arxiv: 2605.09136 · v1 · submitted 2026-05-09 · 💰 econ.TH

Recognition: 2 theorem links

· Lean Theorem

On the Possibility of Informationally Inefficient Markets Without Noise

Mattthijs Breugem

Pith reviewed 2026-05-12 01:50 UTC · model grok-4.3

classification 💰 econ.TH

keywords rational expectations equilibriumprice informativenessinformation aggregationCRRA utilityGrossman-Stiglitz paradoxnoise-free markets

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The pith

Non-exponential preferences cause only partial information revelation through prices even without noise traders.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that rational investors with any utility other than constant absolute risk aversion will leave some of their private signals unrevealed in the equilibrium price. Market clearing aggregates demands in a space that differs from the Bayesian sufficient statistic, so Jensen's inequality creates a gap and prices stay only partially informative. This mechanism requires no noise traders, supply shocks, or behavioral biases and applies directly to standard CRRA preferences. The finding shows why information still carries positive value in equilibrium and why full revelation is not the generic outcome.

Core claim

Under any non-exponential expected utility preference, including CRRA, the rational expectations equilibrium price only partially reveals the aggregate private information because market clearing aggregates signals differently from the Bayesian sufficient statistic; the resulting Jensen gap prevents full revelation. CARA is the unique preference class that produces linear demand in log-odds and therefore full revelation. The equilibrium is characterized by a contour integration fixed point, and numerical solutions for CRRA confirm that partial revelation, positive trade volume, and positive value of information all survive learning from prices.

What carries the argument

The mismatch between the space in which market clearing aggregates individual demands and the space of the Bayesian sufficient statistic for the asset payoff.

If this is right

Information acquisition retains strictly positive value in equilibrium.
Equilibrium trade volume stays positive with zero exogenous noise.
Partial revelation continues to hold when agents update beliefs from the observed price.
Only constant absolute risk aversion preferences produce fully revealing prices.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Preference heterogeneity by itself can sustain persistent information asymmetry in otherwise rational markets.
Changes in effective risk aversion, for instance through regulation or contract design, could alter how much information prices convey.
Empirical work could test whether cross-sectional variation in measured risk aversion predicts differences in price informativeness across assets.

Load-bearing premise

That market clearing aggregates signals in a space distinct enough from the Bayesian sufficient statistic to produce a material Jensen gap for non-CARA utilities.

What would settle it

A closed-form or numerical demonstration that the rational expectations price is a sufficient statistic for the full vector of private signals under CRRA preferences would falsify the partial-revelation claim.

Figures

Figures reproduced from arXiv: 2605.09136 by Mattthijs Breugem.

**Figure 1.** Figure 1: The knife-edge: revelation deficit 1 − R2 versus signal precision τ in the no-learning equilibrium, for γ ∈ {0.5, 1, 4}. Every finite γ produces strictly positive partial revelation; only CARA (γ = ∞, dash-dotted at zero) achieves full revelation. K = 3, W = 1, G = 15. defined by (12). This section formulates the REE as a fixed point on the price function P[i, j, l], develops a contour-integration solution… view at source ↗

**Figure 2.** Figure 2: Price-function contours in the agent-1 slice at [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** Figure 3: Revelation deficit 1 − R2 at the converged REE. Panel (a): versus signal precision τ for three values of γ. Panel (b): versus γ at τ = 2, comparing REE (circles, G = 20) with the no-learning benchmark (squares, G = 20). The REE deficit exceeds the no-learning deficit at every γ tested: learning from prices amplifies partial revelation. CARA is zero in both panels. 5 Economic Implications This section trans… view at source ↗

**Figure 4.** Figure 4: Price as a function of T ⋆ at the converged REE (G=14, γ =0.5, τ =2). The red solid curve is the CRRA REE price function; the blue dashed curve is the no-learning benchmark; the dotted line is the fully-revealing p = Λ(T ⋆ ). The REE price is closer to full revelation than the no-learning price, but remains distinctly below the FR line: learning from prices narrows the gap but does not close it. 5.1 Trade … view at source ↗

**Figure 5.** Figure 5: Converged REE posteriors and price as functions of [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗

**Figure 6.** Figure 6: Aggregate trade volume Vvol(γ, τ=2) at the converged REE, computed at five values of CRRA risk aversion γ ∈ {0.1, 0.25, 0.5, 1.4, 2.0} (markers); the curve is a monotonic cubic interpolation. Volume is strictly positive everywhere in (0, ∞) and collapses to zero in the CARA limit (γ → ∞). K = 3, Wk = 1, no noise traders. the value V (τ ) exceeds c. Under full revelation, VCARA(τ ) = 0 < c, so no agent acqu… view at source ↗

**Figure 7.** Figure 7: Value of information Vinfo as a function of the fraction λ of informed agents in the market, computed at the no-learning REE for four values of CRRA risk aversion. Vinfo is the certainty-equivalent welfare difference between an informed and an uninformed agent at the equilibrium price. For every finite γ, the value of information is strictly positive across the entire informed/uninformed mix; for CARA (dot… view at source ↗

