Rationality with preference discovery costs

Abstract

Economic theory assumes that preferences are rational. However, experiments have found small violations of transitivity. This paper develops a model of rationality with preference discovery costs. Introspection is costly. Thus, agents may find it optimal to use less than full effort, even though this raises the risk of making a poor choice. This model could potentially explain the intransitivities observed in the data while retaining rationality and optimization.

Appendix: Convergence of effort

Let $N_\mathrm{s}$ be the number of signals the agent receives. Only a few harmless assumptions are required to show convergence: as $N_\mathrm{s}$ approaches infinity, $D( {\theta ,p,w} )$ approaches $D( {\theta ^{*},p,w} )$.

Assumption 1

$\theta ^{*}$ is unique and dwells in a compact subset of $R^{n}$.

Assumption 2

For all $p,w,d\ge 0$, the agent’s initial beliefs $f( {y,p,w,d,\theta } )>0$ when $\theta =\theta ^{*}$.

Assumption 3

There exists a valid instrumental variable for d that is in the agent’s information set.

Before going through the mathematical proof, some intuition is in order. An alternative way to update beliefs is by using nonlinear least squares with instrumental variables (NLLS-IV). We instrument for d because it is endogenous; the agent selects d optimally given his beliefs $\theta$. So if there is an instrument for d, the agent can obtain consistent estimates for $\theta$ using NLLS-IV. Thus, ${\lim }_{N_\mathrm{s} \rightarrow \infty } \theta _{\mathrm{NLLS}\_\mathrm{IV}} =\theta ^{*}$. This implies that ${\lim }_{N_\mathrm{s} \rightarrow \infty } D( {\theta _{\mathrm{NLLS}\_\mathrm{IV}} ,p,w} )=D(\theta ^{*},p,w)$.

By definition of optimality, the optimal beliefs are no worse than any other beliefs. Therefore, if effort under the non-optimal beliefs $\theta _{\mathrm{NLLS}\_\mathrm{IV}}$ converge, then effort under the optimal beliefs will also converge. Now for the math.

Let $\hat{\theta }$ denote the optimal beliefs. The beliefs under NLLS-IV are $\theta _\mathrm{NL}$. The information set I augmented with $N_\mathrm{s}$ signals is $I_{N_\mathrm{s}}$. Since $\theta ^{*}$ is the true $\theta$, if you knew $\theta ^{*}$, then other beliefs $\theta$ would be irrelevant for assessing the expected value of the utility gap. Hence, $\int \nolimits _0^\infty y \,f(y|p,w,d, \theta ^{*},\theta _\mathrm{NL} ,\hat{\theta } ,I_{N_\mathrm{s} } ) \mathrm{d}y=\int \nolimits _0^\infty y f(y|p,w,d, \theta ^{*}) \mathrm{d}y$. First, consider the problem faced by an agent who knows $\theta ^{*}$.

$$\begin{aligned} {\mathop {\max }\limits _{\{ d \}}}\, E[u( {x( {p,w,d} )} )|p,w,d,\theta ^{*},\theta _\mathrm{NL} ,\hat{\theta }, I_{N_\mathrm{s} } ]-v( d ) \end{aligned}$$

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This is equivalent to the following.

$$\begin{aligned} {\mathop {\min }\limits _{\{ d \}}}\, E[ {y( {p,w,d} )|p,w,d,\theta ^{*},\theta _\mathrm{NL} , \hat{\theta }, I_{N_\mathrm{s} } } ]+v( d ) \end{aligned}$$

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Maximizing utility is equivalent to minimizing the utility gap plus the disutility of effort. The problem above is the same as

$$\begin{aligned} {\mathop {\min }\limits _{\{ d \}}}\,E[ {y( {p,w,d} )|p,w,d,\theta ^{*}} ]+v( d ). \end{aligned}$$

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The solution is $d=D(\theta ^{*},p,w)$. Since $D(\theta ^{*},p,w)$ is optimal given p, w,  and $\theta ^{*}$, we know that for any $\theta _0$,

$$\begin{aligned}&E[ {y( {p,w,D(\theta ^{*},p,w)} )|p,w,d,\theta ^{*}} ]+v( {D(\theta ^{*},p,w)} )\nonumber \\&\quad \le E[ {y( {p,w,D( {\theta _0 ,p,w} )} )|p,w,d,\theta ^{*}} ]+v( {D( {\theta _0 ,p,w} )} ). \end{aligned}$$

