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


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...


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?


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

  • $\begingroup$ 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. $\endgroup$
    – ForceBru
    Dec 11, 2018 at 21:09

1 Answer 1


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). 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.