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The 1997 Quarterly Journal of Economics (QJE) paper titled "The Effect of Myopia and Loss Aversion on Risk Taking: An Experimental Test" by Thaler, Tversky, Kahneman and Schwartz, says that an individual with the following utility function

$$U(x)=\begin{cases} x, x\geq0\\ 2.5x, x<0\end{cases}$$ is risk neutral for gains and losses. How can I prove that this individual is risk-averse in mixed gambles? A mixed gamble is a gamble with some positive and negative outcomes.

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    $\begingroup$ Do you know the definition of risk-averse? $\endgroup$
    – Giskard
    Commented Jan 23 at 6:47
  • $\begingroup$ It means that an individual gets a higher utility from choosing a sure outcome than a fair gamble/lottery. $\endgroup$
    – Ramandeep
    Commented Jan 23 at 6:54
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    $\begingroup$ Any sure outcome? Like I choose getting $-100\$$ with 100% instead of losing or winning 10\$ with 50-50%? $\endgroup$
    – Giskard
    Commented Jan 23 at 7:02
  • $\begingroup$ No, it must be that the sure outcome is something greater than the lowest payoff from the gamble considered. So, in your example, the sure outcome has to be something greater than the lowest payoff from the gamble with a 50-50% chance of winning $10. $\endgroup$
    – Ramandeep
    Commented Jan 23 at 7:39
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    $\begingroup$ I recommend looking up the exact definition, it uses a very specific sure outcome. Afterwards I think you can solve this problem by applying the definition. $\endgroup$
    – Giskard
    Commented Jan 23 at 8:02

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I presume that by "fair gamble" $G$ we mean a gamble whose expected value is zero, $E(G) = 0$.

Risk-aversion can be expressed mathematically by showing that the utility function is strictly concave, which by Jensen's Inequality imples $$U[E(G)] > E[U(G)]$$

Given the utility function in our case this means that we have to show

$$U(0) > E[U(G)] \implies 0 > E[U(G)].$$

So we need a way to show that, for the given utility function, the expected utility from any "fair gamble" will be negative.

Intuitively, we should expect that since we see that negative outcomes weigh more heavily than positive ones (due to the $2.5$ mutliplicative factor of what happens to utility when the outcome is negative).

How to show?

Define the fair Gamble providing a set of $m$ payoffs $\{x^G_1,..., x^G_m\}$ and a probability distribution over them $P_G = \{p_1,...p_m\}$. To be a fair gamble it must be the case that

$$E(G) = \sum_{i=1}^n p_ix^G_i = 0.$$

Write the utility function using the indicator function

$$U(x) = [1-I\{x< 0\}]\cdot x + 2.5 \cdot I\{x< 0\})]\cdot x = x + 1.5\cdot I\{x< 0\})]\cdot x $$

Then $$E[U(G)] = \sum_{i=1}^m p_iu(x^G_i) = ...$$

I guess you can take it from here.

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