Correction, 30 October 2026. Proposition 1'' (Partial loading, corrected). Let K >= 2, lambda_1 > lambda_2 >= ... >= lambda_K > 0, and let p = a' V + w with ||a|| = 1 and sigma_w^2 > 0, so that R_p = (1/2) log_2(1 + a' Sigma a / sigma_w^2).
Corollary (fixed rate). At a given rate R_p, D* is fixed and D_p = tr Sigma - rho(a) SNR / (1 + SNR), so Delta is minimized at a = ±e_1. A summary of this correction is on the papers page at https://www.susarb.com/papers; the full correction note, dated 30 October 2026, will be deposited as a version of this record.
Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
March 2026
© 2026 Gregory Caldwell Beier. Licensed under CC BY-NC-ND 4.0.
The price system compresses high-dimensional economic information into scalar signals. This paper applies rate-distortion theory to characterize the distortion. The main result: when the price's compression rate suffices for an optimal codec to allocate positive bits to constraint dimensions, the price system's distortion strictly exceeds the information-theoretic minimum. The excess arises from constraint dimensions - resource interactions, scaling dynamics, supply chain dependencies - that receive less than their optimal bit allocation. The result extends to partial constraint loading (the price carries some constraint information through cross-correlation) and to multi-signal environments (multiple prices observing the same economy). Excess distortion remains strictly positive under partial constraint loading unless the observation direction achieves a knife-edge optimum, and the eigenbasis case (zero constraint loading) provides the minimum excess distortion among unit-norm observation directions at fixed noise variance. Multiple prices reduce excess distortion but do not eliminate it when their effective rank on the constraint subspace falls below K - 1. The results are conservative: the Gaussian source model yields a lower bound on excess distortion for non-Gaussian sources, and the MSE distortion measure yields a lower bound on economically relevant excess distortion under downside-dominant loss. A feedback loop between structural compression and rational inattention is identified: the price system's inability to carry constraint information justifies agents' inattention to those dimensions, which ensures that constraint information does not enter subsequent prices, creating a self-reinforcing equilibrium that supplementary channels can break.
Keywords: rate-distortion theory, price formation, lossy compression, information architecture, constraint dimensions, excess distortion, market efficiency
JEL Codes: D82, D83, C60, G14
Hayek (1945) made the canonical case that prices aggregate dispersed information efficiently. The present paper formalizes a complementary claim: the aggregation is lossy, and the distortion has a specific structure that rate-distortion theory can characterize.
A price is a scalar. The economic information relevant to a transaction is multi-dimensional. The compression from multi-dimensional judgment to scalar price is irreversible (Shannon, 1948). Rate-distortion theory (Shannon, 1959; Berger, 1971) characterizes the minimum distortion achievable at a given compression rate. This paper compares the price system's actual distortion to that minimum.
The core result (Proposition 1) is that the price system operates strictly above the rate-distortion bound on constraint dimensions whenever the price is informative enough for an optimal codec to distribute bits beyond the primary dimension. The excess distortion is a structural consequence of scalar clearing, not a correctable inefficiency. A partial-loading extension (Proposition 1') shows that the result survives when the price carries some constraint-dimension information through cross-correlation, with the excess remaining strictly positive and the eigenbasis case providing the minimum among unit-norm observations. A multi-signal extension (Proposition 3) derives the conditions under which multiple prices collectively eliminate or fail to eliminate the excess. A corollary (Proposition 2) establishes that supplementary channels targeting uncompressed constraint dimensions produce higher marginal returns per bit than further price-precision improvements.
The paper contributes to the literature on information and prices (Grossman and Stiglitz, 1980; Admati, 1985; Vives, 2008), complements rational inattention (Sims, 2003) by characterizing the signal degradation that justifies inattention, and connects to information design (Kamenica and Gentzkow, 2011; Bergemann and Morris, 2016) by characterizing the distortion that makes supplementary channels valuable.
The allocation-relevant information about a transaction has K dimensions. We model the full economic value as V = (V_1, V_2, ..., V_K)' ~ N(mu, Sigma). In the eigenbasis of Sigma, with eigenvalues lambda_1 >= lambda_2 >= ... >= lambda_K > 0, the components are independent. V_1 is the consensus-value dimension. V_2, ..., V_K are constraint dimensions: resource interactions, regulatory trajectories, supply chain dependencies, scaling dynamics.
The price signal is p = a'V + w, where a = (a_1, ..., a_K)' is the observation direction, w ~ N(0, sigma^2_w) is independent of V, and the signal-to-noise ratio is SNR = a' Sigma a / sigma^2_w.
The source model raises an immediate question: does a K-dimensional source with a stable eigenvalue decomposition describe anything real?
What is V? The K dimensions of V are latent factors in the economic information environment. They are not the same as return-based factors in the Fama-French (1993) or APT (Ross, 1976) tradition. Return-based factors (market, size, value, profitability, investment) are downstream objects: projections of the economic information environment onto the return space, recoverable from price data by construction. The constraint dimensions V_2, ..., V_K in this paper are upstream objects: dimensions of the information environment that causally affect returns but may not be spanned by return variation because the price carries little or no information about them.