**Figure 8.** Figure 8: Net value of acquiring a precision-τ signal: V (τ ) − c as a function of the acquisition cost c, at τ = 2. Under CARA (dash-dotted), V = 0, so the net value is −c < 0 for every c > 0: no agent acquires (the paradox). Under CRRA, the net value is positive for c < c⋆ = V (τ ), resolving the paradox without noise traders. The threshold c ⋆ is larger at lower γ. Mechanism 3: heterogeneous signal precision. If … view at source ↗

**Figure 9.** Figure 9: Decomposition of the revelation deficit across the mechanisms tabulated in Table 2. [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗

**Figure 10.** Figure 10: Robustness of the knife-edge. Panel (a): the deficit declines roughly as 1 [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗

**Figure 11.** Figure 11: Convergence of the posterior-function fixed point at [PITH_FULL_IMAGE:figures/full_fig_p041_11.png] view at source ↗

read the original abstract

Noise traders can be dispensed with entirely. Partial revelation of information through prices arises under any non-exponential expected utility preference, including CRRA, without noise traders, random endowments, supply shocks, hedging motives, or behavioral biases. The model contains zero exogenous noise. The mechanism is a mismatch between the space in which market clearing aggregates signals and the Bayesian sufficient statistic. CARA demand is linear in log-odds, so prices aggregate in log-odds space and reveal the statistic exactly. Every other preference aggregates differently; the resulting Jensen gap makes revelation partial. I prove that CARA is the unique fully revealing preference class, characterize the rational expectations equilibrium via a contour integration fixed point, and verify that partial revelation survives learning from prices. The Grossman-Stiglitz paradox is resolved: information acquisition has positive value within the rational class. Numerical solution of the rational expectations fixed point at K = 3 confirms partial revelation, positive trade volume, and positive value of information across the full range of CRRA risk aversion, vanishing only in the CARA limit.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper claims non-CARA preferences alone produce partial revelation without noise by creating an aggregation mismatch with the Bayesian statistic, and it pins CARA as the unique full-revelation case.

read the letter

The core claim is that rational markets can stay informationally inefficient purely because non-exponential utilities aggregate signals differently from the log-odds space that CARA uses. Market clearing therefore leaves a Jensen gap, so prices reveal only part of the information. The author proves CARA is the only class that avoids this gap and sets up the REE as the fixed point of a contour-integral operator that maps candidate price functions back to demand aggregation. Numerical solution at K=3 then shows the fixed point stays partial for CRRA utilities, with positive trade volume and positive value of information that disappears only in the CARA limit. That is the actual new piece: a preference-based resolution of the Grossman-Stiglitz paradox that requires zero exogenous noise, random endowments, or behavioral assumptions. The contour-integral characterization is a neat technical move for handling the fixed-point condition cleanly. The numerical check across risk-aversion values is useful as far as it goes and confirms the mechanism does not collapse immediately. The soft spot is the narrow verification. Only K=3 is solved numerically, and the abstract gives no error bounds or checks on whether the operator remains contractive or the analytic continuation holds for larger signal counts or other parameter ranges. If the fixed point either fails to exist or snaps to full revelation outside that single case, the partial-revelation result shrinks. The stress-test concern about robustness is therefore on target until more cases or a general existence argument is shown. This is for theorists who work on rational expectations equilibria and information aggregation in asset markets. Anyone already comfortable with contour integration and fixed-point arguments in REE models will get the most out of it. The paper deserves a serious referee because it directly attacks a classic open question with a new mechanism and supplies both a proof sketch and concrete numbers. I would send it out for review, but the referees will need to press on the stability of the fixed point beyond K=3.

Referee Report

2 major / 2 minor

Summary. The paper claims that partially revealing rational expectations equilibria (REE) can exist in markets with zero exogenous noise when agents have non-CARA vNM preferences (e.g., CRRA). Market clearing aggregates signals in a space that differs from the Bayesian sufficient statistic, producing a Jensen gap that prevents full revelation except for the CARA class, which is proved unique. The REE is characterized as the fixed point of a contour-integration operator on candidate price functions; partial revelation, positive trade volume, and positive value of information are verified numerically for CRRA at K=3 signals across risk-aversion values, with the result vanishing only in the CARA limit.

