minimum of, are negligible as comared to the random fluctuations of the data oints along the curves. Therefore it is not ossible to estimate the value of r from the considered data. Hoever, the data fits indicate that it ould be in conflict ith the model to claim that r has been varying substantially during the observation eriod. If the differential on the left side of (2) takes a value different from zero it is a sign of unbalance beteen the ork effort and the available free time. For a ositive value of the left side the total societal orth (or societal living standard) is increasing, and for a negative value the orth is decreasing. A basic non-dimensional ealth index I r may therefore be defined relatively by di r I r = V (r) [c d( /r ) + (1 c) d( 1 /r ) ] /r 1 /r (6) 5

6 in hich V (r) is some differentiable function of r. The fraction di r /I r is taken to be a measure of the total exerienced orth of ork time and free time together. Integration of (6) gives the general solution I r = J(r) [ ( ) c ( 1 ) ] 1 c V (r) r r { [ ( = J(r) ex V (r) c log ) ( + (1 c) log 1 )]} r r (7) in hich J(r) is another differentiable function of r. In the final definition of the ealth index I r the to functions V (r) and J(r) ill both be set to 1, relying on an argument of slo variation. 4. Invariance rincile Assume that some increment d(r E) of the exected life in good health at birth is obtained by some risk reducing action. The art (/r) d(r E) is used for ork and has the exerienced orth (/r) d(r E)/ (E) = d(r E)/(r E), and the rest is free time that has the exerienced orth (1 /r) d(r E)/[(1 /r)r E] = d(r E)/(r E). According to the combination rule (6) the total exerienced orth of the time increment d(r E) is then exerienced orth of d(r E) = V (r) d(r E) r E [ dr = V (r) r + de ] E The invariance rincile ostulates that the amount d taken from to ay for the risk reducing activity that roduces the value in (8) should be such that di r /I r under constant ork effort is balanced by the exerienced orth of d(r E). By differentiation of (7) under constant it follos that di r I r = V (r) c d + V (r) { ( ) c log r 1 c r + r(r ) + V (r) [ ( c log ) ( + (1 c) log 1 )] + J } (r) dr (9) V (r) r r J(r) here means derivative ith resect to r. Making the reasonable assumtion that the to functions V (r) and J(r) are sufficiently sloly varying that the terms ith V (r)/v (r) and J (r)/j(r) can be neglected, it follos by adding (8) and (9) and setting to zero that the amount d taken from given by (8) d = 1 c de E + [ 1 c r + c log ( r )] dr c (10) balances the gain by obtaining the increments de of exected life and dr of healthy life. For the actual value r 0 of r and the stationary state here = c(r 0 ) = c(1)/r 0 and = min, (10) becomes d = r [ 0 de 1 min c(1) E + c(1) + r ( )] 0 min c(1) c(1) c (r 0 ) log dr (11) r 0 c(1) The sum of the to equations (8) and (9) leading to (10) can be obtained directly as d(i r r E)/(I r r E) = 0 under constant. The roduct I r r E (setting V (r) = 1 and J(r) = 1) is called the Life Quality Time Allocation Index (LQTAI) (here revised relative to (Ditlevsen & Friis-Hansen 2005) by the factor r). To aly (11) for value assignment to dr an assessment of the value of the derivative c (r 0 ) is needed. Data suort is hardly obtainable. Thus it is necessary to make some normative choice based on some areciable 6

7 assumtion. Such an assumtion is used in (Ditlevsen & Friis-Hansen 2007) setting c(r) = c/r and obtained as seen above by ostulating that is not influenced by variations of r (imlying that the GDP and the total salary changes in the same ace ith the variation of r). For the urose to obtain c (r 0 ) the ostulate may be relaxed to be valid only for small variations of r in the vicinity of r 0. Acceting this ostulate the factor to the logarithm in (11) becomes r 0 c(1) c (r 0 ) = 1 r 0 (12) The to terms on the right side of (11) are the fractions of min that the oulation should agree that the society invests er erson for removing the accident source that causes the exected life to decrease by de and the exected life in good health to decrease by E dr, resectively. Dividing by the mean accident rate er erson gives the equivalent time values. After multilication by the salary er time the amounts are obtained in the actual money unit. The first amount is the so-called Imlied Cost of Averting a Fatality (ICAF). The second may be called the Imlied Cost of Averting an Injury (ICAI). To aly these results in a decision roblem for a secific roject, the increments de and dr must be assessed in relation to the roject. This assessment is made by use of engineering modeling of the consequences of the