The distinction matters. If a physical constraint (aquifer depletion beneath a data center) is not priced ex ante, it does not appear in the return covariance until the constraint binds. The standard factor model based on returns cannot recover a dimension that generates no return variation during the estimation window. Augmenting the factor analysis with physical data - satellite gravimetric measurements, grid load data, environmental monitoring, supply chain mapping - potentially recovers dimensions that the return-based model misses. Connor and Korajczyk (1986) established the statistical framework for PCA of asset returns; the extension proposed here applies PCA to returns augmented with physical observables. The identification claim is that the augmented factors carry economically meaningful information about physical constraints beyond what the return-only factors capture. This is the content of Prediction 1 below.
A connection to the rare disaster literature clarifies the identification. Barro (2006) and Gabaix (2012) showed that even small probabilities of large losses can have substantial pricing effects if the disasters are anticipated. The constraint-binding events this paper describes are rare disasters that are NOT anticipated in prices - not because the probability is too small to matter but because the constraint dimension is not in the price's observation space. Prediction 3 (constraint events not anticipated) distinguishes the present paper's constraint dimensions from the rare disaster literature's risk factors: the information is absent from prices for structural reasons (the price cannot carry it), not for probabilistic reasons (the probability is too small).
Are the eigenvalues stable? In the strict sense, no. Covariance structures in economic data are non-stationary (Engle, 2002, on dynamic conditional correlation; Ang and Bekaert, 2002, on regime shifts). The eigenvalue decomposition of Sigma shifts over time. The results do not require eigenvalue stability. They require only K > 1 dimensions of positive variance at any moment - that the economic information relevant to a transaction cannot be captured in a single number. This condition fails only if a single factor explains all economically relevant variation, which the factor literature has rejected for decades.
Can the eigenvalues be estimated? The eigenvalues of the full source covariance - including constraint dimensions not reflected in prices - cannot be estimated from price data alone (this is the content of the main result). Estimation requires auxiliary data. The eigenvalue parameterizations in Section 3.4 are illustrative. The empirical program this paper motivates is the estimation of constraint-dimension eigenvalues from auxiliary data - a precondition for quantifying excess distortion in specific sectors.
In the eigenbasis, with a = (1, 0, ..., 0)', the price loads entirely on V_1. This is the cleanest formal case and the most favorable for the price system (Corollary 1'). The assumption formalizes the claim that prices aggregate consensus value and compress away constraint reasoning.
Recent empirical work establishes that the assumption of zero constraint loading is too strong. Hong, Li, and Xu (2019) found that food company stock prices underreact to drought trends across 31 countries. Engle et al. (2020) constructed climate risk hedging portfolios from asset prices, implying prices carry some climate information. Giglio et al. (2021) documented partial sea-level-rise pricing in coastal real estate. Bolton and Kacperczyk (2021) found carbon emissions priced in the cross-section of stock returns. These studies demonstrate that prices carry some constraint-dimension information. None claims prices fully incorporate it.
The general case is a = (a_1, a_2, ..., a_K)' with a_1^2 >> a_k^2 for k >= 2 but a_k != 0. Proposition 1' below derives excess distortion as a continuous function of the constraint-loading coefficients. The eigenbasis case (a_k = 0 for k >= 2) is the limiting case where excess distortion is minimized among all scalar observations at a given rate.
The mutual information between V and p is R_p = I(V; p) = (1/2) log_2(1 + SNR) bits. Under MSE distortion, the posterior covariance is Sigma_{V|p} = Sigma - (Sigma a a' Sigma)/(a' Sigma a + sigma^2_w). The price system's distortion is:
D_p = tr(Sigma_{V|p}) = sum_{k=1}^{K} lambda_k - sum_{k=1}^{K} (a_k^2 lambda_k^2) / (a' Sigma a + sigma^2_w)
For the eigenbasis case (a = e_1):
Lemma 1. D_p = lambda_1/(1 + SNR_1) + sum_{k=2}^{K} lambda_k, where SNR_1 = lambda_1/sigma^2_w.
Proposition 1 (Excess Distortion). Let K >= 2, lambda_k > 0 for all k, and a = e_1. Define the threshold rate R_thresh = (1/2) log_2(lambda_1/lambda_2). The excess distortion Delta = D_p - D*(R_p) satisfies:
(i) Delta = 0 when R_p <= R_thresh (the price's rate is too low for the optimal codec to reach any constraint dimension).
(ii) Delta > 0 when R_p > R_thresh.
(iii) In the positive-excess regime, d(Delta)/d(lambda_k) = 1 - theta*/lambda_k > 0 for each active constraint dimension k >= 2.
