Question

Question 8: Consider a decision-maker with utility function u(x) = x^0.8 , where x>0 denotes the decision-maker’s wealth.

a. Determine the decision-maker’s attitude towards risk. In other words, is this decision-maker risk-neutral, risk-averse or risk-loving? Provide a justification of your answer.

Solution: We have ?''(x) = -0.16x^-1.2 <0 . Hence, the decision-maker is risk-averse.

Please explain solution. How did he get the answer?

Answer 1

A decision maker is risk averse if it prefers sure outcomes with lower risk even if they give lower payoff compared to the uncertain lottery which might be providing higher payoff but with uncertainty.

Mathematically, $\succeq$ is risk averse if for any lottery p, $[Ep] \succeq p$

where $[Ep]$ denotes the lottery which yields prize p with certainty.

Risk aversion is related to the concavity of the vNM utility function.

Let $\succeq$ be a preference relation on lottery space L(Z) represented by the vNM utility function u. $\succeq$ is risk averse if and only if u is concave.

Concavity of a function f:

A real-valued function f on an interval is concave if, for any x and y in the interval and for any ,

Property:  A differentiable function f is strictly concave on an interval if and only if its derivative function f ′ is strictly monotonically decreasing on that interval i.e. f'' < 0

Given our example,

u(x) = x^0.8

=> u'(x) = 0.8x^_-0.2

=> u''(x) = -0.16x^_-0.12 which is negative for all x > 0,

Thus u(x) is concave and our decision maker is risk-averse.