Behavioral Choice Under Certainty

• Opportunity Costs
• Sunk Costs
• Menu Dependence and Decoy Effect
• Loss aversion and the endowment effect

Individuals use as reference points:

1. Other's choices/consumption
2. Past “utility”
3. Anticipated “utility”
   (i.e., what they expect, not expected utility theory)

Standard theory rules this out: if $\mathbf{x}$ is chosen when $\mathbf{y}$ is available, then adding a third option $\mathbf{z}$ should never make $\mathbf{y}$ the choice. The menu is supposed to be a container, not an ingredient.

Option 2 is the decoy: it costs the same as Option 3 and gives strictly less, so nobody should take it. Yet its presence makes Option 3 look like a bargain and pulls choices away from Option 1.

This model is a simplified version of Köszegi & Rabin (2006).

• There are two goods: $X$ and $Y$.
• There is a reference basket: $r=(\underline x,\underline y)$.
• $u(x)$ consumption utility of good $X$.
• $u(y)$ consumption utility of good $Y$.
• $\mu(\Delta u)$ is the gain-loss utility function.
• The consumer overall utility for the basket $(x,y)$ is:

• $\mu(u(x)-u(\underline x))$ is the gain or loss in utility when the consumer deviates from the $X$-reference consumption level.
• $\mu(u(y)-u(\underline y))$ is the gain or loss in utility when the consumer deviates from $Y$-reference consumption level.
• The gain-loss utility function is given by: $$\mu(\Delta u)\equiv L\cdot \min(\Delta u, 0)+G\cdot \max(\Delta u, 0).$$
• $\mu(\Delta u)$ is the gain-loss utility function.
• $L$ is the slope of the gain-loss utility for utility losses.
• $G$ is the slope of the gain-loss utility for utility gains.

Let $u$ be either a utility for wealth or a gain-loss utility function. The point zero is either the initial wealth or the reference point, and we assume $u(0)=0$.

The loss-gain utility is a component of the overall utility. Its utility/disutility depends on how the consumption-utility of a good differs from the consumption-utility of its reference level.

A loss-gain utility $\mu$ as defined above displays loss aversion if and only if $L\gt G\gt 0$.

Kahneman, Knetsch & Thaler (1990) ran markets in an economics class at Cornell. Half the students, chosen at random, received an object; the other half could buy one. Since the objects were assigned at random, half of those who value them most start without one, so standard theory predicts that half of the 22 objects change hands.

The median mug owner would not sell for less than $5.25, while the median buyer would not pay more than $2.25–$2.75. Median selling prices were about twice median buying prices.

A token is worth whatever the experimenter says it is worth; a mug is worth whatever its owner decides. The endowment effect shows up for the mugs and pens, not for the tokens.

Seventy-seven students at Simon Fraser University were randomly split into three groups and asked, at each price from $0.25 to $9.25, what they would do.

Sellers and Choosers face the same choice—leave with the mug or with the money—yet Choosers price the mug like Buyers. The low volume of trade comes from owners' reluctance to part with the mug, not from buyers' reluctance to part with cash.

Owning does not make the object more attractive: students given a pen did not rate pens as more attractive than the others did, yet 56% of them kept the pen over two chocolate bars, against 24% of those not given one.

Let good $X$ be a mug, $x\in\set{0,1}$, and good $Y$ be money, with $u(y)=y$. Write $m=u(1)-u(0)\gt 0$ for the consumption utility of the mug.

• Seller—owns the mug, so the reference is $\underline x=1$. Selling at price $p$ is a loss of $m$ in $X$ and a gain of $p$ in $Y$: $$\begin{aligned}\Delta U&=-(1+L)\,m+(1+G)\,p\ge 0\\ \iff p&\ge\frac{1+L}{1+G}\,m=\mathrm{WTA}.\end{aligned}$$
• Buyer—has no mug, so the reference is $\underline x=0$. Buying at price $p$ is a gain of $m$ in $X$ and a loss of $p$ in $Y$: $$\begin{aligned}\Delta U&=(1+G)\,m-(1+L)\,p\ge 0\\ \iff p&\le\frac{1+G}{1+L}\,m=\mathrm{WTP}.\end{aligned}$$

Therefore $$\frac{\mathrm{WTA}}{\mathrm{WTP}}=\paren{\frac{1+L}{1+G}}^2\gt 1\iff L\gt G.$$

With the $\mu$ of the numerical example ($L=1$, $G=\tfrac12$), $\mathrm{WTA}/\mathrm{WTP}=\paren{2/\tfrac32}^2=\tfrac{16}{9}\approx 1.8$—close to the factor of about 2 in the mug markets. Without loss aversion ($L=G$) the gap vanishes: $\mathrm{WTA}=\mathrm{WTP}=m$. A Chooser gains either the mug or the money, so is indifferent at $p=m$, between the two—the same order as $\$2.87\lt\$3.12\lt\$7.12$.

