Consider two random variables (lotteries) ˜ω1 and ˜ω2. Let ˜ω2 be dominated by ˜ω1 in the senseof the rst-degree stochastic dominance. In class we argued that the expected utility of ˜ω2 isstrictly less than of ˜ω1, i.e. E[u(˜ω2)] − E[u(˜ω1)] < 0 for any increasing utility function u(w). Explain why expected payo of ˜ω2 is also necessarily less than of ˜ω1, i.e. whyE[˜ω2] < E[˜ω1]. Give an example showing that the reverse is not true. That is, give an example of tworandom variables, such that one random variable gives a better expected payo than theother, i.e. E[˜ω1] > E[˜ω2], but ˜ω2 is NOT rst-degree stochastically dominated by ˜ω1.2
Question
Consider two random variables (lotteries) ˜ω1 and ˜ω2. Let ˜ω2 be dominated by ˜ω1 in the senseof the rst-degree stochastic dominance. In class we argued that the expected utility of ˜ω2 isstrictly less than of ˜ω1, i.e. E[u(˜ω2)] − E[u(˜ω1)] < 0 for any increasing utility function u(w). Explain why expected payo of ˜ω2 is also necessarily less than of ˜ω1, i.e. whyE[˜ω2] < E[˜ω1]. Give an example showing that the reverse is not true. That is, give an example of tworandom variables, such that one random variable gives a better expected payo than theother, i.e. E[˜ω1] > E[˜ω2], but ˜ω2 is NOT rst-degree stochastically dominated by ˜ω1.2
Solution
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The expected payoff of ω2 is necessarily less than that of ω1 because first-degree stochastic dominance implies that the cumulative distribution function (CDF) of ω2 is always to the right of the CDF of ω1. This means that for any given outcome, the probability of ω2 being less than or equal to that outcome is always greater than or equal to the probability of ω1 being less than or equal to that outcome. Since the expected value is the sum of all possible outcomes weighted by their probabilities, and ω2 is more likely to have lower outcomes, the expected value of ω2 is less than that of ω1.
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Consider two lotteries, ω1 and ω2, where ω1 has a 50% chance of paying 3, and ω2 has a 100% chance of paying 2, which is the same as the expected payoff of ω2. However, ω2 is not first-degree stochastically dominated by ω1 because the CDF of ω2 is not always to the right of the CDF of ω1. In fact, for outcomes less than 2, the CDF of ω2 is to the right of the CDF of ω1.
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