Sunk ‘Decision Points’: a theory of the endowment effect and present bias

Abstract

This paper presents a very simple model in which situational cues associated with a particular consumption good compel an agent—who may have otherwise been “thinking about” something else—to consider the decision to consume that good. Within this framework, it is shown how an endowment effect and a present bias can arise through a common mechanism. The analysis points to a novel, contributing role for inattention (of a particular form) in understanding both of these behavioral anomalies while also speaking to evidence that they are often cue-induced.

A Appendix

1.1 A.1 Generalization to durable goods

Now, assume that the good is durable. Then, the benchmark expected lifetime utility and the expected lifetime utility with a cue at t are unchanged: ${\overline{U}}=\sum _{t=0}^{\infty }\delta ^t(1-\pi )=\frac{1-\pi }{1-\delta }$, $U[t]={\overline{U}}-(1-\pi )\delta ^t$. The expected lifetime utility for the endowment $e(\tau )$ is now $U[e(\tau )]={\overline{U}}+\sum _{k=0}^{\infty }\pi \delta ^{\tau +k}(1-\pi )^k e={\overline{U}}+\frac{e\pi \delta ^{\tau }}{1-\delta (1-\pi )}$, where $\pi \cdot (1-\delta (1-\pi ))^{-1}$ is the expected excess discount from the time of the endowment to the time of its eventual consumption (i.e., at the first cue on or after $\tau$).

For a scenario that involves a cue at t and an endowment $e(\tau )$, we have

$$\begin{aligned} {U[t;e(\tau )]\;=\;\left\{ \begin{array}{ccl} {\overline{U}}-(1-\pi )\delta ^t+e \delta ^{\tau }&{}&{}\text {if}\ \,t=\tau ,\\ {\overline{U}}-(1-\pi )\delta ^t+{\displaystyle \frac{e\pi \delta ^{\tau }}{1-\delta (1-\pi )}}&{}\;\;\;\;&{}\text {if}\ \,t\ne \tau . \end{array}\right. } \end{aligned}$$

It is readily verifiable that Proposition 1, part (i) of Proposition 2, Corollary 1, and Propositions 3–6 do not need to be changed to accommodate durable goods because they hold exactly as they did for perishable goods. For part (ii) of Proposition 2, the result holds except we need to modify the measured present bias factor for non-cueing goods, as it is now given by $\beta =\pi (1-\delta (1-\pi ))^{-1}<1$. This expression can be derived from the indifference condition $U[0;e(\tau )]=U[0;D_\tau e(0)]$ using the above expression for $U[t;e(\tau )]$.$\square$

1.2 A.2 Generalization to cue persistence

Now, assume cues induce decision points “tomorrow,” i.e., a cue in t induces a decision point in $t+1$. This extension is nontrivial only if today’s decision point remains uncertain, so for simplicity, assume that a cue in t does not affect the decision point probability in t.

In this case, $U[0]={\overline{U}}-\delta (1-\pi )$. Therefore, as with Proposition 1, there is still a WTA–WTP disparity, except now $v^A(e)-v^P(e)={\overline{U}}-U[0]=\delta (1-\pi )$.

To show that part (i) of Proposition 2 carries over, use

$$\begin{aligned} U[0,\tau ;e(\tau )]= & {} {\overline{U}}-\delta (1-\pi )(1+\delta ^{\tau })+(\pi +\delta (1-\pi ))\delta ^{\tau }e,\\ U[0;D_{\tau }e(0)]= & {} {\overline{U}}-\delta (1-\pi )+(\pi +\delta (1-\pi ))D_{\tau } e. \end{aligned}$$

Solving for the elicited discount function from the indifference condition gives $D_\tau =\beta \delta ^{\tau }$, where the elicited present bias for received goods is now

$$\begin{aligned} {\beta =\left( 1-\frac{\delta (1-\pi )}{(\pi +\delta (1-\pi ))e}\right) <1.} \end{aligned}$$

Here, $\beta$ is increasing in e, which establishes the magnitude effect from Corollary 1.

For part (ii) of Proposition 2, we assume $\tau \ge 2$ to avoid complications from interference between the decision point induced by elicitation and the future endowment’s acquisition. Using the above expressions, we again get $D_\tau =\beta \delta ^{\tau }$, except for non-cueing goods, we have

$$\begin{aligned} {\beta =\frac{\pi }{\pi +\delta (1-\delta )(1-\pi )^2}<1.} \end{aligned}$$

For Proposition 3, as before, we have $v^A(e)=v^P(e)=U[0;(a+e)(0)]-U[0;a(0)]$, so that the auxiliary good still eliminates the WTA–WTP disparity.

