To give an example, say we start with 100 dollars and we enter a lottery. With probability $\pi$, this 100 dollars is reduced by 2 dollars. Otherwise our endowed 100 dollars does not change. Let's say our consumer's utility function is described by $u(c_h)=ln(c_h)$.
It's clear that the consumer is risk-averse, but how would I calculate the degree of risk aversion? How do you find the risk premium here?
Thanks.
The gamble:
$$g = \left(\pi \circ 98, (1-\pi)\circ100\right)$$
The expected value of the gamble:
$$\mathbb{E}[g]= \pi\cdot 98 + (1-\pi)\cdot 100$$
Expected utility:
$$u(g) = \pi \cdot \ln (98) + (1-\pi)\cdot \ln (100)$$
Risk premium is such that:
$$\ln \left(\mathbb{E}[g]-P\right) = \pi \cdot \ln (98) + (1-\pi)\cdot \ln (100)$$
Where $P$ is the risk premium. Solve the above equation for $P$. You can measure the degree of absolute risk aversion as follows:
$$ARA = -\frac{u''(c_h)}{u'(c_h)}$$
And the degree of relative risk aversion as:
$$RRA = - \frac{c_hu''(c_h)}{u'(c_h)}$$
