Lottery is between:

Option A: a certain choice of £5

Option B: £10 with probability 0.1 and £1 with probability 0.9

The probability of receiving £10 increases in each subsequent choice.

How do I calculate the CRRA bounds of this, using $U(x) = \frac{x^{(1-r)}}{1-r}$?


Let me rewrite your Option B as follows: £10 with probability $p$ and £1 with probability $1-p$ where $p\in[0,1]$ varies across the items in the list of options.

Let $\bar p$ be the point where the subject's choice switches. For example, subject chooses Option A for all $p<\bar p$ and Option B for all $p>\bar p$. Then we can use the indifference condition \begin{equation} U(5)=\bar pU(10)+(1-\bar p)U(1) \end{equation} to infer the risk aversion parameter $r$.

In experiments where $p$ is discretized, we can take $p_0,p_1$ as the values that are immediately before and after the switching point in choice list. For example, subject chooses Option A for all $p\le p_0$ and Option B for all $p>p_1$. Then the bounds on the risk aversion parameter can be inferred from the following inequalities: \begin{equation} U(5)\ge p_0U(10)+(1-p_0)U(1) \quad\text{and}\quad U(5)\le p_1U(10)+(1-p_1)U(1). \end{equation}

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    $\begingroup$ +1 NB To infer $r$, we also need to know your initial level of wealth. $\endgroup$ – user17900 Jan 2 at 12:19
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    $\begingroup$ @afreelunch: You're right theoretically. In actual experimental settings, however, since subjects' initial wealth is difficult to observe, I believe experimenters typically assume it to be zero (or assume utility is reference-dependent, where the reference point being zero). $\endgroup$ – Herr K. Jan 2 at 20:06

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