<p>Let me rewrite your Option B as follows: £10 with probability <span class="math-container">$p$</span> and £1 with probability <span class="math-container">$1-p$</span> where <span class="math-container">$p\in[0,1]$</span> varies across the items in the list of options. </p>

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

<p>In experiments where <span class="math-container">$p$</span> is discretized, we can take <span class="math-container">$p_0,p_1$</span> as the values that are immediately before and after the switching point in choice list. For example, subject chooses Option A for all <span class="math-container">$p\le p_0$</span> and Option B for all <span class="math-container">$p&gt;p_1$</span>. Then the bounds on the risk aversion parameter can be inferred from the following inequalities: 
<span class="math-container">\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}</span></p>

CC BY-SA 4.0
2020-01-02T20:06:23.870Z
@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).
Herr K.
CC BY-SA 4.0
2022-03-15T13:50:55.810Z
Isn't it possible to factor our wealth if changes in wealth due to lotteries are relative? Why not design the stakes as say %-points changes? Alternatively assuming payoffs affect total wealth only by a small amount using a taylor approx. around the lottery and factor out wealth as well?
T123
CC BY-SA 4.0
2019-11-28T19:25:24.940Z
2019-11-29T02:37:54.660Z
2019-11-29T02:37:54.660Z
Herr K.
<p>Lottery is between:</p>

<p>Option A: a certain choice of £5</p>

<p>Option B: £10 with probability 0.1 and £1 with probability 0.9</p>

<p>The probability of receiving £10 increases in each subsequent choice.</p>

<p>How do I calculate the CRRA bounds of this, using <span class="math-container">$U(x) = \frac{x^{(1-r)}}{1-r}$</span>?</p>

<microeconomics><utility><risk>
How to calculate CRRA bounds from Holt and Laury (2002) type lottery?
Poppyisswag
# How to calculate CRRA bounds from Holt and Laury (2002) type lottery?

(Asked by: username_0 on 2019-11-28)

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 $U(x) = \frac{x^{(1-r)}}{1-r}$?

**Tags:** microeconomics, utility, risk

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## 1 answers

### Answer by username_1 on 2019-11-28. Score: 1

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}

> **Comments:**
> **username_1**: @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).  
> **username_2**: Isn't it possible to factor our wealth if changes in wealth due to lotteries are relative? Why not design the stakes as say %-points changes? Alternatively assuming payoffs affect total wealth only by a small amount using a taylor approx. around the lottery and factor out wealth as well?  

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