PRIORITY FILING - INITIAL DRAFT
A fully revised and formatted version will follow.
Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
March 19, 2026
From the Prolegomena to the Unification of Economics, Computer Science, Mathematics, and Physics by Gregory Caldwell Beier (forthcoming 2026).
© 2026 Gregory Caldwell Beier. Licensed under CC BY-NC-ND 4.0.
Proposition 2 of the companion paper (Beier, 2026b) establishes that the augmented equilibrium (with the Disclosure Incentive) provides uninformed agents with the same total scalar precision as the standard Grossman-Stiglitz equilibrium, but with a different composition - a portion entering as structured public reasoning rather than through the noisy price signal. The exact precision equivalence is a consequence of the CARA-normal framework: Gaussian signals, exponential utility, and linear price functions produce a trader indifference condition whose functional form is invariant to the composition of public information. This paper asks what happens when these assumptions are relaxed. The central finding is that the exact equivalence is not robust - it is a Gaussian coincidence that breaks under non-Gaussian signals, non-CARA utility, and nonlinear price functions. But the compositional benefit IS robust: in every framework examined, the augmented equilibrium provides uninformed agents with qualitatively richer information (structured reasoning alongside the price) even when the total scalar precision differs. Three results formalize this. Proposition BE-1 shows that under non-Gaussian signals with excess kurtosis (fat tails), the augmented equilibrium provides strictly higher uninformed welfare than the standard equilibrium, because the disclosed signal carries information about tail events that the Gaussian-optimal price cannot. Proposition BE-2 shows that under ambiguity-averse preferences, the augmented equilibrium provides uninformed agents with strictly higher expected utility, because ambiguity-averse agents value the reduction in model uncertainty that disclosure provides beyond the scalar precision. Proposition BE-3 establishes the general principle: for any information structure in which the disclosed signal is not a garbling of the price signal (which it is not, because the disclosure preserves dimensions the price compresses away), the augmented equilibrium is strictly more Blackwell-informative for uninformed agents, and therefore weakly preferred under any decision problem.
Keywords: compositional improvement, Gaussian coincidence, non-Gaussian, ambiguity aversion, Blackwell informativeness, robust welfare, augmented equilibrium
JEL Codes: D82, D83, G14, C62
The companion paper (Beier, 2026b, Proposition 2) proves the following: in the CARA-normal augmented equilibrium, uninformed agents receive total precision tau_U = tau_v + tau_p(lambda_GS*) in the standard economy and tau_U = tau_v + lambda_D* * tau_epsilon + tau_p(lambda_T*) in the augmented economy, and the two are equal because the trader indifference condition forces tau_p(lambda_GS*) = lambda_D* * tau_epsilon + tau_p(lambda_T*).
The exact equality is a striking artifact of the Gaussian framework. It says the Disclosure Incentive doesn't increase the total information available to uninformed agents - it changes the composition while preserving the total. The compositional change is economically meaningful (structured reasoning versus noisy price inference), but the total scalar precision is unchanged.
This paper shows that the exact equality is a special property of the CARA-normal framework and identifies what replaces it in more general settings. The answer is that the total scalar precision generally changes (it can increase or decrease depending on the framework), but the compositional benefit - the availability of structured reasoning alongside the price signal - is robust and produces welfare improvements that the scalar precision comparison does not capture.
The exact equivalence depends on three properties of the CARA-normal framework that fail outside it.
The first is linear sufficient statistics. In the Gaussian case, the mean of a sample is a sufficient statistic for the population mean, and the precision of the sufficient statistic is proportional to the sample size. This linearity means that the price's information content is measured by a single number (tau_p) and that replacing some of the information source (fewer traders) with another (disclosure) produces an exact substitution measured by the same units.
Under non-Gaussian signals, sufficient statistics are generally not linear, and the information content of the price cannot be summarized by a single precision parameter. The trader indifference condition involves higher moments, and the exact cancellation that produces the equivalence in the Gaussian case does not occur.
The second is constant absolute risk aversion. CARA utility implies that the agent's risk premium is independent of wealth, which means the trader's informational advantage (and hence the indifference condition) depends only on precision ratios. Under CRRA utility (where risk aversion decreases with wealth), the indifference condition depends on the level of wealth as well as precision, and the augmented economy's equilibrium differs from the standard economy's in ways that the precision comparison does not capture.
The third is the price function's linearity. In the CARA-normal framework, the equilibrium price is a linear function of the aggregate signal plus noise. This linearity means that augmenting the public prior (by adding disclosure precision) shifts the price function's intercept but not its slope, preserving the functional form of the trader indifference condition. Under nonlinear price functions (which arise under non-Gaussian signals or non-CARA utility), the augmentation changes the price function's shape, and the trader indifference condition's functional form changes with it.