(iv) Delta decomposes as Delta = S_constraint - C_over, where S_constraint = sum_{k in S, k >= 2}(lambda_k - theta*) is the constraint-dimension savings and C_over = theta* - lambda_1 2^{-2R_p} is the over-compression cost on dimension 1. Both are strictly positive in the positive-excess regime.
(v) As SNR_1 -> infinity: Delta -> sum_{k=2}^{K} lambda_k.
Proof. The optimal distortion at rate R_p is D* = sum_{k=1}^{K} min(lambda_k, theta*), where theta* solves R_p = sum_{k=1}^{K} max(0, (1/2) log_2(lambda_k/theta*)) (reverse water-filling; Berger, 1971; Cover and Thomas, 2006).
Part (i): When R_p <= R_thresh, theta* >= lambda_2. Only dimension 1 is active. The optimal allocation matches the price system. D* = D_p.
Part (ii): When R_p > R_thresh, theta* < lambda_2. The optimal codec allocates positive bits to at least dimension 2. The price system is a feasible point with distortion D_p at rate R_p. By uniqueness of the Gaussian RD solution (strict convexity; Cover and Thomas, 2006, Section 10.3.3), D_p > D*.
Part (iii): For k in S = {j : lambda_j > theta*} with |S| elements, implicit differentiation gives d(theta*)/d(lambda_k) = theta*/(|S| lambda_k). Then d(D*)/d(lambda_k) = theta*/lambda_k and d(D_p)/d(lambda_k) = 1, so d(Delta)/d(lambda_k) = 1 - theta*/lambda_k > 0.
Part (iv): The price achieves residual variance lambda_1 2^{-2R_p} on dimension 1. The optimal codec achieves theta* on each active dimension. Since the price over-compresses dimension 1 (lambda_1 2^{-2R_p} < theta* when R_p > R_thresh), the decomposition follows. Positivity of both terms is by construction. S_constraint > C_over follows from Delta > 0.
Part (v): As SNR_1 -> infinity, R_p -> infinity, theta* -> 0, D* -> 0, D_p -> sum_{k=2}^{K} lambda_k. QED.
Decomposition magnitudes across the three parameterizations (SNR_1 = 19, R_p = 2.16 bits):
| Parameterization | Delta | S_constraint | C_over | S/C ratio |
|---|---|---|---|---|
| Consensus-dominated | 0.09 | 0.29 | 0.21 | 1.4 |
| Moderate constraint | 1.07 | 1.88 | 0.81 | 2.3 |
| Constraint-heavy | 2.06 | 3.42 | 1.36 | 2.5 |
The consensus-dominated case produces negligible excess distortion (Delta = 0.09, less than 1% of total source variance), confirming that the mechanism has little purchase when constraint dimensions carry small variance. At SNR_1 = 19, even the consensus-dominated spectrum has R_p > R_thresh (the optimal codec can reach dimension 2), so the excess is technically positive but economically immaterial. In all three parameterizations, the constraint-dimension savings exceed the over-compression cost by a factor of roughly 2 to 3. The dominant source of excess distortion is the waste on constraint dimensions, not the over-compression on the consensus dimension. This confirms that the excess distortion is primarily a problem of missing dimensions, not misallocated bits within the covered dimension.
Proposition 1' (Partial Loading). Let a = (a_1, a_2, ..., a_K)' with ||a|| = 1 and a_1^2 = 1 - epsilon for small epsilon >= 0 (so the price loads primarily on V_1 with partial loading epsilon on constraint dimensions). The excess distortion Delta(epsilon) = D_p(epsilon) - D*(R_p(epsilon)) satisfies:
(i) Delta(0) = Delta from Proposition 1 (the eigenbasis case).
(ii) Delta(epsilon) > 0 for all epsilon in [0, epsilon*), where epsilon* is the unique loading at which the price's observation direction achieves the reverse water-filling allocation. The condition epsilon* > 0 holds whenever the eigenbasis Delta > 0.
(iii) Delta(epsilon*) = 0 only at the knife-edge where a_k^2 lambda_k is proportional to max(0, lambda_k - theta*) for all k. This condition has measure zero in the space of unit observation directions, so Delta > 0 generically.
Proof. Part (i) is immediate. For part (ii): at any epsilon in [0, epsilon*), the price system's observation direction a(epsilon) differs from the optimal codec's rate allocation. The price system is a feasible encoder at rate R_p(epsilon) with distortion D_p(epsilon). The rate-distortion solution D*(R_p(epsilon)) is the unique minimum over all encoders at that rate (by strict convexity of the Gaussian RD problem; Cover and Thomas, 2006, Section 10.3.3). The price system's encoder - which observes a single linear combination a(epsilon)'V - belongs to the feasible set but does not coincide with the optimal encoder unless the observation direction happens to produce the reverse water-filling allocation on all K dimensions. At epsilon < epsilon*, the observation direction differs from this optimum. By uniqueness of the optimum, D_p(epsilon) > D*(R_p(epsilon)), hence Delta(epsilon) > 0.
