<p>You can interprete <span class="math-container">$c_1 \in \{0,1\}$</span> as "I have the tea" and "I don't have the tea", and this tea is valued <span class="math-container">$v=13$</span>. <span class="math-container">$c_2$</span> is just a wealth transfer, i.e., it is <span class="math-container">$-p$</span> if the tea is bought and <span class="math-container">$0$</span> if the purchase is rejected.
Gain-loss utility is assesed separately in each dimension. Such that 
<span class="math-container">$$n(c|r) = n_1(c_1|r_1) + n_2(c_2|r_2).$$</span></p>

<p>In your case the reference point is <span class="math-container">$\widehat r =(1,-30)$</span>. In words, the buyer expects to purchase the good at price 30.
In a PE, it must be that given this reference point
<span class="math-container">$$U (\widehat r| \widehat r) \geq U ( c | \widehat r) \quad \forall c.$$</span>
That is, the buyer prefers to execute his plan.</p>

<p>This leads to utility
<span class="math-container">$$U (\widehat r| \widehat r) = v * 1 - 30 + 0 = 13-30=-17.$$</span>
Suppose instead that the buyer does not buy the good. In pure strategies and with a fixed price, there are only two options "buy at price 30", <span class="math-container">$(1,30)$</span> and not buying, <span class="math-container">$(0,0)$</span>.
That is, she does not spend <span class="math-container">$p$</span> of her wealth and does not get the good with value <span class="math-container">$v$</span>,
<span class="math-container">$$U ((0,0)| \widehat r) = v * 0 - 0 \:+ \quad \lambda(v*0-v*1) \:+ \quad (0-(-30)), $$</span>
where the first part is the consumption utility from not buying, zero.
The second part is the loss-utility from not having the good weighted by the loss-aversion parameter <span class="math-container">$\lambda$</span>. He does not have the good, so he feels a loss of <span class="math-container">$v$</span>, the value of the good. The third part is the gain-utility from saving 30 bucks.
Hence,
<span class="math-container">$$U ((0,0)| \widehat r) =30 - \lambda v = -22$$</span>
with the given parameters. We can verify the given PE.</p>

<p><span class="math-container">$v$</span> is the value of the good. Think of this setting as separable consumption utility
<span class="math-container">$$m (c) = m_1 (c_1) + m_2(c_2)$$</span>
with <span class="math-container">$m_1 (c_1) = v c_1$</span>, i.e., if you have the good you garner <span class="math-container">$v$</span> and otherwise zero. 
The money just enters linearly <span class="math-container">$m_2 (c_2) = c_2$</span>. Gain-loss utility is always evaluated with respect to consumption utility, see the paper. That is, 
actually 
<span class="math-container">$$n_i (c_i|r_i)  = \mu [m_i (c_i) - m_i(r_i)]$$</span>
with the given "universal gain-loss function".</p>

<p>EDIT: <strong>To make this useful for all</strong>, let me mention the paper corresponding to this model: <a href="https://onlinelibrary.wiley.com/doi/abs/10.3982/TE1274" rel="nofollow noreferrer">Heidhues and Kőszegi (TE, 2014)</a>. See how they define the product and the money dimension, <span class="math-container">$k^v =v b$</span>, where <span class="math-container">$b$</span> is the <span class="math-container">$c_1$</span> of your model (buying or not buying), and <span class="math-container">$k^p=-bp$</span>. This paper has interesting insights on the impact of consumer loss aversion on pricing and it supersedes their earlier paper.</p>

CC BY-SA 4.0
2019-03-18T12:20:45.647Z
2019-03-20T10:10:03.133Z
2019-03-20T10:10:03.133Z
Bayesian
<p>First of all, that's a homework question, and I <em>will</em> try to make it as useful to future readers as possible. </p>

<h2>Problem</h2>

<p>So, I was given a problem about the model described by Botond Kőszegi and Matthew Rabin in "A Model of Reference-Dependent Preference" in 2006:</p>

