Journée de la Chaire Santé. Valuing Life as an Asset, as a Statistic and at Gunpoint

Transcription

1 Journée de la Chaire Santé Valuing Life as an Asset, as a Statistic and at Gunpoint Julien Hugonnier 1 Florian Pelgrin 2 Pascal St-Amour 3 1 École Polytechnique Fédérale de Lausanne, CEPR and Swiss Finance Institute 2 EDHEC Business School, CIRANO and Institut Louis Bachelier 3 University of Lausanne, Faculty of Business and Economics (HEC) and Swiss Finance Institute Université Paris-Dauphine, March 30, /27

2 1. Introduction Motivation and outline Different valuation methods to evaluate the price of human life Human capital life value : Prejudice caused to society by the death/injury of an individual (occupational, end-users wrongful death litigation) Present value of the net cash flow associated with human capital (asset pricing view) v j h,t = E T m ( ) s 1 t D t+s 1 + r s=0 where D t+s denotes the net dividend at time t + s marketed labor income minus all expenses to maintain human capital. 2/27

3 Value of a statistical life : Based on individual Willingness-To-Pay (WTP) to avoid small increases in exposure to death risk Aggregation of individual WTP Collective WTP to save one unidentified (i.e. statistical) life. Example : Suppose a population of size n and a change = 1/n in death risk exposure. All agents are individually willing to pay v i ( ) = The empirical VSL is the collective WTP : 1000 v s = v i ( ) = v i = 1MM$. i=1 On the other hand, the theoretical VSL is the negative of the MRS between the probability of death and wealth/the marginal willingness-to-pay and is not observable! 3/27

4 Table : Comparison HK value and VSL (in $) Average HK life value Average VSL Poor Fair Good Very Good Excellent Mean Median Empirical literature [300, 900]K$ [4.2, 13.7]M$ [Huggett and Kaplan, 2016] [Robinson and Hammitt, 2016] 4/27

5 Table : Comparison HK value and VSL (in $) Average HK life value Average VSL Poor Fair Good Very Good Excellent Mean Median Empirical literature [300, 900]K$ [4.2, 13.7]M$ [Huggett and Kaplan, 2016] [Robinson and Hammitt, 2016] VSL is times larger than the HK value of life!...how can we explain and assess this large discrepancy of valuation methods? 4/27

6 Main research questions and contributions 1 Can we provide a reasonable metric for the value of life against which the two alternatives can be gauged? 5/27

7 Main research questions and contributions 1 Can we provide a reasonable metric for the value of life against which the two alternatives can be gauged? Define third life value as benchmark : Gunpoint Value (GPV) overview 5/27

8 Main research questions and contributions 1 Can we provide a reasonable metric for the value of life against which the two alternatives can be gauged? Define third life value as benchmark : Gunpoint Value (GPV) overview 2 Can a common theoretical and empirical framework help in rationalizing the differences between the HK value and the VSL? 5/27

9 Main research questions and contributions 1 Can we provide a reasonable metric for the value of life against which the two alternatives can be gauged? Define third life value as benchmark : Gunpoint Value (GPV) overview 2 Can a common theoretical and empirical framework help in rationalizing the differences between the HK value and the VSL? Provide common theoretical framework for HK, WTP, GPV and VSL. 5/27

10 Main research questions and contributions 1 Can we provide a reasonable metric for the value of life against which the two alternatives can be gauged? Define third life value as benchmark : Gunpoint Value (GPV) overview 2 Can a common theoretical and empirical framework help in rationalizing the differences between the HK value and the VSL? Provide common theoretical framework for HK, WTP, GPV and VSL. Closed-form solutions for HK, WTP, VSL and GPV values of life to evaluate : Role of preferences, technological, distributional parameters. Role of wealth, human capital Shape of WTP. Aggregation issues. 5/27

11 Main research questions and contributions 1 Can we provide a reasonable metric for the value of life against which the two alternatives can be gauged? Define third life value as benchmark : Gunpoint Value (GPV) overview 2 Can a common theoretical and empirical framework help in rationalizing the differences between the HK value and the VSL? Provide common theoretical framework for HK, WTP, GPV and VSL. Closed-form solutions for HK, WTP, VSL and GPV values of life to evaluate : Role of preferences, technological, distributional parameters. Role of wealth, human capital Shape of WTP. Aggregation issues. Structurally estimate WTP, three values with common data set. 5/27

12 3 What lessons can we learn about the interpretation and applicability of the alternative measures in pricing the economic value of a human life? overview 6/27

