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action actuarially fair alternatives analysis approaches assumed assumption axiom behavior under uncertainty benefit-premium ratio budget certainly chap choice function choose commodity bundles conditional distribution consequences consider consumption cost-of-living index critical level critical marginal utility curve decision decreasing defined definition demand functions derived dW/dP Econometrica Engel curve example expected utility expected value expected-utility hypothesis expenditure finite follows gamble given Hence holds Houthakker implies index number indifference curve indifference maps individual insurance payment Jacob Marschak L. J. Savage law of large Lemma Marschak mathematical maximize measure minimax monotone increasing observations optimal policy ordering outcome parameters possible postinsurance income potential surprise preferred probability distribution problem random variable relation relative risk aversion Revealed Preference risky assets Section secure asset situation statistical sufficiently large Suppose theorem theory of choice tion utility function utility of income utility theory wealth zero