By D. Cox, E. Snell
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Additional resources for Applied Sstatistics - Principles and Examples
11 Formal utility theory requires more assumptions, but the discussion here is kept informal. 2 Indifference curves Indifference curves depict combinations of choices that give the same utility. They are constructed by plotting a two-dimensional utility function, that gives utility as a function of combinations of two choices, for example saving and consuming, or apples and pears. 01W22 . W1 and W2 can be thought of as wealth now and wealth next period. An indifference curve is a collection of points with the same value of U (W ).
The first is U (E[W ]), the utility of expected wealth. That equals the 400 we just calculated and that lies on the utility curve. 01 × 1502 = 525 so that E[U (W )] = (225 + 525)/2 = 375 E[U (W )] is a probability weighted, straight line interpolation between the two points U(150) and U(50) on the utility curve. 4. We see that E[U (W )] < U (E[W ]); this expression is an application of Jensen’s inequality that has a much wider validity. The difference between E[U (W )] and U (E[W ]) reflects risk aversion.
This is used to formulate measures of risk aversion, or risk aversion coefficients. One of the oldest and best known is the Arrow–Pratt absolute risk aversion coefficient: ARA(W ) = − U ′′ (W ) U ′ (W ) where U ′ (W ) and U ′′ (W ) are the first and second derivative of the utility function, respectively. The corresponding relative risk aversion coefficient is obtained by multiplying by W: RRA(W ) = −W U ′′ (W ) U ′ (W ) For the utility function U (W ) = α + βW − γ W 2 the coefficients are: ARA(W ) = 2γ β − 2γ W RRA(W ) = 2γ W β − 2γ W Both coefficients increase with W and this is generally considered to be an undesirable property of utility functions.
Applied Sstatistics - Principles and Examples by D. Cox, E. Snell