Specifications of Social Welfare in Economic Studies of Climate Policy: Overview of Criteria and Related Policy Insights

Environmental and Resource Economics

Volumn 58, Issue 1, 2014, Pages 1-33

  Botzen, W J Wouter ^a   van den Bergh, Jeroen C J M ^a,b,c  

^a VU UNIVERSITY AMSTERDAM   (Netherlands)

^b INSTITUCIÓ CATALANA DE RECERCA I ESTUDIS AVANÇATS ICREA   (Spain)

^c INSTITUT DE CIÈNCIA I TECNOLOGIA AMBIENTALS   (Spain)

Author keywords

Climate change; Cost benefit analysis; Discounting; Integrated assessment modelling; Mitigation; Social welfare function; Uncertainty

Indexed keywords

CLIMATE CHANGE; COST BENEFIT ANALYSIS; OPTIMIZATION; UNCERTAINTY ANALYSIS;

DISCOUNTING; INTEGRATED ASSESSMENT MODELLING; MITIGATION; SOCIAL WELFARE FUNCTIONS; UNCERTAINTY;

ECONOMIC ANALYSIS;

CLIMATE CHANGE; CLIMATE MODELING; COST-BENEFIT ANALYSIS; ENVIRONMENTAL POLICY; INTEGRATED APPROACH; MITIGATION; UNCERTAINTY ANALYSIS; WELFARE IMPACT; WELFARE REFORM;

EID: 84897963334     PISSN: 09246460     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10640-013-9738-8     Document Type: Review

References (165)