For part (iii): the reverse water-filling allocation assigns rate r_k* = max(0, (1/2) log_2(lambda_k/theta*)) to dimension k. The price system's effective allocation to dimension k is determined by (a_k^2 lambda_k^2)/(a' Sigma a + sigma^2_w), which must match 2^{-2r_k*} lambda_k for each active dimension. This requires a_k^2 lambda_k proportional to max(0, lambda_k - theta*) - a system of K equations in K - 1 free parameters (since ||a|| = 1). The solution set has measure zero in R^{K-1}. QED.
Remark on monotonicity. The numerical evidence (table below) shows that Delta(epsilon) does not decrease monotonically when the observation direction is varied under the stated parametrization (unit-norm a with fixed noise variance). Increasing epsilon shifts observation weight from the high-variance consensus dimension to lower-variance constraint dimensions, which reduces both the total signal power a'Sigma a and the compression rate R_p. The rate reduction raises the water level theta*, increasing D*. Meanwhile D_p also increases because less variance reduction is achieved on the dominant dimension. Both effects push Delta upward. The eigenbasis case (epsilon = 0) therefore yields the minimum Delta among unit-norm observation directions at the same noise variance - consistent with Corollary 1', which establishes that eigenbasis alignment minimizes D_p at any given rate. The paper claims positivity of Delta (proved) but not monotone decrease in epsilon.
Quantitative illustration. For the constraint-heavy parameterization (lambda = (10, 4, 3, 2, 1.5, 1)) with SNR_1 = 19, sigma^2_w = lambda_1/SNR_1 fixed, varying the constraint loading by distributing epsilon equally across constraint dimensions (a_k^2 = epsilon/5 for k >= 2):
| epsilon (constraint loading) | R_p (bits) | D_p | D*(R_p) | Delta | Delta/Delta(0) |
|---|---|---|---|---|---|
| 0.00 (eigenbasis) | 2.16 | 12.00 | 9.95 | 2.05 | 100.0% |
| 0.05 | 2.13 | 12.10 | 10.02 | 2.08 | 101.5% |
| 0.10 | 2.11 | 12.21 | 10.09 | 2.12 | 103.3% |
| 0.20 | 2.05 | 12.45 | 10.24 | 2.21 | 107.6% |
| 0.40 | 1.91 | 13.09 | 10.62 | 2.48 | 120.6% |
The table reveals that under the stated parametrization, partial constraint loading increases rather than decreases excess distortion. Moving observation weight from the high-variance consensus dimension to lower-variance constraint dimensions reduces total signal power, lowers the compression rate, and raises the water level - effects that dominate the small variance reduction gained on constraint dimensions. The eigenbasis case is not merely conservative but optimal among unit-norm observation directions at fixed noise variance. The result strengthens the paper's central claim: prices that partially load on constraint dimensions (as the empirical literature documents) operate at higher excess distortion than the eigenbasis case predicts, not lower. Proposition 1' guarantees Delta > 0 for all epsilon < epsilon*; the quantitative finding is that Delta also increases over this range.
Corollary 1'. For V ~ N(mu, Sigma) with Sigma positive definite, K >= 2, and p = a'V + w: the excess distortion Delta > 0 whenever the residual covariance Sigma_{V|p} has at least two eigenvalues exceeding the water level theta* at rate R_p. The eigenbasis-aligned price signal minimizes D_p among scalar observations at any given mutual information rate (Rayleigh-quotient argument: a proportional to the first eigenvector maximizes the MSE reduction from a rank-one update). Since D* depends only on the eigenvalue spectrum and the rate, minimizing D_p at fixed rate minimizes Delta. At fixed noise variance (rather than fixed rate), the result is stronger: moving the observation direction away from the first eigenvector both raises D_p and lowers R_p (by reducing the signal power a'Sigma a), and the lower rate raises D* through the water level. Both effects increase Delta. Proposition 1 therefore provides a lower bound on Delta for any scalar price signal, whether compared at the same rate or at the same noise variance.
Proposition 3 (Multi-Price Excess Distortion). Let p = (p_1, ..., p_M)' = AV + w be M price signals with A an M x K observation matrix and w ~ N(0, Sigma_w). The multi-price mutual information is R_M = I(V; p). The multi-price distortion is D_M = tr(Sigma_{V|p}). Then:
(i) Delta_M = D_M - D*(R_M) = 0 if and only if the posterior covariance Sigma_{V|p} has eigenvalues matching the reverse water-filling solution at rate R_M. (By the same strict-convexity uniqueness argument as Proposition 1, part (ii): the multi-signal encoder is a feasible point in the rate-distortion optimization, and it achieves the minimum if and only if its allocation matches the unique optimum.)
(ii) If rank(A Sigma^{1/2}) >= K and Sigma_w is nonsingular, then as SNR scales up uniformly, R_M -> infinity and Delta_M -> 0. With enough independent price signals at sufficient precision, the price system can in principle eliminate excess distortion.