13 3 What lessons can we learn about the interpretation and applicability of the alternative measures in pricing the economic value of a human life? overview HK and GPV directly compute the value of a whole life, rather than using a linear extrapolation to obtain a unitary life value (VSL) ; 6/27

14 3 What lessons can we learn about the interpretation and applicability of the alternative measures in pricing the economic value of a human life? overview HK and GPV directly compute the value of a whole life, rather than using a linear extrapolation to obtain a unitary life value (VSL) ; VSL should not be interpreted as the value of a given human being (Schelling, 1968) rather a local measure of a rate of substitution between wealth and life ; 6/27

15 3 What lessons can we learn about the interpretation and applicability of the alternative measures in pricing the economic value of a human life? overview HK and GPV directly compute the value of a whole life, rather than using a linear extrapolation to obtain a unitary life value (VSL) ; VSL should not be interpreted as the value of a given human being (Schelling, 1968) rather a local measure of a rate of substitution between wealth and life ; VSL is appropriate when computing a collective value on small indiscriminate reductions on mortality for which society will ultimately end up paying the costs (e.g., public s safety) ; 6/27

16 3 What lessons can we learn about the interpretation and applicability of the alternative measures in pricing the economic value of a human life? overview HK and GPV directly compute the value of a whole life, rather than using a linear extrapolation to obtain a unitary life value (VSL) ; VSL should not be interpreted as the value of a given human being (Schelling, 1968) rather a local measure of a rate of substitution between wealth and life ; VSL is appropriate when computing a collective value on small indiscriminate reductions on mortality for which society will ultimately end up paying the costs (e.g., public s safety) ; HK and GPV appear the better alternatives for wrongful death litigation or curative vs terminal care decisions. 6/27

17 Road map 1. Introduction 2. A common framework for life valuation 3. Values of life 4. Structural estimation 5. Discussion 6. Conclusion 7/27

18 2. A common framework for life valuation Economic environment Combine a Grossman-based model with a standard Merton porftolio model 8/27

19 2. A common framework for life valuation Economic environment Combine a Grossman-based model with a standard Merton porftolio model Planning horizon is limited by a stochastic age at death T m : lim h 0 Pr [T m (t, t + h] T m > t] = λ m such that the probability of death by age t (death risk exposure) is monotone increasing in λ m : P(t) = Pr (T m t) = 1 exp ( λ m t) 8/27

20 2. A common framework for life valuation Economic environment Combine a Grossman-based model with a standard Merton porftolio model Planning horizon is limited by a stochastic age at death T m : lim h 0 Pr [T m (t, t + h] T m > t] = λ m such that the probability of death by age t (death risk exposure) is monotone increasing in λ m : P(t) = Pr (T m t) = 1 exp ( λ m t) Changes in death risk exposure P changes in the instantaneous death intensity λ m 8/27

21 Law of motion H t dh t = [ I α t H 1 α t δh t ] dt φht dq st where dq st is a Poisson depreciation (morbidity) shock with constant intensity λ s0 that further depreciates the health stock by φ (0, 1). 9/27

22 Law of motion H t dh t = [ I α t H 1 α t δh t ] dt φht dq st where dq st is a Poisson depreciation (morbidity) shock with constant intensity λ s0 that further depreciates the health stock by φ (0, 1). Budget constraint and income : Individuals can trade in two risky assets to smooth out shocks to consumption stock and insurance against health depreciation dw t = [rw t + Y t c t I t ] dt + π t σ S [dz t + θdt] Y t = y + βh t, +x t [dq st λ s0 dt], where π t denotes the risky portfolio and x t the units of an actuarially-fair insurance. 9/27

23 Preferences Stochastic Differential Utility (Duffie and Epstein, 1992) : Disentangle risk aversion γ from intertemporal elasticity of substitution ε ; Minimum subsistence consumption a ; Preference for life over death ; V m 0 ; Tm U t = E t (f (c τ, U τ ) γ σ τ (U) 2 ) dτ, 2U τ t where the age of death T m is the first occurrence of a Poisson process with constant intensity λ m and the Kreps-Porteus aggregator is : ( f (c t, U t ) = ρ U (ct ) 1 1 ) ε t a /ε U t 10/27