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One such $\theta _0$ is $\theta _\mathrm{NL}$. Since the NLLS-IV estimator is consistent, ${\lim }_{N_\mathrm{s} \rightarrow \infty } \theta _\mathrm{NL} =\theta ^{*}$. Thus, ${\lim }_{N_\mathrm{s} \rightarrow \infty } D( {\theta _\mathrm{NL} ,p,w} )=D(\theta ^{*},p,w)$ and ${\mathop {\hbox {lim}}\nolimits _{N_\mathrm{s} \rightarrow \infty }} E[ y( {p,w,D( {\theta _\mathrm{NL} ,p,w} )} )|p,w,d,I_{N_\mathrm{s} } ]+v( {D( {\theta _\mathrm{NL} ,p,w} )} )=E[ {y( {p,w,D( {\theta ^{*},p,w} )} )|p,w,d,\theta ^{*}} ]+v( {D( {\theta ^{*},p,w} )} )$. Another possible $\theta _0$ is the optimal beliefs $\hat{\theta }$. Therefore, we know

$$\begin{aligned}&{\mathop {\lim }\limits _{N_\mathrm{s} \rightarrow \infty }}\, E[ {y( {p,w,D( {\theta _\mathrm{NL} ,p,w} )} )|p,w,d,I_{N_\mathrm{s} } } ]+v( {D( {\theta _\mathrm{NL} ,p,w} )} )\nonumber \\&\quad \le {\mathop {\lim }\limits _{N_\mathrm{s} \rightarrow \infty }} E[ {y( {p,w,D( {\hat{\theta },p,w} )} )|p,w,d,I_{N_\mathrm{s} } } ]+v( {D( {\hat{\theta },p,w} )} ). \end{aligned}$$

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However, because $\hat{\theta }$ is the optimal beliefs, we know that for all $N_\mathrm{s}$,

$$\begin{aligned}&E[ {y( {p,w,D( {\theta _\mathrm{NL} ,p,w} )} )|p,w,d,I_{N_\mathrm{s} } } ]+v( {D( {\theta _\mathrm{NL} ,p,w} )} )\nonumber \\&\quad \ge E[ {y( {p,w,D( {\hat{\theta },p,w} )} )|p,w,d,I_{N_\mathrm{s} } } ]+v( {D( {\hat{\theta },p,w} )} ). \end{aligned}$$

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Together, the two equations above imply

$$\begin{aligned}&{\mathop {\hbox {lim}}\limits _{N_\mathrm{s} \rightarrow \infty }}\, E[ {y( {p,w,D( {\theta _\mathrm{NL} ,p,w} )} )|p,w,d,I_{N_\mathrm{s} } } ]+v( {D( {\theta _\mathrm{NL} ,p,w} )} )\nonumber \\&\quad ={\mathop {\hbox {lim}}\limits _{N_\mathrm{s} \rightarrow \infty }} E[ {y( {p,w,D( {\hat{\theta } ,p,w} )} )|p,w,d,I_{N_\mathrm{s} } } ]+v( {D( {\hat{\theta },p,w} )} ). \end{aligned}$$

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Since ${\mathop {\hbox {lim}}\nolimits _{N_\mathrm{s} \rightarrow \infty }} E[ {y( {p,w,D( {\theta _\mathrm{NL} ,p,w} )} )|p,w,d,I_{N_\mathrm{s} } } ]+v( {D( {\theta _\mathrm{NL} ,p,w} )} )=E[ {y( {p,w,D( {\theta ^{*},p,w} )} )|p,w,d,\theta ^{*}} ]+v( {D( {\theta ^{*},p,w} )} )$,

$$\begin{aligned}&{\mathop {\lim }\limits _{N_\mathrm{s} \rightarrow \infty }}\, E[ {y( {p,w,D( {\hat{\theta },p,w} )} )|p,w,d,I_{N_\mathrm{s} } } ]+v( {D( {\hat{\theta } ,p,w} )} )\nonumber \\&\quad ={\mathop {\lim }\limits _{N_\mathrm{s} \rightarrow \infty }} E[ {y( {p,w,D( {\hat{\theta } ,p,w} )} )|p,w,d,\theta ^{*}} ]+v( {D( {\hat{\theta },p,w} )} )\nonumber \\&\quad =E[ {y( {p,w,D( {\theta ^{*},p,w} )} )|p,w,d,\theta ^{*}} ]+v( {D( {\theta ^{*},p,w} )} ). \end{aligned}$$

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Therefore, ${\mathop {\hbox {lim}}\nolimits _{N_\mathrm{s} \rightarrow \infty }} D( {\hat{\theta },p,w} ) \epsilon D( {\theta ^{*},p,w} )$. Note that there is no requirement that $D( {\theta ^{*},p,w} )$ has to be a function; more generally, it can be a nonempty-valued correspondence.