For Proposition 4, the indifference condition used to solve for the elicited discount function is $U[0,\tau ;a(0),(e+a)(\tau )]=U[0,\tau ;(a+D_{\tau }e)(0),a(\tau )]$, where

$$\begin{aligned} U[0,\tau ;a(0),(e+a)(\tau )]= & {} \,{\overline{U}}-\delta (1-\pi )(1+\delta ^{\tau })+(a+(e+a)\delta ^{\tau })(\pi +\delta (1-\pi )),\\ U[0,\tau ;(a+D_{\tau }e)(0),a(\tau )]= & {} \,{\overline{U}}-\delta (1-\pi )(1+\delta ^{\tau })+(a+D_{\tau }e+a\delta ^{\tau })(\pi +\delta (1-\pi )). \end{aligned}$$

This implies $D_\tau =\delta ^\tau$, as desired.

Noting that the elicited present bias factor for received goods in this setting is given by $\beta =\delta (1-\pi )[(\pi +\delta (1-\pi ))e]^{-1}$, Proposition 5 still holds, because

$$\begin{aligned} {\frac{v^P(e)}{v^A(e)}=\frac{U[0;e(0)]-{\overline{U}}}{U[0;e(0)]-U[0]}=1-\frac{\delta (1-\pi )}{(\pi +\delta (1-\pi ))e}.} \end{aligned}$$

Proposition 6 still holds because, as before, $v^P(e)=-v^C(e)$.

1.3 A.3 Time inconsistency

As previewed in Sect. 4.2, the next result establishes necessary and sufficient conditions for time-inconsistent preferences. In doing so, we see that preferences can only shift from a later option to an earlier option when the earlier option becomes available.

Proposition 7

Suppose preferences are elicited at $t=0$ and again at $t_0$, with $t_1>t_0>0$:

(i) If $e_1(t_1)$ is strictly preferred to $e_0(t_0)$ at time-zero elicitation, then $e_0(t_0)$ will become strictly preferred to $e_1(t_1)$ at time-$t_0$ elicitation if and only if $(e_0-\delta ^{t_1-t_0}e_1)/(1-\pi )\in S$, where $S\equiv (-\delta ^{t_1-t_0},1-\delta ^{t_1-t_0})$ for received goods and $S\equiv (-e_1,0)$ for non-cueing goods.

(ii) If $e_0(t_0)$ is strictly preferred to $e_1(t_1)$ at time-zero elicitation, then $e_0(t_0)$ will remain strictly preferred to $e_1(t_1)$ at time-$t_0$ elicitation.

Proof

Let $x=\frac{e_0-\delta ^{t_1-t_0}e_1}{1-\pi }$. For received goods, the time-zero indifference condition, $U[0,t_1;e_1(t_1)]=U[0,t_0;e_0(t_0)]$, reduces to $x=1-\delta ^{t_1-t_0}$, implying $e_0(t_0)$ is strictly preferred if $x>1-\delta ^{t_1-t_0}$ and $e_1(t_1)$ is strictly preferred if $x<1-\delta ^{t_1-t_0}$; the time-$t_0$ indifference condition, $U[t_0,t_1;e_1(t_1)]=U[t_0;e_0(t_0)]$, reduces to $x=-\delta ^{t_1-t_0}$, implying $e_0(t_0)$ is strictly preferred if $x>-\delta ^{t_1-t_0}$ and $e_1(t_1)$ is strictly preferred if $x<-\delta ^{t_1-t_0}$. For non-cueing goods, the time-zero indifference condition, $U[0;e_1(t_1)]=U[0;e_0(t_0)]$, reduces to $x=0$, implying $e_0(t_0)$ is strictly preferred if $x>0$ and $e_1(t_1)$ is strictly preferred if $x<0$; the time-$t_0$ indifference condition, $U[t_0;e_1(t_1)]=U[t_0;e_0(t_0)]$, reduces to $x=-e_1$, implying $e_0(t_0)$ is strictly preferred if $x>-e_1$ and $e_1(t_1)$ is strictly preferred if $x<-e_1$. Parts (i) and (ii) then follow directly from the strict preference conditions derived above. $\square$