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Econ-563 · economics · Economics

Inflation-expectation formation

LLM status Open unverified

Statement

Original subject group: Inflation and monetary policy

Audit classification: Explicit published open question. Formation and relevance of expectations are explicit historical questions; the randomized design is editorial. Verification concerns the cited version, not first priority or proof that this exact editorial specification remains unsolved on October 8, 2026.

Definitions and precise research target

Editorial mathematical specification.
For household $i$, let $m_{it}$ denote its subjective mean of next-year inflation, $p_{it}$ its personally experienced price change, $n_{it}$ the inflation information it receives, and $a_{it}$ measured attention. Specify the belief-transition model \[ m_{i,t+1}=G_\theta(m_{it},p_{it},n_{it},a_{it},h_{it})+\epsilon_{i,t+1}, \] where $h_{it}$ contains predetermined household characteristics, $\theta$ is a finite-dimensional parameter, and $\epsilon_{i,t+1}$ is a mean-zero innovation conditional on the stated information. The experimental design independently randomizes informative messages and attention incentives; personally experienced prices require a separate valid instrument or an explicit partial-identification strategy. Identify the causal effects of information content, exposure to personally salient prices, and attention on both the mean and dispersion of subjective inflation distributions. Test whether one estimated transition rule transports to different inflation regimes and communication formats, using held-out households and occasions. Distinguish persistent updating from temporary survey-response priming and reconcile forecasts at different horizons. To establish macroeconomic relevance, connect beliefs to actual consumption or pricing through independently identified outcome equations rather than assuming that survey revisions imply behavior. The target is an empirically disciplined formation mechanism and its policy responsiveness within a declared population; unrestricted household-specific updating rules do not constitute a solution.

Each household has a subjective probability measure $Q_{it}$ on next-year cumulative inflation; $m_{it}=\int x\,dQ_{it}(x)$ and $v_{it}=\int(x-m_{it})^2dQ_{it}(x)$, with finite second moments. Cross-sectional disagreement $\operatorname{Var}_i(m_{it})$ differs from average subjective uncertainty $E_i[v_{it}]$. Define own-price exposure by a fixed household consumption basket so that changes in basket weights do not silently alter treatment. Factorial assignment randomizes message content and attention incentives independently; an incentive is not the same as exogenously setting attention unless its only effect is through measured attention. Conditioning the innovation on information including randomized treatment identifies the transition only under stated restrictions on unobserved household effects and dynamic panel attrition.

Variants and assumption changes

The following are editorial variants, not additional published open conjectures.
  1. Persistent learning. Measure beliefs immediately and at several later horizons after the same message, with no repeated message treatment. Identify decay of the causal revision and distinguish learning from short-lived survey priming.
  2. Distributional updating. Model the full subjective density using a fixed parametric family, not only its mean. Estimate separately changes in disagreement and individual uncertainty and their effects on actual spending.

References and source audit

Formation and relevance of expectations are explicit historical questions; the randomized design is editorial.
  1. Janet L. Yellen. 2016. Macroeconomic Research After the Crisis. Section Inflation Dynamics. https://www.federalreserve.gov/newsevents/speech/yellen20161014a.htm.
  2. European Central Bank. Undated / date unverified. Forecasting and business cycle analysis. Understanding the inflation process, last selected area (expectation measurement and determinants). https://www.ecb.europa.eu/pub/economic-research/research_agenda/forecasting/html/index.en.html.

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-563-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-563

Inflation expectations: separating attention from updating and uncertainty from disagreement

Question and model

The question asks how information and attention change subjective inflation distributions, while distinguishing actual belief revision from survey priming and disagreement from subjective uncertainty. We solve a Gaussian-updating benchmark in which randomized signal precision supplies a concrete identification restriction. Yellen [yellen] and the ECB [ecb] motivate the broader formation agenda; neither is being credited with the experiment or proofs below.

Within a predetermined population stratum, household $i$ has prior belief $N(m_i,v)$ about a common inflation quantity, with $v>0$. A message reports a signal $n$ with a known Gaussian likelihood variance $t>0$. Independently assigned incentive $q$ changes an attention indicator $A_i\in\{0,1\}$, with $\Pr(A_i=1\mid q)=p_q$. Conditional on the stratum and incentive, attention is independent of the signal realization and prior mean. An attentive household applies Bayes' rule; an inattentive household retains its prior. Recorded posterior means may include reporting error with conditional mean zero given $(m_i,n,q,t)$.

Completing the square in the prior times likelihood gives, for an attentive household, posterior mean $m_i+k_t(n-m_i)$ and variance $v(1-k_t)$, where $k_t=v/(v+t)$. Thus the recorded revision has conditional mean \[ E[\Delta m_i\mid m_i,n,q,t]=b_{qt}(n-m_i),\qquad b_{qt}=p_q\frac{v}{v+t}. \tag{1} \] Estimating this slope requires nondegenerate variation in $n-m_i$. An incentive is not itself attention: (1) imposes the exclusion that it changes only the probability of processing the signal.

