Question

Consider an individual who maximizes her expected utility having Bernoulli utility function

• A. constant
• B. increasing
• C. decreasing
• D. uncertain

Hint:

In utility theory, relative risk aversion increases with wealth for exponential utility functions like the one given here.

Answer 1

The Correct Option is B

Step 1: Understanding the utility function.
The utility function is given by \( u(w) = \alpha - \beta e^{-rw} \), where \( \alpha \) and \( \beta \) are constants, and \( w \) represents wealth. The individual maximizes her expected utility.
Step 2: Relative Risk Aversion.
The relative risk aversion (RRA) is given by the formula: \[ RRA(w) = -\frac{w u''(w)}{u'(w)}. \] For the given utility function \( u(w) = \alpha - \beta e^{-rw} \), we calculate the first and second derivatives: \[ u'(w) = \beta r e^{-rw}, \quad u''(w) = -\beta r^2 e^{-rw}. \] Substituting into the RRA formula: \[ RRA(w) = \frac{w \beta r^2 e^{-rw}}{\beta r e^{-rw}} = r w. \] Since \( w>0 \), the RRA is increasing with respect to wealth. Therefore, the individual displays increasing relative risk aversion.