Savage State-Dependent Expected Utility

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Savage State-Dependent Expected Utility

• Lecture IV

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Savages state dependent expected utility

• Savages subjective expected utility theory takes
  as the object of choice state dependent outcomes.

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• Writing the expected utility of the gamble out,
  we have
• What is changing? The probability or the states?

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• Savage proved a similar result to the von
  Neumann-Morgenstern modelif agents preferences
  obey certain axioms they have an expected utlity
  representation through both the probabilities of
  each outcome and the utility function.
• Specifically, the probabilities in the expected
  value framework may be endogenous.

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Axiomatization of StateDependent Expected
Utility

• Start with some notation.
• In the statepreference format we speak of the
  utility function mapping from state-space into
  preferences

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• Next, we want to introduce the notation of
  replacement
• when the return in state s is replaced with y.

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• We are interested in making the comparison
• Note that c,d?RS or are two vectors of outcomes
  or payoffs while y,w?R are two scalar incomes.

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• For example, take an investment

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• The independence axiom for state-dependent
  expected utility requires that
• Intuitively, the ordering remains the same if we
  replace on of the states with a common payoff.

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• Theorem 8.5.1 Assume that there are at least
  three states, S? 3. Utility function u has a
  state-dependent expected utility representation
  iff it obeys the independence axiom.

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Cardinal Utility Systems

• A second article Friedman and L.J. Savage (1952)
  The Expected-Utility Hypothesis and the
  Measurability of Utility Journal of Polictical
  Economy 60(6) 46374 followed the Friedman and
  Savage article we analyzed last class. This
  article responded to a comment by William Baumol
  that raised questions about the cardinality of
  utility measures.

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• Most of our current microeconomic theory is based
  on ordinal utility measurement where the actual
  levels of the utility are not important only the
  order or relative rank.
• Cardinality assigns importance to the actual
  numbers.

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• The Cardinal Coordinate Independence axiom is a
  stronger version of the independence axiom
  discussed above.
• Proposition 8.6.1 Cardional coordinate
  independence implies independence.
• Theorem 8.6.2 Utility function u has a
  state-independent expected utility representation
  iff it obeys the cardinal coordinate independence
  axiom.