24 Optimal allocation V, c, I, π, x Theorem Optima closed-form allocations are given by : c t = a + A(λ m )N(W t, H t ) π t = θ N(W t, H t ) γσ S x t = φp(h t ) ( ) I t = α 1 α 1 α B 1 α P(H t ) V t (W t, H t, λ m ) = Θ(λ m )N(W t, H t ) N(W, H) = }{{} W + (y a)/r + P(H) }{{}}{{} Fin. Wealth NPV of fixed inc. stream Shadow value = BH Marginal value of N : Θ(λ m ) = ρa(λ m0 ) 1 1 ε 0 MPC : A(λ m ) = ερ + (1 ε) ( r λ m + 0.5θ 2 /γ ) 0 11/27

25 3. Values of life Human capital value of life Proposition The HK value of life v h,t = v h (W t, H t, P 0 ) is the expected discounted present value over stochastic horizon T m of labor revenue flows, net of investment costs, T m v h,t = E t 0 = E t T 0 m m t,τ [Y (H τ ) I τ ] dτ m t,τ [y + (βh τ I τ )] dτ where m t,τ = m τ /m t with m t = exp ( rt θz t 0.5θ 2 t ), and writes v h (H, λ m ) = C 0 y r + C 1P(H) with C 0 = r r + λ m and C 1 = r (αb) α 1 α r + λ m (αb) α 1 α. 12/27

26 Willingness to pay Definition The willingness to pay v = v(w, H, P 0, ) to avoid a permanent change [P 0, 1 P 0 ] in death risk exposure P solves V (W v, H, P 0 ) = V (W, H, P 0 + ). > 0 : Indifference between paying the equivalent variation v > 0 at base risk and not paying but facing higher death risk < 0 : Indifference between receiving compensation v > 0 and foregoing lower death risk exposure. 13/27

27 Proposition The willingness to pay to avoid an admissible change A m is : [ ] v(w, H, λ m, ) = 1 Θ(λ m) N(W, H) Θ(λ m ) an increasing and concave function of that is bounded by : [ inf v(w, H, λ m, ) = 1 Θ(0) ] N(W, H) A m Θ(λ m ) sup v(w, H, λ m, ) = N(W, H) A m with λ m = λ m + δ. 14/27

28 Value of a statistical life Proposition The value of a statistical life v s = v s (W, H, P 0 ) is the negative of the MRS between the probability of death and wealth computed from the indirect utility evaluated at base risk P 0 : v s = V P(W, H, P) V W (W, H, P). P=P0 and is given by v s (W, H, λ m ) = 1 N(W, H) A(λ m ) where A(λ m ) is the MPC and N the net total wealth. Equivalently, the VSL is also the marginal willingness to pay : v s (W, H, P 0 ) = v(w, H, P 0, ) v(w, H, P 0, ) = lim. =0 0 15/27

29 Theoretical VSL vs Empirical VSL Definition The empirical value of a statistical life, vs e by : = v e s (W, H, P 0, ) is given v e s (W, H, P 0, ) = v (W, H, P 0, ) for small increment = 1/n where n is the size of the population considered. As 0, v e s (W, H, P 0, ) v s (W, H, P 0 ) ; The bias v e s v s depends on the curvature of the WTP and. 16/27

30 Gunpoint value of life Proposition The gunpoint value v g = v g (W, H, P 0 ) is the WTP to avoid certain, instantaneous death and it solves : V (W v g, H, P 0 ) = V m where V m is the utility at death, and is given by v g (W, H) = N(W, H) W + y a r + BH. Unless y/r is large, v g (W, H) v h (W, H, λ m ) 0 ; v g (W, H) = A(λ m )v s (W, H, λ m ) < v s (W, H, λ m ) ; g(c t a) = g(v s,t ) = g(v g,t ). 17/27

31 To summarize return 18/27

32 4. Structural estimation Econometric model where Y j = B(θ)X j + u j Y j = [c j, π j, x j, I j, Y j ] X j = [1, W j, H j ] Data : PSID 2013 Health : Poor to Excellent using self-reported status (household head). Financial wealth = risky (stocks in publicly held corporations, mutual funds, investment trusts, private annuities, IRA s or pension plans) plus riskless assets (checking accounts plus bonds plus remaining IRA s and pension). 19/27

33 Estimation of structural parameters Parameter Value Parameter Value a. Law of motion health α δ (0.3720) (0.0060) φ c b. Sickness and death intensities λ s λ m (0.0108) (0.0089) d. Preferences γ ε (1.4497) (0.3724) a c ρ c 20/27

34 Value of Statistical Life vs HK Value Wealth quintile level Health level Poor Fair Good Very Good Excellent a. Value of Statistical Life v s All - mean median b. Human Capital Value of Life v h All - mean median /27

35 Gunpoint Value of Life vs HK Value Wealth quintile level Health level Poor Fair Good Very Good Excellent a. Gunpoint Value of Life v g All - mean median b. Human Capital Value of Life v h All - mean median /27