1. Ackerman F, DeCanio SJ, Howarth RB, Sheeran K (2009) Limitations of integrated assessment models of climate change. Clim Change 95: 297-315.
2. Ackerman F, Stanton EA, Bueno R (2013) Epstein-Zin utility in DICE: Is risk aversion irrelevant to climate policy? Environ Resour Econ 56: 73-84.
3. Ackerman F, Stanton EA (2012) Climate risks and carbon prices: revising the social cost of carbon. Economics: The Open Access. Open-Assessment E-Journal 6: 1-25.
4. Anthoff D, Tol RSJ (2010) On international equity weights and national decision making on climate change. J Environ Econ Manag 60: 14-20.
5. Anthoff D, Rose S, Tol RSJ, Waldhoff S (2011) The time evolution of the social cost of carbon: An application of FUND. Economics: The Open-Access. Open-Assessment E-Journal 44: 1-21.
6. Anthoff D, Tol RSJ (2013) Climate policy under fat-tailed risk: An application of FUND. Ann Oper Res.
7. Arrow KJ (1951) Social choice and individual values. Yale University Press, Yale.
8. Arrow KJ (1965) Aspects of the theory of risk-bearing. Yrjö Jahnsson's Lectures, Helsinki.
9. Arrow KJ, Hurwicz L (1972) An optimality criterion for decision-making under ignorance. In: Carter CF, Ford JL (eds) Uncertainty and expectations in economics: Essays in honour of G. L. S. Shackle. Basil Blackwell, Oxford.
10. Arrow KJ (1973) Rawls's principle of just saving. Swed J Econ 75: 323-335.
11. Arrow KJ, Fisher AC (1974) Environmental preservation, uncertainty, and irreversibility. Q J Econ 88(2): 312-319.
12. Arrow KJ, Cropper M, Gollier C, Groom B, Heal G, Newell R, Nordhaus W, Pindyck R, Pizer W, Portney P, Sterner T, Tol RSJ, Weitzman M (2013) Determining benefits and costs for future generations. Science 341: 349-350.
13. Asgary A, Levy J (2009) A review of the implications of prospect theory of natural hazards and disaster planning. Int J Environ Res 3(3): 379-394.
14. Atkinson G, Mourato S (2008) Environmental cost-benefit analysis. Ann Rev Environ Resour 33: 317-344.
15. 2 emissions really optimal? Environ Resour Econ 11(3-4): 301-315.
16. Barbera S, Jackson M (1988) Maximin, leximin, and the protective criterion: characterizations and comparisons. J Econ Theory 46: 34-44.
17. Binmore K (2009) Rational decisions. Princeton University Press, Princeton.
18. Birge JR, Rosa CH (1996) Incorporating investment uncertainty into greenhouse policy models. Energy J 17(1): 79-90.
19. Botzen WJW, van den Bergh JCJM (2009) Bounded rationality, climate risks and insurance: is there a market for natural disasters? Land Econ 85(2): 266-279.
20. Botzen WJW, van den Bergh JCJM (2012) How sensitive is Nordhaus to Weitzman? Climate policy in DICE with an alternative damage function. Econ Lett 117: 372-374.
21. Broome J (1992) Counting the cost of global warming. White Horse Press, Cambridge.
22. De Bruin KC, Dellink RB, Tol RSJ (2009) AD-DICE: An implementation of adaptation in the DICE model. Clim Change 95: 63-81.
23. Bruni L, Porta PL (Eds) (2005) Economics and Happiness. Oxford University Press, Oxford.
24. Chichilnisky G, Heal G, Beltratti A (1995) The green golden rule. Econ Lett 49: 175-179.
25. Chichilnisky G (1996) An axiomatic approach to sustainable development. Social Choice Welf 13(2): 219-248.
26. Chichilnisky G (2000) An axiomatic approach to choice under uncertainty with catastrophic risks. Environ Res Econ 22: 221-231.
27. Cline WR (1992) The economics of global warming. Institute for International Economics, Washington.
28. Cohen MA, Vandenbergh MP (2008) Consumption, happiness, and climate change. RFF Discussion Paper No. 08-39.
29. Crost B, Traeger C (2012) Risk and aversion in integrated assessment of climate change. SSRN Working paper 1 December 2012. http://ssrn. com/abstract=2270482 or doi: 10. 2139/ssrn. 2270482.
30. Daly HE (1992) Steady-state economics: Second edition with new essays. Earthscan, London.
31. Daly HE, Costanza R (1992) Natural capital and sustainable development. Conserv Biol 6(1): 37-46.
32. Dasgupta P (1974) On some alternative criteria for justice between generations. J Publ Econ 3(4): 405-423.
33. Dasgupta P, Heal G (1979) Economic theory and exhaustible resources. Cambridge University Press, Cambridge.
34. Dasgupta P (2008) Discounting climate change. J Risk Uncertain 37: 141-169.
35. Dasgupta P (2011) The ethics of intergenerational distribution: reply and response to John E. Roemer. Environ Resour Econ 50: 475-493.
