Modelo Arrow Pratt

RTHE ARROW-PRATT INDEX OF RELATIVE RISK AVERSION

A NOTE

PAUL C. KETTLER

The Arrow-Pratt index of relative risk aversion concepts of elasticity and marginal utility.

1 The index is used bymany authors writing in

rU (c)

combines the important economic

relation to utility theory, e.g., Karatzas and Shreve [KS98, p. 179, p. 188]. Kenneth Joseph Arrow, the Stanford economicsprofessor emeritus, Nobel laureate in economics in 1972, and member of the American Academy of Sciences, was the co-creator. Along with the late Gerard Debreu, himself a Nobel laureate in 1983, fellowmember of the American Academy, and recipient of the French Legion of Honor, Arrow produced the rst rigorous proof of the existence of a market clearing equilibrium, given certain restrictiveassumptions, [Wikipedia]. See their seminal paper on this subject [AD54]. Specically,

rU (c) is the negative of the elasticity of marginal utility, where typically U (c) is

a utility function of a rateof consumption. The independent variable could also be a function of total consumption, or of wealth, or of another variable, and frequently is stochastic. Marginal utility is simply

U (c).

Inpractical terms one thinks of the index as the fractional change in

this marginal utility resulting from a fractional change in the independent variable, inverting the sign. Equivalently, the indexis the negative of the ratio of the logarithmic dierentials of marginal utility to the independent variable. The negative sign is included so that the index will be positive in its ordinary ranges.In this mode a higher index value is associated with a greater fractional change in marginal utility for a given fractional change in the independent variable. The higher index value identies anindividual of greater risk aversion, one who chooses to avoid situations providing the greater fractional loss in utility. In general the elasticity of any function

y(x) : R+ → R+ x dy dx = , x y dx...