<blockquote>
  <p>A person deals with two goods: tea (<span class="math-container">$c_1 \in \{0, 1\}$</span>), and money (<span class="math-container">$c_2 \in R$</span>).
  <span class="math-container">$$
c_1 = \begin{cases}1, \mbox{if he buys tea}\\ 0, \mbox{if he does not buy tea} \end{cases}
$$</span>
  His consumption utility is given by: <span class="math-container">$$m(c) = v \times c_1 + c_2$$</span>
  His gain-loss utility is given by:
  <span class="math-container">$$
n(x | r) = \begin{cases} (x - r), \mbox{if } x \ge r\\ \lambda \times (x - r), \mbox{if } x \lt r \end{cases}
$$</span>
  Tea costs <span class="math-container">$P_H = 30$</span>; also, let <span class="math-container">$v = 13, \lambda = 4$</span>.</p>
</blockquote>

<p>There are multiple tasks, but I'm struggling even with the first one:</p>

<blockquote>
  <p>Prove that the reference point <span class="math-container">$r = (c_1, c_2) = (1, -30)$</span> with probability 1 is his <em>personal equilibrium</em>.</p>
  
  <p>Do not forget that in the gain-loss utility in case of deviation from expectations, the lost utility should be considered - as it looks in the consumption utility.</p>
</blockquote>

<p><strong>Confusion 1</strong>: What does the second sentence even mean and how do I consider the lost utility??</p>

<h2>My attempt</h2>

<p>First of all, let's define what we're looking for. According to page 1143 of the paper:</p>

<blockquote>
  <p>A selection <span class="math-container">$\{F_l \in D_l\}_{l \in R}$</span> is a <em>personal equilibrium</em> (PE) if for all <span class="math-container">$l \in R$</span> and <span class="math-container">$F'_l \in D_l$</span> <span class="math-container">$U(F_l | \int F_l dQ(l)) \ge U(F'_l  | \int F_l dQ(l))$</span>.</p>
</blockquote>

<p>So, provided that the probability is 1, I can get rid of the integrals and prove that <span class="math-container">$U(r | r) \ge U(x | r) \forall x \ne r$</span>, or, given that there are only two possible choices (right?), that <span class="math-container">$U((1, -30) | (1, -30)) \ge U((0, 0) | (1, -30))$</span>.</p>

<p><strong>Confusion 2</strong>: does he really have only two choices and is the second choice <span class="math-container">$(0, 0)$</span>?</p>

<p>Next, compute <span class="math-container">$U(r | r)$</span>. As per p. 1138 of the same work:</p>

<blockquote>
  <p>We assume that overall utility has two components: <span class="math-container">$u(c|r) = m(c) + n(c|r)$</span>, where <span class="math-container">$m(c)$</span> is "consumption utility" typically stressed in economics, and <span class="math-container">$n(c|r)$</span> is "gain-loss utility."</p>
</blockquote>

<p><span class="math-container">$$
U(r | r) = U((1, -30) | (1, -30)) = m(1, -30) + n((1, -30) | (1, -30))
$$</span></p>

<p>As per the same page:</p>

<blockquote>
  <p>We also assume that gain-loss utility is separable: <span class="math-container">$n(c|r) = \sum_{k=1}^K n_k(c_k | r_k)$</span>.</p>
</blockquote>

<p>So I can do:</p>

<p><span class="math-container">$$
m(1, -30) = v \times 1 - 30 = 13 - 30 = -17\\
n((1, -30) | (1, -30)) = n(1 | 1) + n(-30 | -30) = (1 - 1) + (-30 + 30) = 0\\
\mbox{Thus, } U(r | r) = -17 + 0 = -17
$$</span></p>

<p>Now for the second option:</p>

<p><span class="math-container">$$
U((0, 0) | (1, -30)) = m(0, 0) + n((0, 0) | (1, -30)) = 0 + n(0 | 1) + n(0 | -30) = \lambda \times (0 - 1) + (0 + 30) = 30 - \lambda = 30 - 4 = 26\\
\mbox{Thus, } U(x | r) = 0 + 26 = 26
$$</span></p>

<p><strong>Confusion 3</strong>: So, I got <span class="math-container">$U(r | r) \lt U(x | r)$</span>, which is quite the opposite of what I need to prove...</p>

<h2>UPDATE</h2>

<p>I'm being told by a fellow student that one should multiply the second part of the gain-loss utility by <span class="math-container">$v$</span> (!) to get:</p>

<p><span class="math-container">$$
n(x | r) = \begin{cases} (x - r), \mbox{if } x \ge r\\ v \times \lambda \times (x - r), \mbox{if } x \lt r \end{cases}
$$</span></p>

<p>So that the computation of <span class="math-container">$U(x | r)$</span> I get:</p>

