Attitudes Towards Risk

14.123 Microeconomic Theory III Muha met Yildiz

Model

 X = R = w eal th l e vel

 Lottery = cdf F (pdf f )

 Utility function u : R → R

 U ( F ) ≡ E F ( u ) ≡ ∫ u ( x )d F ( x )

 E F ( x ) ≡ ∫ x d F ( x )

Attitudes Towards Risk

DM is

 risk aver se if E F ( u ) ≤ u ( E F ( x )) (  F )

 strictly risk averse if E F ( u )< u ( E F ( x )) (  “risky” F )

 risk neutral if E F ( u )= u ( E F ( x )) (  F )

 risk seeking if E F ( u ) ≥ u ( E F ( x )) (  F ) DM is

 risk aver se if u is concave

 strictly risk averse if u is strictl y conca v e

 risk neutral if u is linear

 risk seeking if u is convex

Certainty Equivalence



CE ( F ) = u ⁻ ¹( U ( F ))= u ⁻ ¹( E F ( u ))

 DM is

 risk av erse if CE ( F ) ≤ E F ( x ) for all F ;

 risk neutral if CE ( F ) = E F ( x ) for all F ;

 risk seeking if CE ( F ) ≥ E F ( x ) for all F .

 Take DM1 and DM2 with u 1 and u 2.

 DM1 is more risk averse than DM2

  u 1 is mo re concave t han u 2 ,

  u 1 = φ ◦ u 2 for so me concave function φ ,

  CE 1 ( F ) ≡ u 1 ⁻ ¹( E F ( u 1 )) ≤ u 2 ⁻ ¹( E F ( u 2 )) ≡ CE 2 ( F )

Measures of Risk Aversion

 absolute ri sk aver sion:

r A ( x ) = - u ′′ ( x )/ u ′ ( x )

 constant absolute risk aver sion (CARA)

u ( x ) =- e - α x

 If x ~ N( μ , σ ²), CE ( F ) = μ - ασ ²/2

 relative risk aversion:

r R ( x ) = - xu ′′ ( x )/ u ′ ( x )

 constant relative risk aversion (CRRA)

u ( x )= - x 1- ρ /(1- ρ ),

 When ρ =1, u ( x ) = l og( x ).