we-solve (beta)

Econ-641 · economics · Economics

Stability of reference dependence

LLM status Open unverified

Statement

Original identifier: B2 (atlas item 62). Classification: Behavioral identification and stability research program. Original tractability label: Medium (editorial, not a complexity result).

Definitions and assumptions

For individual $i$, domain $d$ in a specified finite collection, and occasion $t$, let $x$ be a scalar consumption outcome expressed in a specified domain-specific utility scale. Let $r_{idt}$ denote a deterministic reference point and consider the illustrative utility \[ U_{idt}(x)=u_d(x)+\eta_{id}\,v_{\lambda_{id}}(x-r_{idt}),\quad v_\lambda(z)=\begin{cases}z,&z\geq0,\\ \lambda z,&z<0,\end{cases} \] where $\eta_{id}\geq0$ and $\lambda_{id}\geq1$. Specify $u_d$, choice noise, information sets, and scale normalization. Expectations-based reference distributions require integration of gain--loss utility over that distribution; they cannot generally be replaced by its mean.

Formal research question

Given longitudinal choices and randomized changes to status quo, elicited expectations, or goals, identify the reference map $r_{idt}=g_d(H_{it},Z_{idt};\gamma_d)$ and the joint distribution of $(\eta_{id},\lambda_{id})$. Here $H_{it}$ is observed history and $Z_{idt}$ is randomized reference manipulation. Characterize the experimental variation and exclusions sufficient to distinguish changes in $g_d$ from changes in curvature or choice noise. For a declared equivalence tolerance $\varepsilon>0$, assess whether cross-domain normalized parameters satisfy \[ \Pr\!\left(\max_{d,d'}|\lambda_{id}-\lambda_{id'}|\leq\varepsilon\right)\geq1-\alpha, \] where $\alpha\in(0,1)$ is a specified population exception rate.

Interpretation and scope

Reference dependence is a family of models, not a uniquely identified primitive inferred from any observed endowment effect. Monetary and nonmonetary domains need comparable normalizations before coefficient equality has content. Randomly assigning a possession may change information, transaction costs, or beliefs about quality, violating an exclusion that it changes only the reference point. Designs should elicit expectations before choice, avoid treating endogenous expectations as instruments, and allow reference points to adapt over time. Stability of individual rankings is weaker than stability of cardinal parameters. An empirical contribution could produce a confidence set for the fraction of individuals satisfying the displayed tolerance condition, or demonstrate observational equivalence under existing protocols. The model shown here intentionally isolates reference dependence; it is not the complete prospect-theory treatment of probability weighting.

Source audit and status

The three references are verified foundational models: [S1] addresses risky choice, [S2] riskless reference dependence, and [S3] expectations-based reference points. Their distinct constructions should not be collapsed into one formula. They establish useful representations and predictions, not a universal invariant loss-aversion coefficient. The formal stability and identification targets above are editorial extensions; the audit does not claim that these old publications certify their open status as of 2026.

References

    },leftmargin=*]

  1. Daniel Kahneman and Amos Tversky (1979). Prospect Theory: An Analysis of Decision under Risk. Econometrica 47(2), 263--291. https://doi.org/10.2307/1914185

    Source role: Foundational descriptive model. Locator: Author bibliography; original model of risky choice.

  2. Amos Tversky and Daniel Kahneman (1991). Loss Aversion in Riskless Choice: A Reference-Dependent Model. Quarterly Journal of Economics 106(4), 1039--1061. https://doi.org/10.2307/2937956

    Source role: Model of reference-dependent riskless choice. Locator: Abstract and article metadata.

  3. Botond Kőszegi and Matthew Rabin (2006). A Model of Reference-Dependent Preferences. Quarterly Journal of Economics 121(4), 1133--1165. https://doi.org/10.1093/qje/121.4.1133

    Source role: Endogenous expectations-based reference point. Locator: Abstract and article metadata.

Common conventions: Source mathematical conventions

Unless an entry states otherwise, agent and firm sets are finite. State and action spaces are measurable, strategies and outcome maps are measurable, probability laws are specified, and expectations exist for the targets under discussion. Infinite-horizon discount factors lie in $(0,1)$ and discounted objectives are finite. Norms, tolerances and complexity input sizes must be specified for computational comparisons. Constants or primitives that appear locally are inputs, not empirical facts asserted by this catalogue.