36. den Elzen MGJ, Lucas PL, van Vuuren DP (2008) Regional abatement action and costs under allocation schemes for emission allowances for achieving low CO2-equivalent concentrations. Clim Change 90: 243-268.
37. Diamond P (1965) The evaluation of infinite utility streams. Econometrica 33: 170-177.
38. Diamond PA (1967) Individualistic ethics, and interpersonal comparison of utility: comment. J Pol Econ 75(5): 765-766.
39. Dietz S (2012) The treatment of risk and uncertainty in the US Social Cost of Carbon for regulatory impact analysis. Economics: The Open Access, Open-Assessment E-Journal 6(2012-18): 1-12.
40. Dietz S, Asheim GB (2012) Climate policy under sustainable discounted utilitarianism. J Environ Econ Manag 63: 321-335.
41. Dixit AK, Pindyck RS (1994) Investment under uncertainty. Princeton University Press, Princeton.
42. Edenhofer O, Hirth L, Knopf B, Pahle M, Schlömer, Schmid S, Ueckerdt F (2013) On the economics of renewable energy resources. Energy Econ. doi: 10. 1016/j. eneco. 2013. 09. 015.
43. Ellsberg D (1961) Risk, ambiguity, and the Savage axioms. Q J Econ 75(4): 643-669.
44. Epstein LG, Zin SE (1989) Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework. Econometrica 57(4): 937-969.
45. Epstein LG, Zin SE (1991) Substitution, risk aversion, and the temporal behavior of consumption and asset returns: An empirical analysis. J Pol Econ 99(2): 263-286.
46. Estrada F, Gay C, Conde C (2012) A methodology for the risk assessment of climate variability and change under uncertainty. A case study: coffee production in Veracruz, Mexico. Clim Change 113(2): 455-479.
47. Etchart-Vincent N (2009) Probability weighting and the 'level' and 'spacing' of outcomes: An experimental study over losses. J Risk Uncertain 39: 45-63.
48. Fankhauser S (1995) Valuing climate change: The economics of the greenhouse. EarthScan, London.
49. Fankhauser S, Tol RSJ, Pearce DW (1997) The aggregation of climate change damages: A welfare theoretic approach. Environ Resour Econ 10: 249-266.
50. Fiddaman TS (2002) Exploring policy options with a behavioral climate-economy model. Syst Dyn Rev 18(2): 243-267.
51. Fisher AC (2000) Investment under uncertainty and option value in environmental economics. Resour Energy Econ 22(3): 197-204.
52. Fisher AC, Narain U (2003) Global warming, endogenous risk, and irreversibility. Environ Resour Econ 25: 395-416.
53. Froyn CB (2005) Decision criteria, scientific uncertainty, and the global warming controversy. Mitigation and adaptation strategies for climate change 10: 183-211.
54. Füssel H-M (2013) Social welfare functions in optimizing climate-economy models: A critical review. Working paper. Potsdam Institute for Climate Impact Research (PIK), Potsdam. http://www. hcs. harvard. edu/~hejc/papers/review_swf_fuessel_prot. pdf.
55. Gerlagh R (1998) The efficient and sustainable use of environmental resource systems. VU University, Amsterdam, Doctoral Dissertation.
56. Gilboa I, Schmeidler D (1989) Maxmin expected utility with non-unique prior. J Math Econ 18: 141-153.
57. Gilboa I, Postlewaite A, Schmeidler D (2009) Is it always rational to satisfy Savage's axioms? Econ Philos 25(3): 285-296.
58. Gollier C, Jullien B, Treich N (2000) Scientific progress and irreversibility: An economic interpretation of the precautionary principle. J Publ Econ 75: 229-253.
59. Gollier C (2010) Ecological discounting. J Econ Theory 145: 812-829.
60. Ha-Duong M (1998) Quasi-option value and climate policy choices. Energy Econ 20(5-6): 599-620.
61. Harsanyi J (1955) Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility. J Pol Econ 63(4): 309-321.
62. Hartwick JM (1977) Intergenerational equity and the investing of rents from exhaustible resources. Am Econ Rev 67(5): 972-974.
63. Heal GM (1997) Discounting and climate change: An editorial comment. Clim Change 37: 335-343.
64. Heal G, Kriström B (2002) Uncertainty and climate change. Environ Resour Econ 22(1-2): 3-39.
65. Henry C, Henry M (2002) Formalization and application of the precautionary principle. http://www. columbia. edu/cu/economics/discpapr/DP0102-22. pdf. Columbia University Department of Economics Discussion Paper Series.
66. Hoel M, Sterner T (2007) Discounting and relative prices. Clim Change 84: 265-280.
67. Hof AF, van Vuuren DP, den Elzen MGJ (2010) A quantitative minimax regret approach to climate change: Does discounting still matter? Ecol Econ 70(1): 43-51.