Capuchins who had learned to trade tokens for food chose between two experimenters. Each experimenter first showed some apple pieces, then took a token and gave apple pieces—sometimes more, sometimes fewer than shown. Each cell below reads pieces shown → pieces given.

Within each experiment the two experimenters deliver the same payoffs; only the display differs. Standard theory gives no reason to favor either one.

• Reference dependence—in experiment 2 the second piece is a bonus from E1 but a penalty from E2.
• Loss aversion—a sure loss (experiment 3) repels more than a half chance of a loss plus a half chance of a gain (experiment 2): 79% against 71%.

The displayed apple sets what the monkey expects—the third kind of reference point above—and getting less than that is coded as a loss. In the mug markets the reference is what one owns; here it is what one was shown.

The reference basket $r=(\bar x,\bar y)$ splits the good space into four regions. Click a region to reveal the reference-dependent consumer's $\mathrm{MRS}$ there.

A consumer looks for the optimal basket $(x,y)$ on the budget line $p_X\,x+p_Y\,y=I$.

• $\tfrac{1}{p_X}$ is the amount of good $X$ one dollar buys.
• Spending that dollar on $X$ raises utility by about $U_X\cdot\tfrac{1}{p_X}$—the marginal utility per dollar of $X$.
• Likewise, one dollar spent on $Y$ raises utility by about $U_Y\cdot\tfrac{1}{p_Y}$.
• If $U_X\cdot\tfrac{1}{p_X}\gt U_Y\cdot\tfrac{1}{p_Y}$, the consumer should take one dollar away from good $Y$ and spend it on good $X$: utility rises.
• If $U_X\cdot\tfrac{1}{p_X}\lt U_Y\cdot\tfrac{1}{p_Y}$, shift one dollar the other way, from $X$ to $Y$.
• The consumer keeps shifting dollars until no move raises utility.

That happens in one of two ways:

• Smooth interior optimum—the marginal utilities per dollar are equal: $$\frac{U_X}{p_X}=\frac{U_Y}{p_Y}\quad\Longleftrightarrow\quad \mathrm{MRS}=\frac{U_X}{U_Y}=\frac{p_X}{p_Y}.$$
• At a kink—where the optimal basket has one good at its reference level ($x=\bar x$ or $y=\bar y$), so the indifference curve bends—the $\mathrm{MRS}$ is not a single number: it takes a one-sided value on each side. The consumer stops as soon as the price ratio lies between them, $$\mathrm{MRS}^{-}\;\ge\;\frac{p_X}{p_Y}\;\ge\;\mathrm{MRS}^{+},$$ because from a kink, shifting a dollar either way lowers the marginal utility gained. Here $\mathrm{MRS}^{+}$ is the slope along the indifference curve when the consumer increases the quantity of $X$ and decreases the quantity of $Y$, while $\mathrm{MRS}^{-}$ is the slope when the consumer decreases $X$ and increases $Y$.

Consider a traditional ($T$) consumer with utility $u_T(x,y)=\sqrt{x}+\sqrt{y}$, a consumer ($R$) with reference dependent utility, $$u_R(x,y\mid 20,80)=\sqrt{x}+\sqrt{y}+\mu(\sqrt{x}-\sqrt{20})+\mu(\sqrt{y}-\sqrt{80}),$$ where:

• $\mu(\Delta u) = \min (\Delta u,0)+\tfrac12\cdot \max (\Delta u,0)$.
• $(20,80)$ is the reference basket.

We assume that both consumers' incomes are $I_T=I_R=120$ and prices initially are $p_X=2$ and $p_Y=1$.

At these prices, the optimal choice for these consumers is to choose the basket $(20,80)$, which (by chance) coincides with the reference basket of the $R$ consumer.

Now suppose the price of $X$ rises to $p_X=3$ (with $p_Y=1$ and $I=120$ unchanged), so the budget line pivots about its $Y$-intercept $(0,120)$.

• The traditional consumer moves to $(10,90)$—substituting toward the now relatively cheaper good $Y$.
• The reference-dependent consumer moves to $(\tfrac{40}{3},80)\approx(13.3,80)$: it keeps $Y$ at its reference level $80$—dropping below the reference would hurt—so the whole adjustment falls on $X$.

Budget lines before ($2x+y=120$, solid) and after ($3x+y=120$, dashed) the price change, with the indifference curves through the initial basket $A=(20,80)$ and the final basket $B=(10,90)$.

The same budget change. The indifference curves are kinked along the reference lines $x=\bar x$ and $y=\bar y$ (dotted). The consumer moves from $A=(20,80)$ to $B\approx(13.3,80)$, staying on the $y=\bar y$ kink line—$Y$ pinned at its reference level.