Identification. A statistical model $m\in\mathcal M$ implies an observable law $P_m$ and target $\theta(m)$. At law $P$, its identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Point identification holds when this set is a singleton, or equivalently when $P_{m_1}=P_{m_2}$ implies $\theta(m_1)=\theta(m_2)$ for all compatible models. Sharp bounds mean bounds attained, or approached when the boundary is not attained, by compatible models. Writing this set is a definition, not a solution: the substantive task is to establish informative restrictions and derive or estimate the set. An unrestricted-confounding model may give only trivial bounds.

Inference. Identification, estimability and valid uncertainty quantification are separate questions. Fix $\alpha\in(0,1)$. A confidence procedure $C_n$ over a declared law class $\mathcal P$ has asymptotic uniform coverage at level $1-\alpha$ when \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P}\Pr_P\{\theta(P)\in C_n\}\geq1-\alpha. \] For set-identified targets the coverage event must be defined separately, for example coverage of the full identified set. If an entry uses identification-indexed models instead of uniquely indexed laws, the same coverage convention is applied over those models. Pointwise limits do not establish uniform validity, and asymptotic validity does not guarantee finite-sample precision. Sample sizes, dependence structures and weak-identification sequences are part of each application.

Counterfactuals. $Y_i(a)$ is a potential outcome under a specified intervention, including its timing, implementation and versions. It is not $Y_i$ conditional on voluntarily choosing $a$. A full intervention vector permits interference; shorthand scalar treatment notation does not silently impose no interference. Assignment, selection, attrition, anticipation and measurement must be specified. A claim about an instrument's local effect is not automatically a population policy effect.

Economic equilibrium. A counterfactual model includes best responses, constraints, feasibility, market clearing, state transitions and expectations consistent with the stated information. When several equilibria exist, either a declared selector is an input or the target is set-valued. Comparisons across policy regimes must use the stated selection convention. Reduced-form relationships are not presumed invariant after an equilibrium-changing intervention.

Optimization and welfare. Optimization is over the stated feasible set. If attainment is not established, $\sup$ or $\inf$ and $\varepsilon$-optimal choices replace an asserted maximizer or minimizer. For example, with value $V=\sup_{a\in\mathcal A}W(a)<\infty$, an $\varepsilon$-optimal set is $\{a\in\mathcal A:W(a)\geq V-\varepsilon\}$ for $\varepsilon>0$. Benefits, costs, risk and health or environmental outcomes can be aggregated only using declared units and conversion weights. Interpersonal utility weights, the treatment of fiscal transfers and foreign profits, discounting, and the behavioral welfare criterion are explicit inputs. A comparative-static derivative additionally requires existence and appropriate differentiability of the selected counterfactual.

Sources. Local labels [S1], [S2], and so forth restart in every question. Locators identify the relevant abstract, model, section or result at the level actually checked; a bibliographic or abstract-level verification is not represented as inspection of an entire proof. Working papers are distinguished from later publications and changing versions. Source summaries are paraphrased. All URL checks and editorial formulations refer to the audit date stated above.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-641-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-641

Stability of reference dependence: research attempt

Target and status

The catalogue asks whether reference dependence, especially loss aversion, is stable across contexts and time. Its model is \[ U_{i,d,t}(x)=u_d(x)+\eta_{i,d}v_{\lambda_{i,d}}(x-r_{i,d,t}), \qquad v_\lambda(z)=\begin{cases}z,&z\geq0,\\ \lambda z,&z<0,\end{cases} \] with $\eta\geq0$, $\lambda\geq1$, and a reference map $r=g(H,Z;\gamma)$ built from histories and randomized manipulations. Utility normalization, baseline curvature, and choice noise are part of the identification assumptions. A population stability statement is of the form \[ {\mathbb{P}}\left(\max_{d,d'}|\lambda_{i,d}-\lambda_{i,d'}|\leq\varepsilon\right) \geq1-\alpha. \] This attempt derives conditional identification and a rigorous interval-based stability assessment. It cannot establish actual stability without longitudinal choice data and credible manipulations.

What a kink experiment can identify

Consider one individual and domain, normalize $u(x)=x$, and assume the reference $r$ is fixed during a block. If choices are deterministic between certain monetary amounts, every admissible model chooses the larger amount: $U$ is strictly increasing on both sides of $r$. Such observations identify neither $\eta$ nor $\lambda$. One needs informative lotteries, tradeoffs with other goods, or a calibrated stochastic choice model.