(iii) If the effective rank of A restricted to the constraint subspace (dimensions 2 through K) is r_c < K - 1, then Delta_M > 0 regardless of SNR on the covered dimensions. The K - 1 - r_c uncovered constraint dimensions remain at full prior variance.
Discussion. A modern equity has many associated prices: the stock, options at multiple strikes, CDS spreads, bond yields, lending rates. The question is whether these collectively span the constraint dimensions.
The argument that they do not: all market-clearing prices are generated by trading, which aggregates participants' willingness to pay. Willingness to pay is a function of the participant's utility-relevant information, which is dominated by the consensus-value dimension (expected cash flows, discount rate) for any position with a finite holding period. The constraint dimensions - how a technology interacts with an aquifer at scale, how a supply chain's concentration creates fragility under geopolitical stress - affect utility-relevant information only when the constraint actually binds, which may be years or decades away. Before binding, these dimensions contribute negligible variance to returns and therefore negligible signal to any price formed through trading.
The effective rank of the multi-price observation matrix on the constraint subspace is therefore substantially lower than the number of prices. CDS spreads, option-implied volatilities, and credit yields all load on the same few dimensions (default probability, recovery rate, liquidity) rather than on independent constraint dimensions. The multiple prices provide redundant observations of the consensus-value neighborhood, not independent observations of the constraint space. Empirically, PCA of multi-asset price data typically finds 3-5 factors explaining 70-90% of variation (Litterman and Scheinkman, 1991, for interest rates; Connor and Korajczyk, 1986, for equities). The effective rank is far below K when K includes physical constraint dimensions.
Proposition 3 connects to a foundational result in welfare economics. In a theoretical Arrow-Debreu world of complete state-contingent markets, the number of independent price signals equals the dimensionality of the state space (r_c = K), and excess distortion vanishes. Greenwald and Stiglitz (1986) proved that when markets are incomplete and information imperfect, competitive equilibria are generically constrained Pareto-inefficient - there exist interventions that improve welfare even respecting the informational constraints agents face. Proposition 3 provides the information-theoretic characterization of why real-world financial markets are incomplete in this sense: the proliferation of derivative instruments largely spans redundant consensus risks rather than novel physical constraints, so r_c remains well below K even as the number of prices M grows large. Excess distortion is the information-theoretic measure of market incompleteness on the constraint subspace.
A qualification: the effective rank on the constraint subspace is not uniform across asset classes or constraint types. For a large-cap equity with deep options markets, put prices encode left-tail probabilities that may carry constraint-relevant information - the probability of a discontinuous loss from a binding physical constraint. Implied volatility smiles and skews plausibly contain constraint signals that the stock price alone does not. For physical constraints that produce discontinuous outcomes (water curtailment, grid failure), option prices may carry more constraint information than for gradual constraints (regulatory trajectory, demographic shift). The effective rank on the constraint subspace is therefore higher for well-covered equities with deep derivatives markets than for mid-caps with equity prices only, and higher for constraints that manifest as tail events than for constraints that manifest as drift. The single-price analysis (Propositions 1 and 1') applies most directly to assets where derivatives coverage is thin - which includes most assets in the global economy. The multi-price analysis (Proposition 3) applies where coverage is deep, and its quantitative content depends on the effective rank, which varies by asset class and constraint type.
For K = 6 dimensions with SNR_1 = 19 (R_p = 2.16 bits):
| Parameterization | lambda | Constraint share | Delta | CE cost (bps/yr) |
|---|---|---|---|---|
| Consensus-dominated | (10, 1, 0.5, 0.3, 0.2, 0.1) | 17% | 0.09 | 0.9 |
| Moderate constraint | (10, 3, 1.5, 0.8, 0.4, 0.2) | 37% | 1.07 | 11.2 |
| Constraint-heavy | (10, 4, 3, 2, 1.5, 1) | 54% | 2.06 | 21.6 |
The consensus-dominated case produces negligible excess distortion (Delta = 0.09), confirming that the mechanism has little purchase when constraint dimensions carry small variance relative to the consensus dimension. At SNR_1 = 19, even this spectrum enters the positive-excess regime (R_p = 2.16 > R_thresh = 1.66), so Delta is technically positive but economically immaterial. For Delta to be exactly zero, the price would need to be less informative (SNR_1 <= 9 for this spectrum).
Certainty-equivalent cost mapping: CARA utility, gamma = 2, 15% annual portfolio standard deviation. These are illustrative. The functional forms are proved; the magnitudes are parameterization-dependent.
Why is D*(R) the appropriate benchmark for an emergent price system?
The comparison is analogous to comparing an engine's thermal efficiency to the Carnot bound. No physical engine achieves Carnot efficiency. The gap is not evidence of poor design. It characterizes the structural loss inherent in the operating principle and identifies where waste-heat recovery has value. Similarly, the gap between D_p and D*(R_p) identifies where supplementary channels have the highest marginal value.