36 5. Discussion Explaining the discrepancy Disjoint theoretical and empirical frameworks? 23/27

37 5. Discussion Explaining the discrepancy Disjoint theoretical and empirical frameworks? Answer : No! 23/27

38 5. Discussion Explaining the discrepancy Disjoint theoretical and empirical frameworks? Answer : No! Collective WTP vs individual WTP? 23/27

39 5. Discussion Explaining the discrepancy Disjoint theoretical and empirical frameworks? Answer : No! Collective WTP vs individual WTP? Answer : The individual MWTP (theoretical VSL) is well approximated by the collective WTP (empirical VSL) 23/27

40 5. Discussion Explaining the discrepancy Disjoint theoretical and empirical frameworks? Answer : No! Collective WTP vs individual WTP? Answer : The individual MWTP (theoretical VSL) is well approximated by the collective WTP (empirical VSL) Diminishing MWTP? 23/27

41 5. Discussion Explaining the discrepancy Disjoint theoretical and empirical frameworks? Answer : No! Collective WTP vs individual WTP? Answer : The individual MWTP (theoretical VSL) is well approximated by the collective WTP (empirical VSL) Diminishing MWTP? Answer : Yes! Strongly diminishing MWTP does explain that VSL is much larger than HK value and GPV of life. 23/27

42 5. Discussion Explaining the discrepancy Disjoint theoretical and empirical frameworks? Answer : No! Collective WTP vs individual WTP? Answer : The individual MWTP (theoretical VSL) is well approximated by the collective WTP (empirical VSL) Diminishing MWTP? Answer : Yes! Strongly diminishing MWTP does explain that VSL is much larger than HK value and GPV of life. 23/27

43 Back to basics? VSL should not be interpreted as the value of a given human being (Schelling, 1968) 24/27

44 Back to basics? VSL should not be interpreted as the value of a given human being (Schelling, 1968) Rather a local measure of a rate of substitution between wealth and life 24/27

45 Back to basics? VSL should not be interpreted as the value of a given human being (Schelling, 1968) Rather a local measure of a rate of substitution between wealth and life Empirical VSL measures adequately an individual MWTP when changes are small 24/27

46 Back to basics? VSL should not be interpreted as the value of a given human being (Schelling, 1968) Rather a local measure of a rate of substitution between wealth and life Empirical VSL measures adequately an individual MWTP when changes are small VSL is appropriate when computing a collective value on small indiscriminate reductions on mortality for which society will ultimately end up paying the costs (e.g., public s safety). 24/27

47 Back to basics? VSL should not be interpreted as the value of a given human being (Schelling, 1968) Rather a local measure of a rate of substitution between wealth and life Empirical VSL measures adequately an individual MWTP when changes are small VSL is appropriate when computing a collective value on small indiscriminate reductions on mortality for which society will ultimately end up paying the costs (e.g., public s safety). HK and GPV appear the better alternatives for wrongful death litigation or curative vs terminal care decisions. 24/27

48 Extensions Endogenous mortality and morbidity 1 λ m (H t ) = lim τ 0 τ P t [t < T m t + τ] = λ m0 + λ m1 H ξm t λ s (H t ) = η + λ s0 η 1 + λ s1 H ξs t Ageing : Time-varying parameters λ m,t, λ s,t, φ t, δ t or β t. SHARE data Immortal Life Value : WTA a compensation to renounce to perpetual life Results remain applicable and are robust. 25/27

49 6. Conclusion Findings Questions HK VSL GPV Theoretical links? - Common framework Dynamic human capital model - Willingness to pay Incr. concave, bounded - Life valuations ENPV(Div.) MWTP Limiting WTP ENPV(excess cons.) Role of primitives? - Technological - Depreciation risk - Mortality risk x - Preferences x x Robust. of reduced-form findings? Yes, VSL HK GPV - Struct. est. life values 420 K$ 8.35 M$ 447 K$ Reasons for differences? - Different model, data? No - Model specific? No - Assumptions? Yes, curvature of WTP 26/27

50 Overview of Gunpoint Value Hicksian Equivalent Variation (EV) : (Maximal) willingness to pay (WTP) to avoid unfavorable event (death). Highwaymen question : What is the amount you would be willing to pay in order to survive in a credible your money or your life highwayman threat or, equivalently, how much would you value your own life? Gunpoint Value of Life (GPV), i.e. the equivalent variation that leaves the agent indifferent between remaining alive and certain death. Return 27/27