Certainty equivalent and risk aversion parameter given utility function

Question

I know this is rather easy but I do not have any example or material to really work from that I could find online.

Consider an agent with a utility index over wealth given by $$v(w) = \frac{w^{1 - \gamma} - 1}{1 - \gamma}, \gamma > 0$$

Say the agent has $10$ dollars in wealth and also owns a lottery that pays $0$ dollars with probability $2/3$ and pays $12$ with probability $1/3$. What is the certainty equivalent value and risk compensation of this lottery to an agent with risk aversion parameter $\gamma = 2$?

So for $\gamma = 2$, we have $$v(w) = \frac{w-1}{w}$$ so the expectd value of the lottery is $4$ dollars. My question is how do we calculate the Certainty equivalent and risk premium. I could not find a good example anywhere. It seems like we get at undefined part in the equation when we calculate $E[v(L)]$ where $L$ is the lottery, thus I don't see how we can find the Certainty equivalent...

Answer 1

For the certainty equivalent and the "risk compensation" (which I am interpreting as probability premium because it's the only thing that intuitively makes sense in this context to me; feel free to correct me), think more intuitively about the concepts. The certainty equivalent is the amount of cold hard cash you'd be indifferent to taking in lieu of the uncertain outcome.

In the good outcome with $\frac{1}{3}$ probability, you'd end up with a wealth of $10 + 12$, and in the the bad state with probability $\frac{2}{3}$ where the lottery does not give you anything, you still end up with wealth of $10$. Thus, we are looking for wealth that I will denote $w^c$ where

$$v(w_c) = \mathbb{E}(v(w)) \implies \frac{w_c - 1}{w_c} = \left(\frac{22-1}{22}\right)\cdot \frac{1}{3} + \left(\frac{10-1}{10}\right) \cdot \frac{2}{3}$$

Solve for $w_c$ and that will be the certainty equivalent. You'll notice that trying to solve this out, we get:

$$\frac{w_c - 1}{w_c} = \frac{101}{110}$$ $$\implies \boxed{w_c = \frac{110}{9} \approx 12.2}$$

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As for probability premium, it is the shift in probability towards the better lottery outcome that would be needed to make the expected utility of the new lottery equal to the utility of the expected value of the old lottery.

$$\mathbb{E}_{\text{new}}(v(w)) = v(\mathbb{E}_{\text{old}}(w))$$ $$\implies \left(\frac{1}{3} + \pi\right) \cdot \left(\frac{22-1}{22}\right) + \left(\frac{2}{3} - \pi\right) \cdot \left(\frac{10-1}{10}\right) = \left(\frac{22 \cdot \frac{1}{3} + 10 \cdot \frac{2}{3} -1}{22 \cdot \frac{1}{3} + 10 \cdot \frac{2}{3}}\right)$$ $$\implies \frac{101}{110} + \frac{6}{110} \pi = \frac{13}{14}$$ $$\implies \pi = \frac{8}{770} \approx 0.01$$

(assuming I calculated my fractions correctly)

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Edit: I see that the question was asking for a risk premium, not probability premium. Notice that the expected wealth due to the lottery is 14, but the certainty equivalent is 12.222...

The difference between the two is the risk premium.