<p><span class="math-container">$$
U((0, 0) | (1, -30)) = \lambda \times v \times (0 - 1) + (0 + 30) = -22
$$</span></p>

<p>...and the same for <span class="math-container">$U((1, -30) | (1, -30))$</span>.</p>

<p>They also say that this is what the "Do not forget that in the gain-loss utility in case of deviation from expectations, the lost utility should be considered - as it looks in the consumption utility." part of the task tells me to do. Unfortunately, I still have no idea what this part means or why I should multiply <em>only</em> the second part of the function by <em>exactly</em> <span class="math-container">$v$</span>. I mean, why would you just change the function out of nowhere?</p>

<h2>Question</h2>

<p>Where am I wrong? And why do I need to multiply by <span class="math-container">$v$</span>, and do I <em>actually</em> need to do this?</p>

CC BY-SA 4.0
2018-12-11T21:09:12.720Z
So... a small update: it turned out that one _should_ indeed multiply by `v` here: this is stated in the answers to the assignment. _Why_ this is the case, though, remains a mystery.
ForceBru
<behavioral-economics><loss-aversion>
Kőszegi - Rabin (2006) model problem
ForceBru
# Kőszegi - Rabin (2006) model problem

(Asked by: username_0 on 2018-10-18)

First of all, that's a homework question, and I *will* try to make it as useful to future readers as possible.

## Problem

So, I was given a problem about the model described by Botond Kőszegi and Matthew Rabin in "A Model of Reference-Dependent Preference" in 2006:

> A person deals with two goods: tea ($c\_1 \in \{0, 1\}$), and money ($c\_2 \in R$).
> $$
> c\_1 = \begin{cases}1, \mbox{if he buys tea}\\ 0, \mbox{if he does not buy tea} \end{cases}
> $$
> His consumption utility is given by: $$m(c) = v \times c\_1 + c\_2$$
> His gain-loss utility is given by:
> $$
> n(x | r) = \begin{cases} (x - r), \mbox{if } x \ge r\\ \lambda \times (x - r), \mbox{if } x \lt r \end{cases}
> $$
> Tea costs $P\_H = 30$; also, let $v = 13, \lambda = 4$.

There are multiple tasks, but I'm struggling even with the first one:

> Prove that the reference point $r = (c\_1, c\_2) = (1, -30)$ with probability 1 is his *personal equilibrium*.
>
> Do not forget that in the gain-loss utility in case of deviation from expectations, the lost utility should be considered - as it looks in the consumption utility.

**Confusion 1**: What does the second sentence even mean and how do I consider the lost utility??

## My attempt

First of all, let's define what we're looking for. According to page 1143 of the paper:

> A selection $\{F\_l \in D\_l\}\_{l \in R}$ is a *personal equilibrium* (PE) if for all $l \in R$ and $F'\_l \in D\_l$ $U(F\_l | \int F\_l dQ(l)) \ge U(F'\_l | \int F\_l dQ(l))$.

So, provided that the probability is 1, I can get rid of the integrals and prove that $U(r | r) \ge U(x | r) \forall x \ne r$, or, given that there are only two possible choices (right?), that $U((1, -30) | (1, -30)) \ge U((0, 0) | (1, -30))$.

**Confusion 2**: does he really have only two choices and is the second choice $(0, 0)$?

Next, compute $U(r | r)$. As per p. 1138 of the same work:

> We assume that overall utility has two components: $u(c|r) = m(c) + n(c|r)$, where $m(c)$ is "consumption utility" typically stressed in economics, and $n(c|r)$ is "gain-loss utility."

$$
U(r | r) = U((1, -30) | (1, -30)) = m(1, -30) + n((1, -30) | (1, -30))
$$

As per the same page:

> We also assume that gain-loss utility is separable: $n(c|r) = \sum\_{k=1}^K n\_k(c\_k | r\_k)$.