For a concrete identification calculation, suppose a binary comparison of $x$ to a fixed anchor $y$ follows \[ {\mathbb{P}}(x\text{ chosen over }y) =\frac{\exp((U(x)-U(y))/s)}{1+\exp((U(x)-U(y))/s)},\quad s>0. \tag{1} \] Assume the probability curve is observed over an interval straddling $r$, and $s$ is common across those comparisons. Its log odds $L(x)$ have slopes \[ m_+=\frac{1+\eta}{s}\quad(x>r),\qquad m_-=\frac{1+\eta\lambda}{s}\quad(x<r). \tag{2} \] If $s$ is known and $\eta>0$, they identify \[ \eta=s m_+-1,\qquad \lambda=\frac{s m_--1}{s m_+-1}. \tag{3} \] The reference is the unique slope-change location if $\eta(\lambda-1)>0$. This follows directly from the piecewise-affine formula for $U$; a fixed anchor contributes only an intercept. In a model with a known smooth baseline $u$, subtract its known derivative before applying the same argument.

If $s$ is unknown, the slope ratio identifies only \[ \frac{m_-}{m_+}=\frac{1+\eta\lambda}{1+\eta}. \] For every $s>1/m_+$, equation (3) supplies an observationally equivalent pair when $m_->m_+$. For example, observed slopes $(m_+,m_-)=(2,3)$ can arise from $(s,\eta,\lambda)=(1,1,2)$ or $(2,3,5/3)$. These generate the same entire log-odds difference curve for comparisons within this piecewise-linear model, because their normalized utility slopes and kink agree. Thus apparent changes in a fitted loss-aversion coefficient can reflect changes in noise or gain--loss weight unless their scales are fixed.

At $\eta=0$, $\lambda$ has no effect at all. At $\lambda=1$, the reference enters utility only as the additive constant $-\eta r$ and cancels from within-block comparisons. If all observed $x,y$ lie above $r$, neither the loss slope nor the reference location is identified. Furthermore, \[ \frac{\partial\lambda}{\partial m_-}=\frac{s}{s m_+-1}=\frac{s}{\eta} \] in (3), displaying weak identification as the gain--loss weight approaches zero. Precise choice probabilities need not produce a precise estimate of $\lambda$ in that region.

Identifying a reference location in each block identifies $\gamma$ only if distinct admissible parameter values give distinct reference maps on the observed support. Randomization of $Z$ supplies variation, but not this injectivity condition. Nor does it establish exclusion: changing a status quo can change information, transaction costs, or perceived quality in addition to the reference.

A distribution of references is not its mean

Suppose ${\mathbb{E}}|R|<\infty$ and the person integrates the random reference $R$ before the choice shock is realized, so the following evaluated utility is finite: \[ \overline U(x)=x+\eta{\mathbb{E}}[v_\lambda(x-R)]. \tag{4} \] This assumption differs from mixing logit probabilities over unobserved realized references. With a continuous reference distribution, bounded differentiation gives \[ \overline U'(x)=1+\eta\{1+(\lambda-1)[1-F_R(x)]\}. \tag{5} \] If a density exists, $\overline U''(x)=-\eta(\lambda-1)f_R(x)$. Hence dispersed references can smooth a sharp individual kink into apparent curvature. When $\eta(\lambda-1)>0$ is known, the distribution can formally be recovered from a calibrated derivative curve as \[ F_R(x)=1-\frac{\overline U'(x)-1-\eta}{\eta(\lambda-1)}. \] An unknown baseline curvature would confound this recovery. At atoms, use one-sided derivatives; the slope jumps encode their masses.

For a direct check, let $R=-1$ or $1$ with equal probability. Its mean is zero, but at $x=0$, \[ {\mathbb{E}} v_\lambda(-R)=\frac{1-\lambda}{2}\ne v_\lambda(0)=0 \] when $\lambda>1$. This is not merely an irrelevant utility-level shift: between $x=0$ and $x=1/2$, expected gain--loss utility increases by $(1+\lambda)/4$, whereas replacing the reference by its mean gives $1/2$. With $\lambda=2$, these increments are $3/4$ and $1/2$. Thus the catalogue's random-reference variant requires the distributional integral or a justified special case permitting reduction to a point reference.

