Risk Premia and Certainty Equivalents
- VII.A.1: Distinguish among risk-averse, risk-neutral, and risk-loving agents.
- VII.A.2: Compute measures of risk aversion given a utility function.
- VII.A.4: Solve for expected utility, certainty equivalents, and risk premia.
Consider the following utility function: \(U(x_1,x_2) = \pi x_1^r + (1-\pi) x_2^r\) where $x_1$ and $x_2$ are consumption levels in state 1 and 2, respectively which occur with probabilities $\pi = 0.8$ and $1-\pi=0.2$. Assume that there is a lottery that pays nothing in state 1 and 100 dollars in state 2.
Assume $r$ is equal to 0.5.
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Predict using intuition if the expected utility of the lottery is greater than, less than, or equal to the utility of its expected value. Is this person risk-averse, risk-neutral, or risk-seeking? What is the underlying felicity function? Is it concave, linear, or convex?
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Using the graph below, confirm your answer from (a). Be sure to set $r$ and $\pi$ equal to the given values.
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Confirm your answer from (a) algebraically by calculating the expected value of the lottery and the utility of its expected value. Determine the person’s attitude toward risk using the second derivative of the underlying felicity function.
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The US government decides that it wants to give its citizens a choice between having a lottery ticket or getting a direct payment of $X$ dollars. This amount $X$ is also referred to as the certainty equivalent of the lottery. What is the value of $X$ in this case?
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What is the risk premium of the lottery for this individual?
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Now, assume 𝑟 is equal to 1.5. Is this person risk-averse, risk-neutral, or risk-seeking? What is the individual’s expected utility from the lottery? What is the certainty equivalent of the lottery and what is its risk premium? Hint: Use the graph to get intuition for what’s going on.