A specific version of this objection deserves direct engagement: the optimal RD codec requires a centralized encoder that observes the full K-dimensional state V and jointly encodes it. The price system is a decentralized aggregation mechanism where no single agent observes V. Is the comparison a category error? It is not - but the reason matters. The RD bound is not an institutionally feasible benchmark for the price system. It is a physical limit on how much distortion any system at rate R_p must accept, regardless of whether the encoder is centralized or decentralized. The excess distortion measures the informational vacuum that decentralized scalar clearing creates - a vacuum that cannot be filled by better trading algorithms, faster information transmission, or more sophisticated market microstructure, because the vacuum is orthogonal to the dimension the price aggregates. The comparison does not claim the price system should operate as a centralized codec. It identifies the specific dimensions where a supplementary mechanism - necessarily different in kind from the price system - has value.
A qualification the Carnot analogy requires: the Carnot bound is a physical law independent of the analyst's model. The rate-distortion bound depends on the source model. If V is not well-described by the specified source distribution, D*(R) shifts. The results are the Carnot bound conditional on the source model. The Gaussian robustness argument (Section 6) establishes that the Gaussian case is conservative - but the conservatism is relative to sources with the same covariance, not relative to arbitrary departures from the model.
For the Gaussian source under MSE distortion, the rate-distortion bound is achieved by a memoryless test channel. The price system is itself a single-observation encoder. The comparison is between two single-observation systems - one concentrating all capacity on one dimension, one distributing optimally. The gap is a pure allocation mismatch. A potential objection is that prices are not single observations but continuous stochastic processes, and that a one-dimensional signal observed over time can in principle trace a higher-dimensional space. This requires a static source and a noiseless channel. Because physical and regulatory constraints are non-stationary - they evolve as technologies scale, as resource stocks deplete, as regulatory regimes shift - the constraint state space changes faster than a noisy one-dimensional price trajectory can trace it. Time-series observation of prices cannot recover constraints that are continuously changing.
Proposition 2. At the current operating point, the marginal distortion reduction from a supplementary bit allocated to the leading uncompressed constraint dimension strictly exceeds the marginal reduction from an additional bit of price precision, provided lambda_2 > lambda_1/(1 + SNR_1).
Proof. Marginal distortion reduction per bit at allocation r for a Gaussian dimension with residual variance sigma^2: 2 ln(2) sigma^2 2^{-2r}. At r = 0 (uncompressed constraint dimension): 2 ln(2) lambda_2. At r = R_p (consensus dimension): 2 ln(2) lambda_1/(1 + SNR_1). Dominance when lambda_2 > lambda_1/(1 + SNR_1). For SNR_1 = 19: lambda_2 > lambda_1/20. QED.
The result connects to Grossman and Stiglitz (1980). The incentive to produce information about V_1 is destroyed as the price becomes efficient (marginal return per bit declines with SNR_1). But the constraint dimensions are not subject to this paradox. The marginal value of the first bit about V_2 is lambda_2 regardless of price efficiency on V_1, because the price carries zero information about V_2 in the eigenbasis case and negligible information in the partial-loading case. Information production about constraint dimensions is immune to the efficiency-destroys-incentive paradox.
This connects to rational inattention (Sims, 2003). Agents with finite attention optimally attend to the price (which carries V_1) and rationally ignore constraint dimensions (which the price carries little about). The price system's structural compression justifies rational inattention to constraint dimensions, which in turn ensures that constraint information does not enter subsequent prices - a feedback loop in which structural compression and rational inattention reinforce each other. The supplementary channel breaks this loop by making constraint-dimension information available without requiring agents to produce it individually through costly attention.
Remark (Fixed-Point Structure). The feedback loop has the structure of a fixed point in an iterated map. Let Sigma_t denote the price system's information structure at time t, let alpha(Sigma_t) denote agents' optimal attention allocation given that information structure (the rational inattention solution), and let Sigma_{t+1} = F(alpha(Sigma_t)) denote the information structure that results when agents trade on their optimally allocated attention. A fixed point Sigma* = F(alpha(Sigma*)) is a self-consistent information structure where the price system's compression justifies the attention allocation that reproduces the compression. The low-constraint-information equilibrium - where prices carry little constraint information, agents rationally ignore constraints, and the next period's prices carry little constraint information - is plausibly a stable fixed point: small perturbations (one agent paying attention to a constraint) produce small changes in price informativeness that do not shift other agents' attention allocation enough to sustain the perturbation. A supplementary channel that exogenously injects constraint-dimension information operates outside this feedback loop, potentially shifting the system to a different fixed point where constraint information is present in the information environment, agents attend to it, and subsequent prices reflect it. Characterizing the fixed points of this iterated map and their stability properties - including whether multiple equilibria exist and what determines which equilibrium the system occupies - is a natural extension of the rational inattention program to endogenous information structure. This paper identifies the fixed-point structure but does not solve for the equilibria.