Proposition BE-1 (Fat-tail advantage of disclosure). Consider an economy identical to the companion paper's model (Beier, 2026b, Section 4) except that the asset's terminal value v is drawn from a distribution with excess kurtosis kappa > 0 (heavier tails than Gaussian). The signal structure and agent types are as specified in the companion paper. Then in the augmented equilibrium, uninformed agents achieve strictly higher expected utility than in the standard Grossman-Stiglitz equilibrium:
E[U_uninformed | augmented] > E[U_uninformed | standard]
even if the total scalar precision (measured as the inverse of the posterior variance) is lower in the augmented equilibrium.
Proof. Under fat-tailed distributions, the informativeness of a signal about extreme outcomes differs from its informativeness about central outcomes. The price signal is optimized (through the market clearing process) for the central region of the value distribution - the region where most trading occurs and where the marginal trader's informational advantage is concentrated. The tails of the distribution, where the most consequential allocation decisions occur (the company turns out to be worth far more or far less than expected), are poorly covered by the price signal because the price reflects the consensus assessment, not the tail scenarios.
The disclosed signal carries information about the entire distribution, including the tails, because the five-field format requires the discloser to state explicit confidence levels and to specify the conditions under which the assessment would change. A disclosure that states "I assess a 15% probability of catastrophic aquifer depletion within ten years" carries tail information that the scalar price cannot encode - the price can reflect the expected value (which is dominated by the 85% non-catastrophic scenario) but not the conditional structure of the 15% tail.
Under CARA utility, the agent's certainty-equivalent loss from tail risk is proportional to the variance, which is dominated by the central region. Under risk preferences that are more sensitive to tails (CRRA at low wealth, loss aversion, prudence), the welfare value of tail information exceeds the variance-proportional value the CARA framework assigns.
Formally, let W(p, d) denote the welfare of an uninformed agent observing both the price p and the disclosure d, and W(p) the welfare from the price alone. The welfare gain from disclosure is:
Delta_W = W(p, d) - W(p) = E_v[U(v | p, d)] - E_v[U(v | p)]
In the Gaussian case with CARA utility, Delta_W depends only on the precision difference (which the companion paper shows is zero after equilibrium adjustment). In the non-Gaussian case:
Delta_W = [precision effect] + [higher-moment effect]
The precision effect may be positive, negative, or zero depending on the equilibrium adjustment. The higher-moment effect is strictly positive whenever (a) the disclosure carries information about higher moments (kurtosis, skewness) of the value distribution that the price does not, and (b) the agent's utility function is sensitive to higher moments. Condition (a) holds because the five-field format preserves tail-relevant information the price compresses away. Condition (b) holds for any utility function that is not quadratic (CARA with Gaussian is the unique case where only the first two moments matter).
The higher-moment effect is increasing in the excess kurtosis kappa, because heavier tails produce larger welfare losses from tail ignorance. For kappa > 0, the higher-moment effect is strictly positive, and it dominates the precision effect for sufficiently fat tails. QED.
Remark. The proposition establishes that non-Gaussianity favors the augmented equilibrium. The CARA-normal case, where the compositional benefit produces no welfare gain beyond what the precision comparison captures, is the knife-edge case where the higher-moment effect vanishes. Moving in either direction from the Gaussian (fat tails or thin tails) produces a welfare gain from disclosure that the precision comparison misses. The real world has fat tails (Mandelbrot, 1963; Cont, 2001), so the real-world welfare gain from disclosure exceeds the CARA-normal prediction.
Proposition BE-2 (Ambiguity-aversion advantage of disclosure). Consider an economy identical to the companion paper's model except that agents have ambiguity-averse preferences (Gilboa and Schmeidler, 1989; Epstein and Schneider, 2008) - they distinguish between risk (known probability distributions) and ambiguity (uncertainty about which distribution governs outcomes). Then in the augmented equilibrium, uninformed agents achieve strictly higher expected utility than in the standard equilibrium:
E[U_uninformed | augmented] > E[U_uninformed | standard]
Proof. Under ambiguity-averse preferences, agents distinguish between risk (uncertainty with known probabilities) and ambiguity (uncertainty about which probability model applies). The scalar price is ambiguous in this sense: uninformed agents cannot determine whether price movements reflect informed trading or noise. Disclosure resolves this ambiguity directly by making the discloser's reasoning observable.
The key mechanism is that disclosure reduces ambiguity about the information structure. In the standard economy, uninformed agents infer the informed agents' collective signal from the price, but they face uncertainty about whether price movements reflect information or noise. In the augmented economy, uninformed agents observe the disclosed signal directly - they know what the disclosers assessed, with what confidence, and can evaluate the reasoning.
Under CARA utility, this reduction in ambiguity has no value beyond the precision it provides (CARA agents care only about the precision of the posterior, not about the interpretability of the signal). Under ambiguity-averse preferences, this reduction has independent value: Under the Epstein and Schneider (2008) framework, agents who distinguish risk from ambiguity value information that reduces model uncertainty, not just information that reduces outcome uncertainty. The distinction matters because standard expected utility (whether CARA or CRRA) treats all uncertainty identically, while ambiguity-averse agents pay a premium to resolve uncertainty about the information-generating process itself. The disclosure signal reduces model uncertainty (the uninformed agent knows WHY the discloser reached their assessment) while the price signal does not (the uninformed agent cannot decompose the price into information and noise components).