ossible accidents that are taken into account in the cost-benefit analysis. Examles are given in (Ditlevsen 2003, Ditlevsen 2004). Even though it is based on an idea of logically consistent evaluation of exerienced orth, the invariance rincile alied to the ealth index I r to obtain ICAF and ICAI is a normative rincile because it can only be suorted by subjective oinion. Of logical and ethical reasons it is argued that it is the right rincile to aly, but the illingness to ay and current ractice may be different. The ay out is to interret the ICAF and the ICAI as societal values that should enter the ublic accet rules as formulated in Section Public accetance rules To rotect the elfare of its citizens against undue damage and exloitation the ublic sets certain restrictions on otherise free money making activities. The decisions related to the design and oeration of the technical installations needed for these activities are in a free society left to the oners ho aim for otimizing their gains er time unit. Their cost- benefit analyses tyically only include the direct costs to be sent on insurance remiums and damage comensations. These costs are often much less than the societal value of life and limb as obtained in Section 4. Therefore the society must consider the ossibility of this larger loss and rotect itself under the consideration that the society has a ositive interest in the realization of the roject. A ay to ensure this is to require that the comany tax yield to the society is sufficiently large to cover the loss of the society in excess of the oners direct comensation after the occurrence of the damage. The concet of society is indeendent of country borders imlying that in the modeling it is not imortant hether the tax is aid in the one or the other country ithin a region that may embrace several countries. The comany tax rate as ell as other arameters of the model should be considered as local or regional assessment arameters that may vary from region to region around the orld. Accordingly the model defines a rocedure that in the average over all kinds of risky oerations is a guide to formulate ublic accetance criteria. Referring to (1), the net gain rate after loss due to the adverse events is n i=1 (g i λ i µ oi ) here µ oi is the oners mean cost value of the adverse event and g i is the gain er time unit, both associated to the ith adverse event category. The gain g i is reasonably defined so that g 1 /µ o1 = g 2 /µ o2 =... = g n /µ on and n i=1 g i = g. With a comany tax rate of ρ, the requirement of full comensation for damage in any single category then imlies that the oeration is only acceted by the ublic if (g i λ i µ oi )ρ λ i µ i here µ i is 7

8 the societal cost value in excess of the oner s exected cost value µ oi. The accetance rule can be ritten on the dimensionless form ( λ i µ oi ) µ 1 i (13) g i ρ µ oi It is interesting to note that accetance by requiring that the LQI (or the LQTAI) does not decrease by the lanned activity is a milder requirement than (13). In fact, it is shon in (Ditlevsen 2003) that the accetance criterion becomes (13) ith ρ = A model for determining de and dr generated by an accident As a reasonable simlification it is assumed that the robability that any erson exeriences at most one serious injury due to an accident during the life time is close enough to 1 that the ossibility of more than one serious injury er erson can be neglected. Moreover, it is assumed that the time from birth to the injury or the death haens can be modeled as the random variable min{x, Y } here X is an exonentially distributed random variable of mean κ 1 and Y is the random time to natural death (that is, the life time in case the source of the considered fatal event is removed). This model as considered in (Ditlevsen 2003) and generalized slightly in (Ditlevsen 2004) for the time to death. Here it is used for the time to the occurrence of serious injury also. To distinguish beteen death and injury the life time L and the life time in good health L h are modeled as { { min{x, Y } R min{x, Y } both ith robability P f L = and L h = (14) Y RY T 1 X<Y both ith robability 1 P f resectively, here R is the random reduction factor, T RY is the time it takes to recover from the injury, P f is the robability that the accident is fatal given that it occurs, and 1 X<Y is the indicator function for the event X < Y. Without deeer investigations it is assumed that R is uncorrelated ith both max{x, Y } and Y (not excluding stochastic deendence due to a ossible nonlinear relation beteen R and Y ) and that X and Y are indeendent. Then the exectations become, (Ditlevsen 2003), E[L] = P f 1 κ 0 (1 e κt ) f Y (t) dt + (1 P f )E[Y ] (15) E[L h ] = re[l] (1 P f )E[T ]E[1 X<Y ] = (r + dr)e (16) here r = E[R], E = E[L] and here T is assumed to be indeendent of X and Y. The last equality is according to the definition of the mean life in good health. It follos from (15) that E[Y ] E[L] 1 P κe[l] 2 f 2 (1 + V Y 2 ) or de E P 1 f 2 (1 + V Y 2 ) κ E (17) as κ 0. V Y ( 0.2) is the coefficient of variation of Y. This is a simle modification of the result in (Ditlevsen 2003) corresonding to P f = 1 and roved in the same ay. Moreover it follos from (16) that dr = (1 P f )κe[t ] (18) using that E[1 X<Y ] = P(X < Y ) = κ E, (Ditlevsen 2003). 