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Econ-476 · economics · Economics

General equilibrium with limited higher-order beliefs

LLM status Open unverified

Statement

Original subject group: Economic theory, equilibrium and social choice

Audit classification: Published research agenda. The chapter explicitly proposes empirical belief elicitation in §6.2. The treatment design and identification problem are editorial; attenuation is already a source result.

Definitions and precise research target

Model and research target. Let a population of agents $i\in[0,1]$ choose scalar actions in a linear coordination economy, \[ a_i=\theta_i+\alpha E_i[\bar a],\qquad \bar a=\int_0^1a_i\,di,\quad0<\alpha<1. \] Here $\theta_i$ is the payoff-relevant fundamental and $E_i$ denotes the agent's subjective conditional expectation. Under full information and consistent common beliefs, a common additive shock to fundamentals has aggregate multiplier $M_0=(1-\alpha)^{-1}$. Let $Z$ be a randomized public-information treatment about such a shock, and let $b_i^{(k)}$ denote elicited beliefs about other agents' beliefs at order $k$. Define $M_Z=\partial E[\bar a\mid do(\theta+\eta),Z]/\partial\eta$ at $\eta=0$. The research target is to identify how $M_Z/M_0$ varies with belief precision, reasoning depth, and their population distributions. Establish conditions under which a finite collection of action observations and incentivized belief elicitations separately identifies incomplete information and truncated reasoning, or derive sharp observational equivalence sets when it cannot. Estimate the multiplier and its uncertainty under randomized information and stable structural payoffs, and evaluate predictions on untreated economies.

Definitions and maintained assumptions (editorial unless a source definition is named). Use a probability space $(\Omega,\mathcal F,\mathbb P)$, jointly measurable agents $i\in[0,1]$, information sigma-fields $\mathcal I_i$, and square-integrable fundamentals. Let $E_i$ be conditional expectation under a specified subjective probability $\mathbb P_i$; state absolute-continuity and integrability restrictions so these expectations and their cross-agent integrals exist. The equation $a_i=\theta_i+\alpha E_i[\bar a]$ defines an equilibrium only after the information and belief operators have been fixed. Common-prior Bayesian behavior and finite-level reasoning are competing model classes, not identical interpretations of that equation. Define recursively $b_i^{(1)}=E_i[\theta]$ for a common shock and $b_i^{(k+1)}=E_i[\int_0^1 b_j^{(k)}\,dj]$; measured reports additionally pass through a specified error kernel. Replace the informal intervention symbol by an experimentally specified family $\mathbb P_{\eta,Z}$, holding payoffs fixed, and set $M_Z=\left.\frac{d}{d\eta}\mathbb E_{\eta,Z}[\bar a]\right|_{\eta=0}$, when the derivative exists. Point identification means any two maintained models with the same joint law of actions and reports yield the same multiplier.

Variants and assumption changes

  1. Finite levels (editorial). Fix a maximum level $K$, a specified level-zero forecast, and a distribution on $\{0,\ldots,K\}$. Ask which randomized signals and elicited orders identify the level distribution separately from report error and fundamental uncertainty.
  2. Bayesian information (editorial). Fix Gaussian common-prior fundamentals and a family of signal precisions, preserving unlimited reasoning. Compare its observable laws with the finite-level class, reporting the intersection and the multiplier range on that intersection rather than assuming the two mechanisms are distinguishable.

References and source audit

  1. §6.2 printed p.27; §7 pp.28–31. Supports the stated research area; exact displayed specification is editorial. https://economics.mit.edu/sites/default/files/publications/handbook-GE-NBER.pdf
  1. George-Marios Angeletos; Chen Lian. 2022. Dampening General Equilibrium: Incomplete Information and Bounded Rationality. §6.2, printed p.27; §7 Discussion, pp.28–31. https://economics.mit.edu/sites/default/files/publications/handbook-GE-NBER.pdf. DOI: https://doi.org/10.3386/w29776.

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-476-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-476

Which multiplier is dampened? Common-prior accounting and identification of reasoning depth

Problem ID: D84-472. Status: Unresolved; rigorous restricted results.

Question and an intervention distinction

The target seeks to distinguish imprecise information from limited reasoning using actions, belief reports, and randomized information. We establish two results. First, a correctly specified common-prior model does not attenuate the unconditional response to a commonly understood location shift of fundamentals. Second, conditional responses to a realized shock do attenuate, and a finite-level Gaussian class admits sharp depth-distribution sets and an explicit identification design. This is a restricted analysis of the coordination economy, not a complete general-equilibrium welfare model.

Angeletos and Lian [al], Section 6.2, printed p.27, explicitly discuss eliciting beliefs to discriminate different level-one specifications. Their Section 7 addresses broader interpretation. We use that agenda, without attributing the experimental design or the propositions below to their chapter.