The proofs exploit the closed-form Gaussian rate-distortion function. Asset returns are not Gaussian (Cont, 2001). The core mechanism - scalar compression concentrating all capacity on one dimension when distributing would achieve lower total distortion - does not depend on the distributional form.
Among sources with a given covariance, the Gaussian has the highest R(D) (Cover and Thomas, 2006, Chapter 10). The Gaussian D* is therefore a lower bound: D*_Gaussian(R) >= D*_other(R) for non-Gaussian sources with the same covariance. For constraint dimensions, D_p is distribution-invariant (the price carries zero or negligible information regardless of distribution). Hence Delta_nonGaussian >= Delta_Gaussian. The Gaussian case is conservative.
Extensions to specific distributional classes (Rose, 1994; Yang and Zhang, 1999) could sharpen the bounds but are expected to strengthen the result.
MSE is symmetric: overestimating a constraint is penalized equally with underestimating it. Economic costs of constraint-dimension errors are asymmetric. Underestimating water stress (investing as if the aquifer is fine when it is not) produces stranded assets and discontinuous loss. Overestimating (avoiding investment when the aquifer is fine) produces foregone opportunity. The costs are not symmetric, not quadratic.
Under an asymmetric distortion measure d(v, v_hat) that penalizes underestimation of constraint severity more heavily than overestimation - a condition we call downside-dominant on constraint dimensions - the optimal codec at rate R allocates more bits to constraint dimensions than under MSE. The intuition: with downside-dominant loss, the cost of leaving a constraint dimension uncompressed (risking catastrophic underestimation) exceeds the MSE cost (which averages overestimation and underestimation symmetrically). The optimal allocation shifts bits from the consensus dimension toward constraints to reduce the probability of the more costly error.
The inequality Delta_asymmetric >= Delta_MSE holds under downside-dominant asymmetry on constraint dimensions. It reverses under the opposite asymmetry (if overestimation of constraint risk is costlier than underestimation - a false-alarm-dominant loss). For the physical constraints this paper considers - water curtailment, grid failure, supply chain disruption - downside dominance is the empirically relevant case: the cost of investing as if the aquifer is fine when it is not (stranded assets, discontinuous loss) far exceeds the cost of avoiding investment when the aquifer turns out to be fine (foregone opportunity).
A formal qualification: rate-distortion theory under non-MSE distortion measures does not generally admit the clean reverse-water-filling solution (Berger, 1971, Chapter 4). The optimal codec's allocation may not decompose dimension-by-dimension. The claim that "D*_asymmetric allocates more capacity to constraints" is an argument about the direction of the allocation shift under downside-dominant loss, not a computation of the asymmetric-loss rate-distortion bound. The formal rate-distortion theory under asymmetric loss is substantially more complex than the MSE case. The MSE results of this paper are therefore conservative with respect to the distortion measure under the empirically relevant asymmetry, but the precise magnitude of the additional excess distortion under asymmetric loss is not computed here.
The voluntary disclosure literature (Grossman, 1981; Milgrom, 1981; Verrecchia, 1983; Dye, 1985) analyzes conditions under which agents reveal private information. This literature has focused primarily on the consensus-value dimension. Proposition 2 identifies a different margin - constraint dimensions where the price carries zero or negligible information and where the marginal value of disclosure is highest. The disclosure calculus on constraint dimensions is structurally different from the consensus dimension: the disclosure is entirely additive rather than partially redundant with the price.
Prediction markets (Hanson, 2003; Wolfers and Zitzewitz, 2004; Arrow et al., 2008) elicit scored probabilistic judgments. A single prediction contract compresses diverse judgments into a scalar (the contract price), subjecting it to the same dimensional mismatch this paper characterizes.
Hanson's (2003) combinatorial information markets address the dimensionality problem by design: a set of contracts on {water curtailment by 2027, by 2028, by 2029} x {Phoenix, Tucson, Las Vegas} x {partial, full} produces a multi-dimensional price surface. In principle, with enough contracts, the dimensionality limitation disappears.
In practice, combinatorial markets face a thin-markets problem: the number of contracts in the product space grows exponentially with the number of dimensions, and liquidity in any individual contract is inversely related to the number of contracts. A combinatorial market with 3 x 3 x 2 = 18 contracts on a specific constraint interaction divides the available liquidity eighteen ways, producing noisy prices on each contract. The dimensional coverage is high but the precision per dimension is low. This is a different trade-off from the scalar price (high precision on one dimension, zero on the rest) but does not obviously dominate when the number of constraint dimensions is large relative to available market-making capital.
The excess-distortion framework provides a quantitative tool for evaluating this trade-off: the optimal allocation of a fixed rate budget R across K dimensions, which is exactly the reverse water-filling problem. The scalar price concentrates all R bits on dimension 1. The combinatorial market distributes bits across many dimensions. Whether the combinatorial market's allocation is closer to the reverse water-filling optimum than the scalar price depends on the eigenvalue spectrum and the liquidity constraints. The framework does not resolve this question but provides the formal structure for analyzing it.