8. Distribution on accident categories The results (17) for de and (18) for dr are for a single accident category. Assume that it is ossible to categorize all essential accidents into n different categories and that the oulation of size N can be assigned to these categories so that N i ersons are assigned to category i = 1,..., n. By assumtion 8

9 N N n = N since no erson can belong to to or more categories. The total -allocation er time unit to accidents is then N d = N 1 ( d) N n ( d) n here ( d) i = r [ κ i c(1) 2 (1 + V Y 2 ) E min P fi + c(1) + 1 ( )] r0 c(1) log E[T i ] min (1 P fi ) (19) r 0 min c(1) according to (10), (17) and (18). Since κ i is the mean fraction er time unit of the oulation associated to category i that suffers from an accident, each erson associated to the ith category gets allocated a time loss equal to the right side of (19). 9. Examle: Ferry on fire This examle illustrates the socio-economic assessment of life and limb by alication to a risk analysis of a fire on a RoRo ferry carrying 150 assengers inclusive cre. Although fires reresent a serious hazard for shis, many studies have shon that fire is the second largest hazards for cre and assengers on shis. Foundering due to collision, grounding or hull structural failure is generally considered being more dominant hazards. Nonetheless, for illustrative uroses the focus is here on assessing the number of injuries, the related duration of injury, and the number of fatalities. The most critical hase, as regards fatalities and injuries of cre and assengers, is not measured directly after the occurrence of the incident. In the hase that follos, the catain may try to revent the incident from evolving into a serious accident by, for examle, intentionally beaching a shi that takes in ater and thereby try to kee it from sinking. Hoever, if such risk mitigating measures fail, evacuation rovides the last oortunity to minimize injuries and fatalities. When an evacuation is initiated the evacuation erformance is very imortant. Clearly an orderly and timely evacuation is crucial. To model the critical situation, a Bayesian Netork construct of the fire scenario is very useful. The netork constructed for this examle considers initiation of fires in the accommodation areas and in the engine room. Also the escalation of the fires is included as a model element. Simulated evacuation times reorted in (Vanem & Skjong 2006) are used. The number of fatalities in the fire scenario is assessed in deendence of the available evacuation time. Similarly, the robability of a erson being injured during the evacuation is modeled artly as a function of the seriousness of the incident and of the available evacuation time. The seriousness of the occurring injuries in deendence of the available evacuation time is assessed on a subjective basis although suorted by the findings in (Hansen & Vinter 2003). Results: The annual frequency of fire in Scandinavian aters is estimated to κ = y 1. Given the occurrence of a fire estimates of the robability of a fatal accident and of injury is P f = and P i = 0.214, resectively. Thus, the robability is that the fire ocurrence does not harm an arbitrary assenger or cre member. Conditional on injury the exected recovery time is estimated to E[T injury] = 0.27 y. Hence, E[T ] = ( )/( ) = y = 22 days. Inserting these values into (19) and using c = 0.084, min = 1.81, r 0 = 0.95, E = 80 y, and V Y = 0.2, the to terms in (19) become d f /κ = 6.68 y and d i /κ = 1.34 y. Multilying these by S = G D P/ min = 33, 340/1.81 = 18, 420 /y their monetary equivalents are mill and mill, resectively. Thus the societal cost of an injury is about 20% of the cost of a fatality. For N = 150 assengers the total exected socio-economic loss of life and limb er fire becomes 150 ( ) = 22.2 mill. The oner loss exectations are estimated to / fatality and / injury. The exected oner loss due to fatalities and injuries caused by a fire occurrence is thus µ o = ( ) 150 = mill. The total annual gain before tax for the RoRo-ferry is mill / y of hich 30% is allocated to cover the fire risk. The to sides of the accetance inequality (13) then become left side: 1.2 ( = , right side: ) ρ ρ=0.3 = (20) Thus the ublic accetance criterion (13) is satisfied. For ρ = 1 the criterion is the so-called LQI (or LQTAI) criterion hich is less restrictive than if ρ < 1. For ρ = 1 the right side of (13) becomes