Significance. If the fixed-point characterization and uniqueness proof hold, the result supplies a preference-based mechanism for informational inefficiency and resolves the Grossman-Stiglitz paradox without invoking noise traders, random endowments, or behavioral biases. The contour-integration approach and numerical confirmation for CRRA utilities would constitute a technically novel contribution to the theory of REE.

major comments (2)

The existence and stability of the contour-integration fixed point for non-CARA utilities is load-bearing for the central claim of partial revelation under any non-exponential preference. The abstract asserts analytic characterization and that the fixed point lies strictly inside the non-revealing set, yet supplies only a numerical solution at K=3; without an analytic proof of existence (e.g., via contractivity of the operator or verification that the required analytic continuation exists for general K and risk-aversion parameters), the result may fail to generalize and the partial-revelation conclusion collapses.
The uniqueness proof that CARA is the only fully revealing class rests on showing that only linear aggregation in log-odds space coincides with the Bayesian sufficient statistic. The manuscript must explicitly rule out other non-CARA utilities for which the demand aggregation happens to produce a zero Jensen gap (or for which the contour-integral operator admits the fully revealing map as a fixed point), as this would constitute a counterexample to the claimed uniqueness.

minor comments (2)

The abstract states that partial revelation 'survives learning from prices,' but the precise updating rule and the sense in which the fixed point remains stable under iterative learning should be stated more explicitly.
Numerical results are reported only for K=3; a brief table or figure showing sensitivity to higher K (or to the CRRA coefficient) would strengthen the verification without altering the main argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below. We agree that additional analytic support for the fixed-point characterization would strengthen the paper and will revise accordingly, while maintaining that the existing numerical evidence and uniqueness argument are robust for the cases considered.

read point-by-point responses

Referee: The existence and stability of the contour-integration fixed point for non-CARA utilities is load-bearing for the central claim of partial revelation under any non-exponential preference. The abstract asserts analytic characterization and that the fixed point lies strictly inside the non-revealing set, yet supplies only a numerical solution at K=3; without an analytic proof of existence (e.g., via contractivity of the operator or verification that the required analytic continuation exists for general K and risk-aversion parameters), the result may fail to generalize and the partial-revelation conclusion collapses.

Authors: We appreciate the referee's emphasis on this foundational aspect. The manuscript analytically characterizes the REE as the fixed point of the contour-integration operator and establishes partial revelation numerically for K=3 across CRRA risk-aversion values. We agree that a general analytic proof of existence and stability (including contractivity or analytic continuation for arbitrary K) is not supplied and would be valuable. In revision, we will add a dedicated subsection analyzing the operator's mapping properties on suitable function spaces, provide conditions under which contractivity holds for non-CARA utilities, and explicitly bound the scope of the K=3 numerical verification while noting that generalization remains an open direction. revision: partial
Referee: The uniqueness proof that CARA is the only fully revealing class rests on showing that only linear aggregation in log-odds space coincides with the Bayesian sufficient statistic. The manuscript must explicitly rule out other non-CARA utilities for which the demand aggregation happens to produce a zero Jensen gap (or for which the contour-integral operator admits the fully revealing map as a fixed point), as this would constitute a counterexample to the claimed uniqueness.

Authors: We agree that the uniqueness result benefits from explicit exclusion of potential counterexamples. The proof shows that full revelation requires the market-clearing aggregation to recover the Bayesian sufficient statistic exactly, which occurs if and only if individual demands are linear in log-odds—a property that holds solely for the CARA class. For non-CARA utilities the resulting nonlinearity generates a nonzero Jensen gap. In the revision we will strengthen the uniqueness theorem with an additional lemma that directly rules out other vNM preference classes capable of producing a zero gap or admitting the fully revealing price as a fixed point of the contour operator, by deriving the necessary functional form of demand from the zero-gap condition. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation follows from model primitives and fixed-point characterization

full rationale

The paper establishes that CARA utility yields linear demand in log-odds space, so market clearing aggregates exactly to the Bayesian sufficient statistic, while non-CARA preferences (including CRRA) induce a Jensen gap due to nonlinear aggregation. It proves uniqueness of the CARA class for full revelation and characterizes the REE as the fixed point of a contour-integral operator derived directly from the market-clearing condition and demand functions. The K=3 numerical solution verifies existence and partial revelation for CRRA but does not constitute the source of the claim; the result is obtained by solving the model's own equations rather than by fitting parameters or renaming inputs. No self-citations appear load-bearing, and no step reduces the central partial-revelation result to a pre-defined quantity by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model rests on standard expected-utility maximization and rational expectations equilibrium concepts drawn from prior literature, plus a new aggregation mismatch mechanism; the numerical checks introduce a discrete signal count K=3 as a modeling choice.

free parameters (1)

CRRA risk aversion coefficient
The paper reports results across the full range but performs numerical verification at specific values; this parameter is varied rather than fitted to external data.

axioms (2)

domain assumption Traders maximize expected utility of wealth
Core modeling assumption stated in the abstract for all preference classes considered.
domain assumption Rational expectations equilibrium exists and is characterized by a fixed-point condition
Invoked to define the equilibrium price function via contour integration.

pith-pipeline@v0.9.0 · 5476 in / 1466 out tokens · 59101 ms · 2026-05-12T01:50:14.441374+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
Theorem 1 (CARA uniqueness for log-odds-linear demand). ... Suppose the optimal demand admits ... x⋆(μ,p,W)=[logitμ−logitp]/α(p,W). Then U is CARA ... U(W)=a−b exp(−α0 W)
IndisputableMonolith/Cost.lean Jcost_pos_of_ne_one / convexity properties echoes
the probability average of a logistic transform is not the logistic transform of the average ... Jensen gap ... is the size of the revelation failure

Reference graph

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