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Econ-613 · economics · Economics

Motivated beliefs in economic decisions

LLM status Open unverified

Statement

Original subject group: Behavioral and experimental economics

Audit classification: Background; proposed extension. Publisher indexing and the authors’ research page verify the 2025 stock-return experiment. Full publisher text remained inaccessible; the general self-image, persistence and savings agenda is editorial.

Definitions and precise research target

Model and research target. Fix an economic decision with uncertain monetary return $R$ and objectively specified public information $I$. Let $Z_i\in\{0,1\}$ randomly assign an ownership or self-image treatment intended to change the desirability of a favorable return without changing available information. At horizon $t$, observe an incentivized subjective return forecast $B_i(Z_i,t)$ and an economic action $A_i(Z_i,t)$, such as a portfolio share. The belief and action effects in target population $P$ are \[\tau_B(t)=\mathbb E_P[B_i(1,t)-B_i(0,t)],\qquad \tau_A(t)=\mathbb E_P[A_i(1,t)-A_i(0,t)].\] Specify the information benchmark $b^*(I)=\mathbb E[R\mid I]$ and examine whether the treatment changes forecast errors relative to it. A motivation interpretation requires ruling out treatment-induced information, altered forecast incentives, or rational inferences about the experiment. To determine whether distorted beliefs cause the action effect, additionally randomize a credible belief-correction intervention or justify an explicit mediation model; correlation between $B_i$ and $A_i$ is insufficient. Estimate persistence over preregistered horizons and heterogeneity across initially winning and losing positions. The broader question is whether these identified mechanisms recur with economically important magnitudes in consequential saving and investment decisions, under comparable information and ownership conditions. It should not be treated as a theorem that all optimistic beliefs are motivated or that the stock-market findings already identify population prevalence.

Definitions and maintained assumptions (editorial unless a source definition is named). A full objective probability law for $(R,I)$ is needed to define $b^*(I)=\mathbb E[R\mid I]$; prices and publicly observed information alone do not provide an objectively known expected return in an actual stock market. In a controlled experiment choose that law explicitly, or call a field benchmark a maintained forecast model. Treatment $Z$ specifies ownership, endowment or self-image incentives and what it changes. Forecast elicitation has a fixed scoring rule and known stakes, since treatment may otherwise change forecast incentives. Potential beliefs and actions are measured at specified horizons before any treatment-dependent stopping or sale; selection into continued ownership must be handled explicitly. The belief-error effect is $\mathbb E[(B(1,t)-b^*(I))-(B(0,t)-b^*(I))]$ under unchanged information. A second intervention targeting beliefs identifies an action response only if its direct utility/information effects are excluded. Winning/losing subgroups should be defined by preassigned exogenous price paths, not treatment-induced portfolios.

Variants and assumption changes

  1. Known-return ownership (editorial). Randomize stock endowment under an announced finite return distribution and identical forecast payoffs. Estimate ownership effects on forecast error and sale decisions.
  2. Field persistence (editorial). Use comparable real equities and a preregistered forecasting benchmark over repeated horizons. Estimate treatment effects and correction responses with explicit attrition and information assumptions; a benchmark-dependent optimism measure is not proof of irrationality.

References and source audit

  1. Publisher-indexed and author abstract verify article/results; full publisher paper inaccessible. No exact broad open-question locator. https://doi.org/10.1016/j.jbankfin.2025.107510
  2. Publisher-indexed and author abstract verify article/results; full publisher paper inaccessible. No exact broad open-question locator. https://sites.google.com/site/carloscuevaherrero/home/research
  1. Carlos Cueva; Iñigo Iturbe-Ormaetxe. 2025. Motivated beliefs about stock returns. Abstract, corroborated by author research page https://sites.google.com/site/carloscuevaherrero/home/research. Full paper was not accessible in this audit.. https://doi.org/10.1016/j.jbankfin.2025.107510. DOI: https://doi.org/10.1016/j.jbankfin.2025.107510.
  2. Carlos Cueva. 2025. Author research page: Motivated beliefs about stock returns. Published, entry Motivated beliefs about stock returns. https://sites.google.com/site/carloscuevaherrero/home/research.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-613-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-613

Motivated return beliefs: distortion, elicitation and a correction experiment

Catalogue problem: D91-609. Status: Unresolved. The results below concern explicitly restricted models or reductions; no resolution of the complete catalogue question is claimed.

Question and scope

The catalogue asks whether randomized ownership or self-image incentives alter forecasts and economic actions under unchanged information, and whether a credible correction can isolate the belief channel. Cueva and Iturbe-Ormaetxe [cueva] report related stock-return experiments. Their author abstract was verified; no broader prevalence claim is inferred from it. We solve an explicit finite-state distortion model, show why identical monetary forecast rewards can still produce treatment-dependent reports, and characterize a factorial correction design.