10 Conclusions A recaitulation is given of the reviously ublished dimensionless analysis of the equilibrium relation beteen the gross domestic roduct, the salary, and the ork time all on average over the entire oulation of a macro economical homogeneous geograhical region. The analysis suggests to measure all monetary quantities in the unit of time and thereby to eliminate the influence of inflation, the currency unit, and the local urchase oer. The theoretically obtained ork and leisure time distributing equilibrium rincile has an emirical back-u that makes it reasonable also to use this equilibrium rincile to define a normative rule for allocating time to life and limb saving measures in risky activities. The allocation rule is generated from invariance of the life quality time allocation index (LQTAI) derived conveniently from the dimensionless analysis behind the equilibrium rincile. The insiration to this construct comes decisively from a study of the literature on the life quality index (LQI) mainly authored by Nathani, Lind, Pandey, and Rackitz, and a concern to get a deeer understanding of hat the basis is for the LQI. The result of the study is a more general mathematical formulation than that of the LQI (in different versions). If the macro economy including the ork time fraction is long time stationary the to indices are almost identical even though their background hilosohies are strongly different. One imortant difference is that the LQTAI is defined so that it is alicable also for a non-stationary macro economy ith varying ork time ratio. A result of the more general formulation is that not only a rincile is obtained for socio-econnomical allocation of time (i.e. money) to fatality mitigation, but also for allocation of reasonable and ethically defendable time for injury mitigation. The concluding examle ith a ferry in fire illustrates the fact that there is a large difference beteen the damage comensation required from the oner of a risky oeration and the societal value of life and limb. This difference motivates the formulation of a ublic accetance rule (reviously ublished) hich is considerably more restrictive than the LQI (or LQTAI) criterion hich states that the gain er time unit must be at least as large as the societal loss value er time unit as obtained by the rincile of LQI invariance. Acknoledgment The first author thanks Otto Mønsteds Fond for conference travel suort. References Ditlevsen, O. (2003). Decision modeling and accetance criteria. Structural Safety 25: Ditlevsen, O. (2004). Life quality index revisited. Structural Safety 26: Ditlevsen, O. (2007). Model of observed stochastic balance beteen ork and free time suorting the LQTAI definition. Structural Safety. In ress. Prerint on htt://.mek.dtu.dk/staff/od/aers.htm until ublication. Ditlevsen, O. & P. Friis-Hansen (2005). Life quality allocation index an equilibrium economy consistent version of the current life quality index. Structural Safety 27: Ditlevsen, O. & P. Friis-Hansen (2007). Life quality index an emirical or a normative concet? International Journal of Risk Assessment and Management, in ress. Friis-Hansen, P. & O. Ditlevsen (2003). Nature reservation accetance model alied to tanker oil sill simulations. Structural Safety 25:1 34. Hansen, H. L. & M. Vinter (2003). Occuational accidents and shi design: Imlications for revention. In World Maritime Conference, San Francisco. Nathani, J. S., N. C. Lind, & M. D. Pandey (1997). Affordable Safety By Choice: The Life Quality Method. Waterloo, Ontario, Canada: Institute for risk Research, University of Waterloo. OECD, S. (2004). OECD Economic Outlook Database, ne.sourceoecd.org/database/oecdeconomicoutlook. Pandey, M. D., J. S. Nathani, & N. C. Lind (2006). The derivation and calibration of the life quality index (LQI) from economical rinciles. Structural Safety 28: Rackitz, R. (2002). Otimization and risk accetability based on the life quality index. Structural Safety 24: Rackitz, R., A. Lentz, & M. Faber (2005). Socio-economically sustainable civil engineering infrastructures by otimization. Structural Safety 27: Vanem, E. & R. Skjong (2006). Designing for safety in assenger shis utilizing advanced evacuation analysis - a risk based aroach. Safety Science 44: