# Kőszegi - Rabin (2006) model problem

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?

• 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 Dec 11 '18 at 21:09

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.