An objective benchmark and an explicit distortion mechanism

The public experiment specifies a finite return distribution $p=(p_1,\ldots,p_J)$, with $p_j>0$, return values $r_j\in{\mathbb{R}}$, and at least two distinct values. Thus the objective mean $b^*=\sum_jp_jr_j$ is known by design. Ownership $Z\in\{0,1\}$ is randomly assigned independently of this law and does not reveal a signal.

A maintained psychological model lets the individual select a subjective distribution $q$ by maximizing \begin{equation}\label{wsref-0-65-71-3a-62-65-6c-69-65-66} \theta Z\sum_jq_jr_j-\kappa\sum_jq_j\log(q_j/p_j), \qquad q_j\geq0,\quad\sum_jq_j=1, \end{equation} where $\kappa>0$, $\theta\geq0$, and $0\log0=0$. The first term is anticipatory desirability of a favorable return; the relative-entropy term penalizes departure from the announced law. This is a behavioral primitive, not a consequence of Bayesian information processing. Choices are evaluated using the resulting beliefs. A forecast report is assumed truthful only under an elicitation protocol justified below.

Theorem (Unique exponential distortion and testable comparative statics): Problem (eq:belief) has unique solution \[ q_j(Z)=\frac{p_j\exp(\theta Zr_j/\kappa)} {\sum_\ell p_\ell\exp(\theta Zr_\ell/\kappa)}. \] Writing $b(t)=\sum_jq_j(t)r_j$ for the tilted mean at $t=\theta Z/\kappa$, one has \[ b(0)=b^*,\qquad b'(t)={\operatorname{Var}}_{q(t)}(R)>0. \] Therefore positive ownership motivation raises the subjective mean, and its effect decreases when $\kappa$ increases. For binary returns $0,1$, \[ \log\frac{b(Z)}{1-b(Z)}=\log\frac{p}{1-p}+\theta Z/\kappa. \]
Proof: Let $K(t)=\sum_jp_je^{tr_j}$ and $q_j^*=p_je^{tr_j}/K(t)$. For any distribution $q$, the objective is \[ \kappa\log K(t)-\kappa\sum_jq_j\log(q_j/q_j^*). \] Relative entropy is nonnegative, with equality only at $q=q^*$. One proof uses $\log x\leq x-1$ on $x=q_j^*/q_j$ for all positive $q_j$, then sums; zero coordinates are handled by continuity, and equality requires every ratio to equal one. Thus the displayed distribution uniquely maximizes. Differentiating the finite sum gives $b(t)=(\log K)'(t)$ and $b'(t)=(\log K)''(t)={\mathbb{E}}_qR^2-({\mathbb{E}}_qR)^2$. Strict positivity follows because every state retains positive mass and returns are not constant. The binary odds formula follows by dividing its two probabilities. ∎

The log-odds formula yields a narrow persistence implication. If homogeneous motivation follows $\theta_t=\theta_0\rho^t$, $\rho\in[0,1]$, with constant $\kappa$ and unchanged objective $p$, then the treated--control log-odds difference is $\theta_0\rho^t/\kappa$. Ratios at successive positive differences equal $\rho$. This implication concerns individual or homogeneous-type probabilities; taking a logit of heterogeneous mean forecasts does not preserve it.

Identical scoring rules need not give comparable reports

A proper score is ordinarily justified under a particular payoff criterion. Ownership can change that criterion by making forecast rewards correlated with other wealth.
Proposition (A hedging distortion with fully correct beliefs): Let objective and subjective beliefs both be Bernoulli $p\in(0,1)$. An individual with CARA utility $u(w)=-e^{-\gamma w}$, $\gamma>0$, holds a payoff $ZaR$, $a>0$, and reports $b\in[0,1]$. The monetary quadratic score is $s\{1-(b-R)^2\}$, $s>0$, and both the asset and score are paid. The unique optimal report lies in $(0,1)$ and solves \[ \log\frac{b}{1-b}+\gamma s(2b-1) =\log\frac p{1-p}-\gamma Za. \] Consequently ownership strictly lowers the optimal reported probability, even though the true subjective probability remains $p$.
Proof: In each state the score is strictly concave in $b$. Composing it with strictly increasing concave CARA utility preserves strict concavity, so expected utility has at most one optimum. Its derivative is positive at zero and negative at one, giving an interior maximizer. At the optimum, after dividing the common positive factor $2s\gamma$, \[ p(1-b)e^{-\gamma\{Za+s[1-(b-1)^2]\}} =(1-p)b e^{-\gamma s(1-b^2)}. \] Taking logs and rearranging gives the equation. Its left side is strictly increasing, with derivative $1/[b(1-b)]+2\gamma s>0$, and ranges from minus to plus infinity. Its right side strictly falls with ownership. The unique report therefore falls. This is a wealth-hedging effect, not motivated optimism. ∎

The sign in this example is pessimistic, which makes the diagnostic especially clear: a treatment effect on a scored report need not be a treatment effect on the underlying belief. Other state-dependent wealth or preferences can change its direction.

