Question

1. Irma has concave utility function U(w) = ln(w). Calculate the arrow-Pratt measure of risk aversion and find the relation between risk aversion and wealth.

Answer 1

GIVEN,

U(W) = ln(W)

First order derivative : U'(W) = d(U)/d(W) = 1/W

Second order derivative : U"(W) = d²U/dW² = -1/W²

There are two measures of Risk Aversion:

1) Absolute Risk Aversion (ARA) :

-U"(W)/U'(W) = -[(-1/W²) / (1/W) ] = 1/W

Now, measuring the relation between absolute risk aversion and wealth :

d(ARA)/dW = -1/W² <0 i.e. for one unit increase in wealth , ARA decreases.

2) Relative Risk Aversion(RRA) :

-W* [U"(W)/U'(W)] = -W*[(-1/W²) / (1/W) ] = 1

Now, measuring the relation between relative risk aversion and wealth :

d(RRA)/dW = 0 i.e. for one unit increase in wealth , RRA is constant.