Theorem (Two precisions identify common prior variance and attention): Fix an incentive arm with $p_q>0$ and two known distinct precision variances $0<t_1<t_2$. Suppose the same $v$ and $p_q$ apply at both precisions and (1) holds. Then the two slopes identify \[ v=\frac{b_{qt_2}t_2-b_{qt_1}t_1}{b_{qt_1}-b_{qt_2}}, \qquad p_q=b_{qt_1}\frac{v+t_1}{v}. \tag{2} \] The compatible slopes satisfy $b_{qt_1}>b_{qt_2}>0$, the displayed $v>0$, and $0<p_q\le1$. Conversely, these conditions are sufficient to realize the slopes in this benchmark. One precision alone does not in general separate $p_q$ from $v$.
Proof: Multiplying (1) by $v+t_j$ yields $b_{qt_j}(v+t_j)=p_qv$. Subtraction gives the first formula, and substitution gives the second. Positive $v,p_q$ and $t_1<t_2$ imply the strict slope ordering. Conversely, if (2) gives permissible $v,p_q$, the defining equality for $v$ makes both products equal to $p_qv$, recovering (1). For one precision and any observed $b\in(0,1)$, choose any $v\ge bt/(1-b)$ and set $p=b(v+t)/v\le1$. Every such choice gives the same slope, proving nonuniqueness. ∎
Zero attention produces zero slopes and leaves $v$ unidentified by this experiment. Nearly equal slopes also make (2) unstable; division does not create precision. If prior variances and attention are accurately measured separately, (1) instead becomes a testable overidentifying restriction. Randomizing additional incentive arms then permits comparisons of $p_q$ while retaining the same prior-variance normalization.

Unobserved heterogeneity defeats the two-number inversion

Suppose three unobserved prior-variance types $v_1,v_2,v_3>0$ are distinct and each has population probability $1/3$. Give all types the same prior mean, and let the experimental signal deviation be independent of type with nonzero variance. Type $j$ attends with probability $3w_j$, where $0<w_j<1/3$, independently of that deviation conditional on type. Attention can therefore depend on type. The two aggregate slopes are \[ b_r=\sum_{j=1}^3 w_j\frac{v_j}{v_j+t_r},\qquad r=1,2, \] and average attention is $p=\sum_jw_j$.
Proposition (Identical aggregate updating, different average attention): For $t_1\ne t_2$, there exist two strictly interior attention-weight vectors with exactly the same two slopes but different average attention. Thus the homogeneous inversion cannot be applied to arbitrary variance mixtures.
Proof: Let $F$ be the two-by-three matrix with entries $v_j/(v_j+t_r)$. The three row vectors obtained by adding $(1,1,1)$ are linearly independent. Indeed, a dependence would make the rational function $a_0+a_1v/(v+t_1)+a_2v/(v+t_2)$ vanish at the three distinct positive $v_j$. After multiplying by $(v+t_1)(v+t_2)$, its numerator is a quadratic with three roots, hence identically zero. Its constant coefficient gives $a_0=0$; the two remaining coefficients give $a_1+a_2=0$ and $a_1t_2+a_2t_1=0$, hence $a_1=a_2=0$. Therefore there is $h\in\ker F$ with $\sum_jh_j\ne0$. Starting at $w_j=1/6$, choose a sufficiently small $\epsilon>0$ so that both $w+\epsilon h$ and $w-\epsilon h$ remain in $(0,1/3)^3$. Their images under $F$ agree, while their sums differ. The prescribed type and attention probabilities realize both models. ∎
This does not say richer data cannot identify heterogeneity. Elicited prior variances, accurate attention measures, more signal precisions, or restrictions on the type distribution may do so. It identifies the exact limitation of using only two aggregate slopes.

Disagreement can rise while individual uncertainty falls

For a fixed incentive/precision arm, let prior means $M$ have finite variance $D$ and mean $\mu$. Conditional on a fixed true inflation quantity $\theta$, let each household receive $N=\theta+\xi$, where $\xi\sim N(0,t)$ is independent of $M$ and of $A\sim\operatorname{Bernoulli}(p)$. Take the same subjective variance $v$ and weight $k=v/(v+t)$ across households. The true posterior mean and subjective posterior variance are \[ M'=(1-Ak)M+AkN,\qquad V'=v(1-Ak). \]
Theorem (Exact dispersion decomposition): Under these assumptions, \[ E[V']=v(1-pk), \] \[ \operatorname{Var}(M')=(1-p)D+p\{(1-k)^2D+k^2t\} +p(1-p)k^2(\theta-\mu)^2. \tag{3} \] In particular, if $D=0$ and $\mu=\theta$, then for $p>0$ individual uncertainty falls on average but disagreement rises from zero to $pk^2t>0$.
Proof: Average the variance formula directly. Conditional on $A=0$, the posterior mean has mean $\mu$ and variance $D$; conditional on $A=1$, it has mean $(1-k)\mu+k\theta$ and variance $(1-k)^2D+k^2t$ by independence. The law of total variance yields the first two weighted terms plus the variance of these two conditional means, namely $p(1-p)k^2(\theta-\mu)^2$. Substitution proves the final assertion. ∎
The identity $\operatorname{Var}(X)=E[V']+\operatorname{Var}(M')$ applies to a population mixture formed by first sampling a household and then drawing from its subjective posterior. This mixture variance is different again from either component. Survey noise adds yet another variance term unless separately measured. With two independent mean-zero elicitation errors conditional on the true mean, the covariance of two reports equals $\operatorname{Var}(M')$; this follows by expanding their covariance and using conditional mean-zero errors and conditional error independence.

Unresolved mechanisms

The precision experiment identifies attention only under common variances, stable processing probabilities, and truthful known signal likelihoods. Incentives can directly alter effort on survey questions, and content can change confidence through channels absent from (1). Persistent belief formation requires later elicitation with no renewed treatment; the current one-period model does not identify persistence. Personally salient price exposure needs its own causal intervention or instrument, and behavioral effects require observed consumption or pricing outcomes with separate identification. The calculations therefore provide restrictions and diagnostic counterexamples for D84-559, rather than a complete empirical formation model.

Completion Estimate

50%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt identifies homogeneous Gaussian attention updating from two signal precisions and derives heterogeneity obstructions and exact disagreement-versus-uncertainty formulas. Full inflation-expectation formation still requires causal own-price exposure, persistent learning versus survey priming, regime and format validation, and independently identified consumption or pricing responses.

References

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