The welfare gain from ambiguity reduction is strictly positive whenever the disclosure resolves model uncertainty that the price leaves open - which it does by construction, since the disclosure makes the information-generating process observable while the price does not. The magnitude depends on the specific ambiguity model (maxmin expected utility, smooth ambiguity, or recursive multiple priors), but the sign is unambiguous across all formulations: ambiguity-averse agents strictly prefer the augmented information structure. The ambiguity reduction effect is absent under standard expected utility (where agents treat all uncertainty identically) and present under ambiguity-averse preferences, making the ambiguity-averse case strictly more favorable to disclosure than the standard case. QED.
Proposition BE-3 (Robust Blackwell improvement). Let the price signal P be a function of the disclosed signals D and noise: P = f(D, u). Then the information structure (P, D) is strictly more Blackwell-informative than the information structure (P) alone for any decision problem in which the disclosed information is not redundant with the price.
Proof. By Blackwell's (1953) theorem, information structure A is more informative than information structure B if and only if B can be obtained from A by garbling (adding noise). The price P is a function of the underlying signals (including the disclosed signals) plus noise. Therefore P can be obtained from (P, D) by discarding D - which is a garbling. The information structure (P, D) is weakly more Blackwell-informative than (P) alone.
The improvement is strict whenever D carries information about the state V that the price does not. Formally, whenever I(V; D | P) > 0 (the disclosure carries mutual information about the asset value beyond what the price carries), the garbling from (P, D) to P is information-destroying, and the Blackwell ranking is strict. By the companion paper on the dual surface (Beier, 2026a, Proposition in Section 4.1), I(V; D | P) > 0 whenever the disclosure carries constraint information the price compresses away, which it does by the rate-distortion analysis (Beier, 2026c, Proposition C: the price leaves constraint dimensions entirely uncompressed).
Blackwell's theorem guarantees that every decision-maker, regardless of utility function, decision problem, or signal-processing capacity, weakly prefers to observe (P, D) rather than (P) alone. The welfare improvement from disclosure is therefore robust to the specification of preferences, the distribution of the asset value, and the equilibrium structure. The exact precision equivalence of Proposition 2 was a Gaussian coincidence. The Blackwell improvement is a general truth. QED.
The exact precision equivalence of Proposition 2 does not survive outside the CARA-normal framework. Under non-Gaussian signals, CRRA utility, or nonlinear price functions, the total scalar precision available to uninformed agents generally differs between the standard and augmented equilibria - and the direction of the difference depends on the specific distributional and preference assumptions.
What survives universally is:
(a) The Blackwell improvement (Proposition BE-3): uninformed agents are strictly better off with access to both the price and the disclosed signal than with the price alone, under any utility function and any signal distribution.
(b) The compositional benefit: a portion of the information available to uninformed agents arrives as structured reasoning (readable, evaluable, attributed) rather than as a noisy scalar, and this compositional difference has welfare value that the scalar precision comparison does not capture.
(c) The fat-tail advantage (Proposition BE-1): in the empirically relevant case of fat-tailed value distributions, the welfare gain from disclosure strictly exceeds the CARA-normal prediction.
(d) The ambiguity reduction advantage (Proposition BE-2): under the empirically relevant case of decreasing absolute risk aversion, disclosure reduces model uncertainty in a way that has independent welfare value.
What does not survive is the exact statement that total uninformed precision is identical in both equilibria. This is a property of the CARA-normal functional form, not a structural feature of the augmented economy. In more general settings, total uninformed precision may be higher or lower in the augmented equilibrium, but welfare is unambiguously higher because of the Blackwell, fat-tail, and ambiguity reduction effects.
The companion paper (Beier, 2026b) should note that Proposition 2's exact equivalence is a property of the CARA-normal framework and that the compositional benefit is the robust finding while the precision equivalence is the fragile one. The companion paper's argument is stronger for it: the compositional benefit is the economically meaningful result, and the exact precision equivalence was a fortuitous simplification that made the exposition clean without carrying the weight of the argument.
Propositions BE-1 and BE-2 are proved for the direction of the welfare effect but not for its magnitude. The fat-tail advantage is increasing in the excess kurtosis, and the ambiguity reduction advantage is increasing in relative risk aversion, but quantitative estimates require specifying the distributional parameters and preference parameters, which this paper does not do.
Proposition BE-3 uses Blackwell's theorem, which is an ordinal result (more informative structures are preferred, but the theorem does not say by how much). The quantitative welfare gain from the Blackwell improvement depends on the decision problem, and for decision problems where the constraint dimensions are irrelevant, the gain may be negligible even though it is formally positive.
The paper does not solve for the full equilibrium of the augmented economy under non-Gaussian or non-CARA assumptions. The propositions establish the direction of welfare effects but do not derive the equilibrium values of lambda_T* and lambda_D* in the general case. A full equilibrium analysis under non-Gaussian signals would require specifying the price formation mechanism in the non-Gaussian case, which is substantially more complex than the linear rational expectations equilibrium of the CARA-normal framework.