ESG rating divergence (Berg, Koelbel, and Rigobon, 2022; average provider correlation approximately 0.54 versus credit rating correlation approximately 0.99) is a predicted consequence of lossy compression of multi-dimensional constraint information into scalar scores. Different rating agencies apply different proprietary methodologies to the same multi-dimensional source, producing divergent scalars because the compression is too lossy for convergence. In the present framework, ESG ratings are a second round of scalar compression applied to information that was already compressed by the price system.
Regulatory mandates (TCFD, SASB, ISSB, EU SFDR) require structured sustainability reporting. Christensen, Hail, and Leuz (2021) survey the growing literature. Krueger, Sautner, and Starks (2020) document institutional investor climate engagement. These mandates structure facts (emissions, water usage, board composition). The marginal value identified in Proposition 2 accrues to information about V_2, ..., V_K - judgment about how facts interact at scale, with calibrated confidence and verifiable predictions. The mandates target measurement of individual quantities; the excess distortion is about the multi-dimensional structure of interactions among quantities.
Propositions 1, 1', and 2 characterize what is lost and where supplementary information has the highest marginal value. The design of the optimal supplementary channel - format, incentive structure, scoring mechanism, institutional arrangement - is a mechanism design problem this paper does not solve. Companion work (Beier, 2026d) develops one candidate. Whether that mechanism or an alternative best exploits the opportunity is an open question. What is not open is that the opportunity exists.
Prediction 1 (Constraint-dimension factors). Factor analysis of asset returns augmented with physical constraint data (satellite measurements, environmental monitoring, infrastructure load data) will identify factors orthogonal to return-based factors that carry significant explanatory power for long-horizon returns. These augmented factors correspond to V_2, ..., V_K. If no such factors exist - if the augmented data adds no explanatory power beyond standard return-based factors - the source model is rejected.
Prediction 2 (Disclosure lag). When structured constraint-dimension assessments are published, asset prices will adjust to the disclosed information with a lag consistent with the information being absent ex ante. The lag distinguishes constraint-dimension information from consensus-dimension information, which is incorporated rapidly.
Prediction 3 (Constraint events not anticipated). Major constraint-relevant events (water curtailments, grid failures, supply chain disruptions from physical concentration) will not be systematically anticipated in price movements before the event. Prices will adjust discontinuously at or after the event. If prices systematically decline before constraint events, the zero/low-constraint-loading assumption is rejected.
Prediction 4 (Dimensional mismatch in ESG). A supplementary channel preserving dimensionality (multi-field assessments rather than aggregated scores) will produce assessments with provider correlations significantly exceeding 0.54 on the same entities.
Prediction 5 (Marginal value ordering). Investors receiving structured constraint-dimension information alongside price data will make allocation decisions with lower ex post distortion than investors receiving only additional price-precision improvements of equivalent information content. This directly tests Proposition 2.
The Gaussian assumption is used throughout. Section 6.1 argues this produces conservative estimates.
The MSE distortion measure understates the economic cost of constraint-dimension errors when losses are asymmetric. Section 6.2 argues MSE is a lower bound on economically relevant excess distortion under downside-dominant loss.
The single-price analysis (Propositions 1 and 1') is extended to the multi-price case (Proposition 3), but the effective-rank argument for why multiple prices do not span the constraint subspace is empirical rather than formal. A formal model of multi-price equilibrium with endogenous observation directions is deferred.
The eigenvalue parameterizations are illustrative. Empirical estimation from auxiliary data is the program this paper motivates but does not complete.
The paper characterizes the opportunity for supplementary channels but does not design one.
The price system is a lossy codec. On its own dimension, it is an optimal single-letter encoder under the Gaussian assumption. The excess distortion arises from its structural inability to allocate capacity to constraint dimensions. The excess is strictly positive in the economically relevant regime, increasing in constraint-dimension variance, and conservative with respect to both the distributional assumption and the distortion measure. Partial constraint loading - prices carrying some constraint information through cross-correlation - increases rather than decreases excess distortion, because shifting observation weight from the high-variance consensus dimension to lower-variance constraint dimensions reduces both signal power and compression rate. The eigenbasis case (zero constraint loading) is the most favorable for the price system, making the formal results a lower bound on real-world excess distortion. Multiple prices reduce but generically do not eliminate the excess unless they collectively span the full constraint subspace - a condition that fails when all prices load on the same consensus-value neighborhood. A supplementary channel targeting the uncompressed dimensions achieves higher marginal return per bit than further price refinement.
The result identifies where supplementary information channels have value. Whether such channels can be built at a cost below the distortion they reduce is a design question. That the distortion exists, that it concentrates on constraint dimensions, and that the marginal return to targeting those dimensions dominates the return to further consensus-dimension refinement - these are information-theoretic results for any multi-dimensional economy coordinating through scalar prices.