68. Horowitz J, Lange A (2009) What's wrong with infinity note on Weitzman's dismal theorem. Working Paper University of Maryland. http://faculty. arec. umd. edu/jhorowitz/weitzman_final. pdf.
69. Howarth RB (2001b) Climate rights and economic modelling. In: Howarth RB, Hall DC, Shogran JF (eds). The long-term economics of climate change: Beyond a doubling of greenhouse gas concentrations. Book series: Advances in the economics of environmental resources. Emerald Group Publishing Limited, Bingley.
70. Howarth RB (1992) Intergenerational justice and the chain of obligation. Environ Values 1: 133-140.
71. Howarth RB, Norgaard RB (1995) Intergenerational choices under global environmental change. In: Bromley D (ed) Handbook of environmental economics. Blackwell Publishers, Oxford, pp 111-138.
72. Howarth RB (1998) An overlapping generations model of climate-economy interactions. Scand J Econ 100: 575-591.
73. Howarth RB (2001a) Intertemporal social choice and climate stabilization. Int J Environ Pollut 15(4): 386-405.
74. Howarth RB (2003) Discounting and uncertainty in climate change policy analysis. Land Econ 79(3): 369-381.
75. Hurwicz L (1951) Optimality criteria for decision-making under ignorance. CowlesCommission Discussion Paper, Statistics No 370.
76. Interagency Working Group on Social Cost of Carbon (2010) Final Rule Technical Support Document (TSD): Energy efficiency program for commercial and industrial equipment: Small electric motors, Appendix 15A: Social Cost of Carbon for Regulatory Impact Analysis Under Executive Order 12866. U. S. Department of Energy, United States Government. http://www1. eere. energy. gov/buildings/appliance_standards/commercial/pdfs/smallmotors_tsd/sem_finalrule_appendix15a. pdf.
77. Interagency Working Group on Social Cost of Carbon (2013) Technical support document: Technical update of the social cost of carbon for regulatory impact analysis under executive order 12866. United States Government. http://www. whitehouse. gov/sites/default/files/omb/inforeg/social_cost_of_carbon_for_ria_2013_update. pdf.
78. IPCC (2007) Climate Change 2007: The physical science basis. In: Solomon S, Qin D, Manning M, Chen Z, Marquis M, Averyt K, Tignor MMB, Miller HL (eds). Contribution of working group I to the fourth assessment report of the intergovernmental panel on climate change. Cambridge University Press, Cambridge.
79. Iverson T (2012) Communicating trade-offs amid controversial science: decision support for climate policy. Ecol Econ 77: 74-90.
80. Iverson T, Perrings C (2012) Precaution and proportionality in the management of global environmental change. Global Environ Change 22: 161-177.
81. Kahneman D, Tversky A (1979) Prospect theory: An analysis of decision under risk. Econometrica 47(2): 263-291.
82. Kahneman D, Krueger AB (2006) Developments in the measurement of subjective well-being. J Econ Perspect 20(1): 3-24.
83. Karp L (2009) Sacrifice, discounting, and climate policy: Five questions. CESifo Working Paper Series No. 2761. Available at SSRN: http://ssrn. com/abstract=1458887.
84. Kaufman N (2012) The bias of integrated assessment models that ignore climate catastrophes. Clim Change 110(3): 575-595.
85. Kelsey D (1993) Choice under partial uncertainty. Int Econ Rev 34(2): 297-308.
86. Kimball MS (1990) Precautionary saving in the small and in the large. Econometrica 58: 53-73.
87. Klibanoff P, Marinacci M, Mukerji S (2005) A smooth model of decision making under ambiguity. Econometrica 73(6): 1849-1892.
88. Klibanoff P, Marinacci M, Mukerji S (2009) Recursive smooth ambiguity preferences. J Econ Theory 144(3): 930-976.
89. Knetsch JL (1990) Environmental policy implications of disparities between willingness to pay and compensation demanded measures of values. J Environ Econ Manag 18: 227-237.
90. Knight FH (1971) Risk, uncertainty, and profit. University of Chicago Press, Chicago.
91. 2 control in the face of uncertainty and learning. In: van Ierland EC (ed) International environmental economics. Elsevier Science Publishers, Amsterdam.
92. Kolstad CD (1996b) Learning and stock effects in environmental regulation: The case of greenhouse gas emissions. J Environ Econ Manag 31(1): 1-18.
93. Kolstad CD (1996a) Fundamental irreversibilities in stock externalities. J Publ Econ 60(2): 221-233.
94. Koopmans TC (1965) On the concept of optimal economic growth. In: Study week on the econometric approach to development planning, Pontificiae Academiae Scientiarum Scripta Varia, vol. 28. North Holland, Amsterdam, pp 225-266.