So I can do:

$$
m(1, -30) = v \times 1 - 30 = 13 - 30 = -17\\
n((1, -30) | (1, -30)) = n(1 | 1) + n(-30 | -30) = (1 - 1) + (-30 + 30) = 0\\
\mbox{Thus, } U(r | r) = -17 + 0 = -17
$$

Now for the second option:

$$
U((0, 0) | (1, -30)) = m(0, 0) + n((0, 0) | (1, -30)) = 0 + n(0 | 1) + n(0 | -30) = \lambda \times (0 - 1) + (0 + 30) = 30 - \lambda = 30 - 4 = 26\\
\mbox{Thus, } U(x | r) = 0 + 26 = 26
$$

**Confusion 3**: So, I got $U(r | r) \lt U(x | r)$, which is quite the opposite of what I need to prove...

## UPDATE

I'm being told by a fellow student that one should multiply the second part of the gain-loss utility by $v$ (!) to get:

$$
n(x | r) = \begin{cases} (x - r), \mbox{if } x \ge r\\ v \times \lambda \times (x - r), \mbox{if } x \lt r \end{cases}
$$

So that the computation of $U(x | r)$ I get:

$$
U((0, 0) | (1, -30)) = \lambda \times v \times (0 - 1) + (0 + 30) = -22
$$

...and the same for $U((1, -30) | (1, -30))$.

They also say that this is what the "Do not forget that in the gain-loss utility in case of deviation from expectations, the lost utility should be considered - as it looks in the consumption utility." part of the task tells me to do. Unfortunately, I still have no idea what this part means or why I should multiply *only* the second part of the function by *exactly* $v$. I mean, why would you just change the function out of nowhere?

## Question

Where am I wrong? And why do I need to multiply by $v$, and do I *actually* need to do this?

**Tags:** behavioral-economics, loss-aversion
> **Comments:**
> **username_0**: So... a small update: it turned out that one _should_ indeed multiply by `v` here: this is stated in the answers to the assignment. _Why_ this is the case, though, remains a mystery.  

---

## 1 answers

### Answer by username_1 on 2019-03-18. Score: 2

You can interprete $c\_1 \in \{0,1\}$ as "I have the tea" and "I don't have the tea", and this tea is valued $v=13$. $c\_2$ is just a wealth transfer, i.e., it is $-p$ if the tea is bought and $0$ if the purchase is rejected.
Gain-loss utility is assesed separately in each dimension. Such that
$$n(c|r) = n\_1(c\_1|r\_1) + n\_2(c\_2|r\_2).$$

In your case the reference point is $\widehat r =(1,-30)$. In words, the buyer expects to purchase the good at price 30.
In a PE, it must be that given this reference point
$$U (\widehat r| \widehat r) \geq U ( c | \widehat r) \quad \forall c.$$
That is, the buyer prefers to execute his plan.

This leads to utility
$$U (\widehat r| \widehat r) = v \* 1 - 30 + 0 = 13-30=-17.$$
Suppose instead that the buyer does not buy the good. In pure strategies and with a fixed price, there are only two options "buy at price 30", $(1,30)$ and not buying, $(0,0)$.
That is, she does not spend $p$ of her wealth and does not get the good with value $v$,
$$U ((0,0)| \widehat r) = v \* 0 - 0 \:+ \quad \lambda(v\*0-v\*1) \:+ \quad (0-(-30)), $$
where the first part is the consumption utility from not buying, zero.
The second part is the loss-utility from not having the good weighted by the loss-aversion parameter $\lambda$. He does not have the good, so he feels a loss of $v$, the value of the good. The third part is the gain-utility from saving 30 bucks.
Hence,
$$U ((0,0)| \widehat r) =30 - \lambda v = -22$$
with the given parameters. We can verify the given PE.

$v$ is the value of the good. Think of this setting as separable consumption utility
$$m (c) = m\_1 (c\_1) + m\_2(c\_2)$$
with $m\_1 (c\_1) = v c\_1$, i.e., if you have the good you garner $v$ and otherwise zero.
The money just enters linearly $m\_2 (c\_2) = c\_2$. Gain-loss utility is always evaluated with respect to consumption utility, see the paper. That is,
actually
$$n\_i (c\_i|r\_i) = \mu [m\_i (c\_i) - m\_i(r\_i)]$$
with the given "universal gain-loss function".

EDIT: **To make this useful for all**, let me mention the paper corresponding to this model: [Heidhues and Kőszegi (TE, 2014)](https://onlinelibrary.wiley.com/doi/abs/10.3982/TE1274). See how they define the product and the money dimension, $k^v =v b$, where $b$ is the $c\_1$ of your model (buying or not buying), and $k^p=-bp$. This paper has interesting insights on the impact of consumer loss aversion on pricing and it supersedes their earlier paper.


---