An unconditional invariance result

Let agents $i\in[0,1]$ have information $\mathcal I_i$ on a common probability space with common prior $P$. Assume every conditional-expectation field below has a jointly measurable version. For a square-integrable field $x_i$, define \[ \bar x=\int_0^1x_i\,di,\qquad (Tx)_i={\mathbb{E}}[\bar x\mid\mathcal I_i]. \] Use the norm $\|x\|^2=\int_0^1{\mathbb{E}}[x_i^2]di$, identifying fields almost everywhere. Assume these fields form a closed subspace preserved by $T$. This explicit regularity condition avoids an unjustified exact law of large numbers for an arbitrary continuum of independent random variables.
Theorem (Existence and mean accounting): If $0<\alpha<1$ and $\theta$ belongs to the specified closed square-integrable field space, the equation $a=\theta+\alpha Ta$ has a unique square-integrable solution. It is \[ a=\sum_{j=0}^{\infty}\alpha^jT^j\theta, \qquad {\mathbb{E}}\bar a=\frac{{\mathbb{E}}\bar\theta}{1-\alpha}. \tag{1} \] For any intervention family $(P_{\eta,Z},\theta_{\eta,Z},T_{\eta,Z})$ satisfying these assumptions separately at every $(\eta,Z)$, with beliefs given by the corresponding common prior and $\alpha$ fixed, \[ {\mathbb{E}}_{\eta,Z}\bar\theta_{\eta,Z}=\mu_Z+\eta \quad\Longrightarrow\quad \frac{d}{d\eta}{\mathbb{E}}_{\eta,Z}\bar a_{\eta,Z} =\frac1{1-\alpha}. \tag{2} \] Information precision may depend arbitrarily on $Z$ or $\eta$.
Proof: Jensen's inequality for conditional expectations gives $\int{\mathbb{E}}[(Tx)_i^2]di\leq{\mathbb{E}}[\bar x^2]\leq\int{\mathbb{E}}[x_i^2]di$. Thus $\|T\|\leq1$. The displayed geometric series converges in the Hilbert norm, because the norm of its $j$th term is at most $\alpha^j\|\theta\|$. Applying the bounded operator $I-\alpha T$ to partial sums and taking limits proves the equation. Two solutions differ by $\alpha T$ of their difference, forcing their norm difference to be zero. Fubini and iterated expectations give ${\mathbb{E}}\int(Ta)_i di={\mathbb{E}}\bar a$. Integrating the equilibrium equation establishes (1). Applying this identity separately at each intervention value gives $(\mu_Z+\eta)/(1-\alpha)$ exactly; differentiation yields (2), without differentiating any belief operator. ∎

This result pinpoints a possible estimand mismatch. A randomized announcement about the realization of an initially uncertain shock is not generally a change in the entire common prior which agents correctly understand. Attenuation in a conditional response is consistent with (1). A claimed information-dependent unconditional multiplier must specify which feature of the theorem fails, for example fixed misspecified beliefs, a direct fundamental mean that changes with treatment, or a different structural payoff equation.

A conditional Gaussian benchmark

Let the common fundamental be $\Theta\sim N(0,\sigma_\theta^2)$, and private signals have marginal law $X_i=\Theta+\varepsilon_i$, with $\varepsilon_i\sim N(0,\sigma_\varepsilon^2)$ independent of $\Theta$. Assume an aggregation construction under which the relevant cross-sectional averages equal their conditional means given $\Theta$. This can alternatively be read as a conditional-population limit; we do not claim it on every ordinary continuum product space. Put \[ \lambda=\frac{\sigma_\theta^2}{\sigma_\theta^2+\sigma_\varepsilon^2}\in(0,1), \qquad {\mathbb{E}}[\Theta\mid X_i]=\lambda X_i. \] The payoff-relevant term in the observable best response is now the posterior mean $\lambda X_i$. This is the catalogue equation with $\theta_i$ set equal to that sufficient statistic, not the unobserved realization $\Theta$.

Unlimited Bayesian reasoning gives $a_i=\lambda X_i+\alpha{\mathbb{E}}[\bar a\mid X_i]$. Substituting a linear solution and applying the uniqueness argument from Theorem 1 yields \[ a_i=\frac{\lambda X_i}{1-\alpha\lambda},\qquad {\mathbb{E}}[\bar a\mid\Theta=\theta]=\frac{\lambda\theta}{1-\alpha\lambda}. \tag{3} \] Indeed, if $a_i=bX_i$, aggregate action is $b\Theta$, and its expectation given $X_i$ is $b\lambda X_i$; solving $b=\lambda+\alpha\lambda b$ verifies (3). The ratio of this conditional slope to full-information $M_0$ is $\lambda(1-\alpha)/(1-\alpha\lambda)<1$. The unconditional average remains zero. Under a correctly anticipated mean shift $\eta$, actions acquire the intercept $\eta/(1-\alpha)$ after centering signals and fundamentals, recovering (2).