95. Koopmans T (1960) Stationary ordinal utility and impatience. Econometrica 28: 287-309.
96. Kopp RE, Mignone BK (2012) The U. S. government's social cost of carbon estimates after their first two years: pathways for improvement. Economics: The Open Access, Open-Assessment E-Journal, 6: 1-41.
97. Kousky C, Kopp RE, Cooke R (2011) Risk premia and the social cost of carbon: A review. Economics: The Open Access, Open-Assessment E-Journal 5(2011-21): 1-24.
98. Krysiak FC, Krysiak D (2006) Sustainability with uncertain future preferences. Environ Resour Econ 33(4): 511-531.
99. Krysiak FC (2009) Sustainability and its relation to efficiency under uncertainty. Econ Theory 41(2): 297-315.
100. Kunreuther H, Heal G, Allen M, Edenhofer O, Field CB, Yohe G (2013) Risk management and climate change. Nature Clim Change. doi: 10. 1038/NCLIMATE1740.
101. Lange A (2003) Climate change and the irreversibility effect: combining expected utility and MaxiMin. Environ Resour Econ 25: 417-434.
102. Lange A, Treich N (2008) Uncertainty, learning and ambiguity in economic models on climate policy: Some classical results and new directions. Clim Change 89(1): 7-21.
103. Laury SK, Morgen-McInnes M, Swarthout JT (2009) Insurance decisions for low-probability losses. J Risk Uncertain 39: 17-44.
104. Lind RC (1995) Intergenerational equity, discounting, and the role of cost-benefit analysis in evaluating global climate policy. Energy Policy 23: 379-389.
105. Llavador H, Roemer JE, Silvestre J (2010) Intergenerational justice when future worlds are uncertain. J Math Econ 46: 728-761.
106. Llavador H, Roemer JE, Silvestre J (2011) A dynamic analysis of human welfare in a warming planet. J Publ Econ 95(11): 1607-1620.
107. Loomes G, Sugden R (1982) Regret theory: An alternative theory of rational choice under uncertainty. Econ J 92(368): 805-824.
108. Loulou R, Kanudia A (1999) Minimax regret strategies for greenhouse gas abatement: methodology and application. Oper Res Lett 25: 219-230.
109. Maddison DJ (1995) A cost-benefit analysis of slowing climate change. Energy Policy 23(4/5): 337-346.
110. Manne A, Mendelsohn R, Richels R (1995) MERGE: A model for evaluating regional and global effects of GHG reduction policies. Energy Policy 23(1): 17-34.
111. Millner A (2013) On welfare frameworks and catastrophic climate risks. J Environ Econ Manag 65: 310-325.
112. Millner A, Dietz S, Heal G (2010) Ambiguity and climate policy. NBER Working Paper 16050. National Bureau of Economic Research, Cambridge.
113. Neumayer E (1999) Global warming: discounting is not the issue, but substitutability is. Energy Policy 27: 33-43.
114. Nordhaus WD (2007) A review of the stern review on the economics of climate change. J Econ Lit XLV: 686-702.
115. Nordhaus WD (1994) Managing the global commons. MIT Press, Cambridge.
116. Nordhaus WD, Yang Z (1996) RICE: A regional dynamic general equilibrium model of optimal climate-change policy. Am Econ Rev 86(4): 741-765.
117. Nordhaus WD, Boyer J (2000) Warming the world: Economic models of global warming. The MIT Press, Cambridge.
118. Nordhaus WD (2008) A question of balance: weighting the options of global warming policies. Yale University Press, New Haven.
119. Nordhaus WD (2011) The economics of tail events with an application to climate change. Rev Environ Econ Policy 5(2): 240-257.
120. Perrings C (1991) Reserved rationality and the precautionary principle: Technological change, time and uncertainty in environmental decision making. In: Constanza RC (ed) Ecological economics: The science and management of sustainability. Columbia University Press, New York.
121. Perrings C (2003) The economics of abrupt climate change. Philos Trans R Soc Lond Ser A Math Phys Sci 361(1810): 2043-2057.
122. Pigou AC (1932) The economics of welfare. Macmillan, London.
123. Pindyck RS (2000) Irreversibilities and the timing of environmental policy. Resour Energy Econ 22(3): 233-260.
124. Pindyck RS (2007) Uncertainty in environmental economics. Rev Environ Econ Policy 1(1): 45-65.
125. Quiggin J (2008) Stern and his critics on discounting and climate change: An editorial essay. Clim Change 89(3-4): 195-205.
126. Rabl A (1996) Discounting of long-term costs: what would future generations prefer us to do? Ecol Econ 17(3): 137-145.
127. Raiffa H (1968) Decision analysis reading. Addison-Wesley, Reading.
128. Ramsey F (1928) A mathematical theory of saving. Econ J 38: 543-549.