Lemma (A sufficient elicitation separation): Suppose payment randomly selects either the asset task or the forecasting task, independently of the return and report, with positive probability for each. When the forecasting task is selected, total wealth apart from the forecast prize is state-independent. A report $b\in[0,1]$ gives a fixed high prize rather than a fixed low prize with probability $1-(b-R)^2$, using an independent lottery. For any expected-utility preferences strictly favoring the high prize and subjective Bernoulli probability $q$, the unique optimal report is $b=q$, regardless of utility curvature and asset endowment in the unselected task.
Proof: Expected utility from the asset-payment branch does not depend on the report. In the forecasting branch it is \[ u_L+(u_H-u_L){\mathbb{E}}_q[1-(b-R)^2], \] with $u_H-u_L>0$ constant across return states. Maximizing is therefore equivalent to minimizing expected squared error. Its derivative is $2(b-q)$ and its second derivative is two, yielding the unique optimum $b=q$. ∎

The lemma states the required payment and utility separations; merely calling a score "incentivized" would not establish them. The exclusive-payment procedure may itself affect how psychologically consequential ownership feels, which must be measured rather than assumed away.

A factorial correction design and an exact decomposition

In addition to randomized ownership $Z$, randomize a correction $C\in\{0,1\}$ independently. For each individual let $b_i(z,c)$ be the subjective mean measured truthfully, and assume complete correction sets \[ b_i(0,1)=b_i(1,1)=b_i^*. \] The corrected benchmark may depend on preassigned public information but is the same across ownership. The correction changes no action payoff, endowment, risk attitude or ownership attachment other than through beliefs.

Let the action be the unique interior maximizer of \[ xb_i(z,c)+\eta_i z x-\gamma_i x^2/2, \qquad\gamma_i>0, \] on a declared action interval containing all four unconstrained optima. Assume the resulting actions and the quantities below are integrable. Here $\eta_i$ is a direct ownership motive separate from belief desirability, and $\gamma_i$ and $\eta_i$ do not change with correction. Heterogeneity may be arbitrarily correlated across these primitives.

Theorem (Correction identifies a weighted belief-channel effect): Define the randomized ownership action contrast in correction arm $c$ by $D_c={\mathbb{E}}[x_i(1,c)-x_i(0,c)]$. Then \[ D_1={\mathbb{E}}[\eta_i/\gamma_i],\qquad D_0-D_1={\mathbb{E}}\left[\frac{b_i(1,0)-b_i(0,0)}{\gamma_i}\right]. \] All terms on the left are identified from the four randomized arm means. With homogeneous $\gamma_i=\gamma$, the uncorrected ownership belief contrast $\tau_B$ satisfies $D_0-D_1=\tau_B/\gamma$. With heterogeneous curvature, division by a mean curvature generally gives the wrong expression.
Proof: Strict concavity gives $x_i(z,c)=\{b_i(z,c)+\eta_i z\}/\gamma_i$. Subtract the ownership actions within each correction arm. Complete correction makes its belief difference zero at $c=1$, leaving $\eta_i/\gamma_i$. Subtract this from the $c=0$ difference and average. Independent assignment and consistency identify each arm mean with the corresponding potential-action mean. The homogeneous formula follows by taking the constant inverse curvature outside expectation; no such factorization is available under arbitrary heterogeneous dependence. ∎

This is an intervention-specific decomposition under a stated action model. It is not a nonparametric natural indirect effect, and it does not follow just from a correlation between reports and actions. For example, if correction also adds a direct payoff term $\zeta_i ZC x$, the same calculation gives \[ D_0-D_1={\mathbb{E}}[(b_i(1,0)-b_i(0,0))/\gamma_i] -{\mathbb{E}}[\zeta_i/\gamma_i]. \] Thus, even with exact belief resetting, the exclusion restriction is essential. If $|{\mathbb{E}}[\zeta_i/\gamma_i]|\leq\varepsilon$ is externally defensible, the measured difference identifies the weighted belief contribution only within plus or minus $\varepsilon$.

Remaining gaps

The finite announced return law makes the information benchmark objective; a field stock-return benchmark instead depends on a maintained forecasting model. The entropy penalty, complete correction, interior quadratic actions and payment separation are restrictions requiring evidence. Attrition must be handled in the original randomized cohort, and winning or losing status must follow preassigned price paths. The persistence formula does not establish persistence in actual data. What is proved here is a coherent distortion mechanism, a concrete elicitation confound, and a correction-based decomposition with its exact exclusion failure. Population magnitudes and broader savings or investment implications remain unresolved.

Completion Estimate

54%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt derives an entropy-based motivated-belief mechanism, proves a scoring-induced hedging confound and an elicitation remedy, and identifies a weighted belief-channel contrast under correction exclusions. Population magnitudes and persistence still require validated belief measurement, credible complete correction, attrition control and field information benchmarks for consequential saving and investment.

References

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