Depth heterogeneity and sharp sets

Define a finite reasoning hierarchy explicitly. A depth-zero agent chooses $a_i^{(0)}=\lambda X_i$, forecasting zero strategic response. A depth-$k$ agent best responds to the hypothetical economy in which everyone uses the depth-$(k-1)$ rule. The actual depth shares $\mu_0,\ldots,\mu_K$ are independent of signals and the fundamental. This is a level model, not a Bayesian equilibrium in which agents correctly forecast the actual mixed-depth population.
Proposition (Depth coefficients): For every $k\leq K$, \[ a_i^{(k)}=v_k(\alpha,\lambda)X_i, \quad v_k(\alpha,\lambda)=\lambda\sum_{j=0}^k(\alpha\lambda)^j. \] The actual aggregate conditional multiplier is \[ B(\alpha,\lambda)=\sum_{k=0}^K\mu_kv_k(\alpha,\lambda) =\lambda\sum_{j=0}^K s_j(\alpha\lambda)^j, \quad s_j=\Pr(k\geq j),\quad s_0=1. \tag{4} \]
Proof: The formula holds at depth zero. If it holds at depth $k-1$, the hypothetical aggregate equals $v_{k-1}\Theta$. Its conditional expectation is $v_{k-1}\lambda X_i$, so $v_k=\lambda+\alpha\lambda v_{k-1}$. Iterating gives the finite sum. Averaging across independent depth types yields the first expression for $B$. Interchanging two finite sums gives the tail-probability representation. ∎

Suppose the observed summaries are aggregate conditional multipliers $B_t$ from treatments with known $(\alpha_t,\lambda_t)$, and first-order belief reports calibrated as $R_i=\lambda_t X_i+\xi_i$ with ${\mathbb{E}}[\xi_i\mid\Theta,t]=0$. Then ${\mathbb{E}}[R_i\mid\Theta,t]=\lambda_t\Theta$ identifies $\lambda_t$ whenever the support of $\Theta$ contains at least two points and its value is observed. Calibration and observation of the fundamental are substantive requirements. The following result deliberately concerns these aggregate summaries, not the richer law of individual actions: noiseless individual action-signal ratios can reveal depth directly.

Theorem (A sharp summary-experiment set and a separating design): For fixed known $K$ and invariant shares $\mu$, the sharp set given these summaries is the polytope \[ \mathcal U=\left\{\mu\geq0:\ \sum_k\mu_k=1,\quad \sum_k\mu_kv_k(\alpha_t,\lambda_t)=B_t\ \text{for every }t\right\}. \tag{5} \] At a new treatment $(\alpha_*,\lambda_*)$, the sharp multiplier interval consists of the minimum and maximum of $\sum_k\mu_kv_k(\alpha_*,\lambda_*)$ over $\mathcal U$. If $K\geq1$, $K$ treatments with distinct nonzero $x_t=\alpha_t\lambda_t$, together with known $\lambda_t$, uniquely identify all shares whenever the model is compatible.
Proof: Necessity of (5) follows from (4). Conversely any vector in (5) defines a population of depth types independent of the same Gaussian signals, with the stipulated calibrated reports, reproducing exactly all observed summaries. This proves sharpness for the stated observation scheme. The simplex is compact and the moment restrictions closed, so extrema of a linear new-treatment multiplier exist when $\mathcal U$ is nonempty. Every intermediate value is attained by a convex combination of extremizers, still in $\mathcal U$.

For identification, subtract one from $B_t/\lambda_t$ and divide by $x_t$: \[ \frac{B_t/\lambda_t-1}{x_t} =s_1+s_2x_t+\cdots+s_Kx_t^{K-1}. \] Two candidate polynomials of degree at most $K-1$ agreeing at $K$ distinct points are equal: their difference would otherwise have more roots than its degree, by repeated polynomial division. Thus all tails are determined. Recover $\mu_0=1-s_1$, $\mu_k=s_k-s_{k+1}$ for $1\leq k<K$, and $\mu_K=s_K$. For $K=0$ the share is already one. ∎

With only one multiplier and $K\geq2$, the polytope can have positive dimension. At an interior share vector, the two constraints $\sum\mu_k=1$ and $\sum\mu_kv_k=B$ leave a nonzero null direction because there are at least three coordinates. Small movements along it preserve positivity. This supplies actual observationally equivalent heterogeneous-depth models. Richer belief orders alone do not necessarily remove this ambiguity: in this Gaussian model truthful fundamental-belief hierarchies satisfy $b_i^{(j)}=\lambda^jX_i$ regardless of the action reasoning depth.

Limits and next empirical step

The identification design requires invariant depth shares across randomized signal precision or strategic-payoff treatments. If information complexity itself changes the shares, the common polynomial restriction is lost. Noisy reports with unrestricted treatment-specific bias can also absorb arbitrary changes in $\lambda$; random assignment alone does not calibrate them. A practical next step is to estimate the report error model in known-payoff calibration tasks and to test the monotone-tail inequalities $1\geq s_1\geq\cdots\geq s_K\geq0$ on held-out treatment cells. The paper establishes no optimal inference rate or external validity to an untreated macroeconomy. It does establish precisely which intervention could produce the proposed attenuation and which cannot.

Completion Estimate

65%

This is a post hoc, heuristic estimate for the full catalogue target. Common prior mean accounting settles the unconditional location shift multiplier, while a Gaussian finite level model yields sharp depth share sets and a separating summary experiment. Full action and report law equivalence, calibrated report errors, stable depth distributions, statistical uncertainty, and prediction in untreated economies remain unresolved.

References

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