129. Rawls J (1971) A theory of justice. Harvard University Press, Cambridge.
130. Roemer JE (2011) The ethics of intertemporal distribution in a warming planet. Environ Resour Econ 48: 363-390.
131. Roy AD (1952) Safety first and the holdings of assets. Econometrica 20(3): 431-449.
132. Savage LJ (1951) The theory of statistical decision. J Am Stat Assoc 46: 55-67.
133. Savage LJ (1954) The foundations of statistics. Wiley, London.
134. Schimmelpfennig D (1995) The option value of renewable energy: The case of climate change. Energy Econ 17(4): 311-317.
135. Schmeidler D (1989) Subjective probability and expected utility without additivity. Econometrica 57(3): 571-87.
136. Schneider MT, Traeger CP, Winkler R (2012) Trading off generations: Equity, discounting, and climate change. Eur Econ Rev 56: 1621-1644.
137. Solow RM (1974) Intergenerational equity and exhaustible resources. Rev Econ Stud 41: 29-45.
138. Starmer C (2000) Developments in non-expected utility theory: The hunt for a descriptive theory of choice under risk. J Econ Lit 38(2): 332-382.
139. Stern N (2007) The economics of climate change: The Stern review. Cambridge University Press, Cambridge.
140. Stern N (2008) Richard T. Ely lecture: The economics of climate change. Am Econ Rev 98(2): 1-37.
141. Tol RSJ (1994) Communication-the damage costs of climate change: A note on tangibles and intangibles, applied to DICE. Energy Policy 22(5): 436-438.
142. Tol RSJ (1997) On the optimal control of carbon dioxide emissions: An application of FUND. Environ Model Assess 2: 151-163.
143. Tol RSJ (1999) Time discounting and optimal control of climate change: An application of FUND. Clim Change 41: 351-362.
144. Tol RSJ (2001) Equitable cost-benefit analysis of climate change policies. Ecol Econ 36: 71-85.
145. Tol RSJ (2002) Welfare specifications and optimal control of climate change: An application of fund. Energy Econ 24: 367-376.
146. Tol RSJ (2003) Is the uncertainty about climate change too large for expected cost-benefit analysis? Clim Change 56(3): 265-289.
147. Tol RSJ, Downing TE, Kuik OJ, Smith JB (2004) Distributional aspects of climate change impacts. Global Environ Change 14: 259-272.
148. Tol RSJ (2005) Emission abatement versus development as strategies to reduce vulnerability to climate change: An application of FUND. Environ Dev Econ 10: 615-629.
149. Tol RSJ (2009a) The economic impact of climate change. J Econ Perspect 23(2): 29-51.
150. Tol RSJ (2009b) The feasibility of low concentration targets: An application of FUND. Energy Econ 31: 121-130.
151. Tóth FL (2000) Intergenerational equity and discounting. Integr Assess 1: 127-139.
152. Traeger CP (2009) Recent developments in the intertemporal modeling of uncertainty. Ann Rev Resour Econ 1: 261-285.
153. Traeger CP (2011) Sustainability, limited substitutability, and non-constant social discount rates. J Environ Econ Manag 62: 215-228.
154. Tversky A, Kahneman D (1992) Advances in prospect theory: cumulative representation of uncertainty. J Risk Uncertain 5(4): 297-323.
155. Ulph A, Ulph D (1997) Global warming, irreversibility and learning. Econ J 107(442): 636-650.
156. van den Bergh JCJM (2004) Optimal climate policy is a utopia: From quantitative to qualitative cost-benefit analysis. Ecol Econ 48: 385-393.
157. van den Bergh JCJM (2010) Safe climate policy is affordable: 12 reasons. Clim Change 101(3): 339-385.
158. von Neumann J, Morgenstern O (1944) The theory of games and economic behavior, 2nd edn. Princeton University Press, Princeton.
159. Weitzman ML (1998) Why the far-distant future should be discounted at its lowest possible rate. J Environ Econ Manag 36: 201-208.
160. Weitzman ML (2009) On modeling and interpreting the economics of catastrophic climate change. Rev Econ Stat 91(1): 1-19.
161. Weitzman ML (2012) GHG targets as insurance against catastrophic climate damages. J Publ Econ Theory 14(2): 221-244.
162. Woodward RT, Bishop RC (1995) Efficiency, sustainability, and global warming. Ecol Econ 14: 101-111.
163. Woodward RT, Bishop RC (1997) How to decide when experts disagree: uncertainty-based choice rules in environmental policy. Land Econ 73(4): 492-507.
164. Woodward RT (2000) Sustainability as intergenerational fairness: Efficiency, uncertainty, and numerical methods. Am J Agri Econ 82: 581-593.
165. Yaari M (1965) Uncertain lifetime, life insurance, and the theory of consumer. Rev Econ Stat 91(1): 1-19.