Sharp stability bounds from individual parameter intervals

Suppose there are $D\geq2$ domains, and for each sampled individual there are simultaneous intervals $\lambda_{i,d}\in[l_{i,d},u_{i,d}]$. They may come from a maintained stochastic-choice model or from partial identification. Define \[ L_i=\max\{0,\max_d l_{i,d}-\min_d u_{i,d}\},\qquad U_i=\max_{d\ne d'}(u_{i,d}-l_{i,d'})_+, \] where $x_+=\max(x,0)$. The maximum pairwise distance equals the range of the domain parameters.
Proposition: Over the Cartesian product of these intervals, the smallest possible parameter range is $L_i$ and the largest is $U_i$. Consequently the fraction stable within $\varepsilon\geq0$ among $n$ sampled individuals lies in the sharp interval \[ \left[\frac1n\sum_i\mathbf1\{U_i\leq\varepsilon\},\quad \frac1n\sum_i\mathbf1\{L_i\leq\varepsilon\}\right]. \tag{6} \]
Proof: Any chosen parameters include one at least $\max_d l_{i,d}$ and one at most $\min_d u_{i,d}$, proving the lower bound on range. If the intervals share a point, choose it in every domain and obtain zero. Otherwise write $a=\min_d u_{i,d}<b=\max_d l_{i,d}$. Every interval intersects $[a,b]$, and selecting in these intersections attains range $b-a$: a domain with upper endpoint $a$ and one with lower endpoint $b$ force the endpoints. These are distinct domains because each interval has ordered endpoints.

Any difference between parameters from distinct domains is bounded above by the corresponding upper-minus-lower endpoints. Selecting endpoints in a maximizing pair attains $U_i$. Therefore every admissible vector is stable exactly when $U_i\leq\varepsilon$, and at least one is stable exactly when $L_i\leq\varepsilon$. When $L_i\leq\varepsilon<U_i$, both stable and unstable vectors exist. Choices can be made independently across individuals, attaining both fractions in (6). ∎

The distinct-domain restriction in $U_i$ matters: using $\max_d u_{i,d}-\min_d l_{i,d}$ can overstate the largest feasible range when both extremes belong to the same domain. For instance, intervals $[1,10]$ and $[5,6]$ have maximum range five, not nine. Extra joint restrictions across domains or people can narrow (6); the bounds remain valid outer bounds but need not then be sharp.

To translate to a population claim, assume individuals are sampled independently and identically, and the simultaneous interval coverage event has probability at least $1-\delta_m$. For a chosen $\delta_s>0$, set \[ h_n=\sqrt{\frac{\log(2/\delta_s)}{2n}}. \] Hoeffding's inequality bounds the difference between the empirical fraction of truly stable individuals and its population probability by $h_n$ with probability at least $1-\delta_s$. Combining this event with interval coverage gives the population interval \[ \left[\max\{0,\widehat p_L-h_n\},\ \min\{1,\widehat p_U+h_n\}\right] \tag{7} \] with coverage at least $1-\delta_m-\delta_s$, where the endpoints in (6) are $\widehat p_L,\widehat p_U$. No independence between coverage and sampling events is needed for the union bound. A lower endpoint at least $1-\alpha$ supports the specified stability criterion under these assumptions. The population exception allowance $\alpha$ is different from the statistical error probabilities $\delta_m,\delta_s$.

Completion Estimate

50%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt derives conditional kink and random-reference identification, exhibits noise-scale and weak-identification failures and gives sharp interval-based bounds for cross-domain stability. Joint individual parameter and reference-process identification from finite longitudinal experiments still requires credible exclusions, comparable scales and empirical coverage procedures for the needed intervals.

Sources and unresolved identification

The inspected primary paper Tversky and Kahneman, Loss Aversion in Riskless Choice: A Reference-Dependent Model (1991), motivates separating reference dependence from ordinary wealth curvature: https://www.sscnet.ucla.edu/polisci/faculty/chwe/austen/tversky1991.pdf. The author-hosted bibliographic record for Kőszegi and Rabin, A Model of Reference-Dependent Preferences (2006), is https://rabin.scholars.harvard.edu/publications/model-reference-dependent-preferences. The calculations above concern the explicitly stated catalogue model and do not claim to establish every feature of either source's framework.

The remaining obstacle is credible joint identification across domains: finitely many binary choices do not by themselves recover an unrestricted distribution of individual parameters and reference processes. Equations (3)--(7) explain which normalizations, experimental variation, and uncertainty bounds would support a stability claim. They provide no empirical evidence that the claim is true, and the full problem remains unresolved.

Go to contribution →