Grossman–Stiglitz Paradox

The Grossman–Stiglitz Paradox is a foundational theorem in financial economics, market microstructure, and business and economics theory, formulated by American economists Sanford J. Grossman and Joseph E. Stiglitz in their landmark 1980 American Economic Review article "On the Impossibility of Informationally Efficient Markets." Challenging the strong form of Eugene F. Fama's (1970) Efficient Market Hypothesis (EMH)—which posited that competitive asset prices P instantaneously and fully reflect all available fundamental information θ—Grossman and Stiglitz demonstrated that when gathering and processing fundamental information carries any positive resource cost c > 0, a competitive rational expectations equilibrium with fully revealing prices cannot exist. If the equilibrium market price P(θ) revealed the private signal θ to every market participant for free, no rational trader would pay the positive research cost c > 0 to become informed, since any uninformed trader could infer θ costlessly by observing the market price P and earn the same gross investment return; yet if zero traders pay c to acquire information (λ = 0), the market price P contains zero fundamental information about θ, creating a profit opportunity for an individual trader to pay c and trade on θ. To resolve this logical contradiction, Grossman and Stiglitz constructed the Noisy Rational Expectations Equilibrium (NREE) framework, proving that aggregate liquidity or supply noise x renders equilibrium prices partially revealing and allows an endogenous fraction λ* ∈ (0, 1) of informed traders to earn just enough extra expected trading returns over uninformed traders to offset the information acquisition cost c.

Hayek's Price System vs. the Information Free-Rider Problem

A central theme of neoclassical economics, articulated by Friedrich A. Hayek (1945) in "The Use of Knowledge in Society" and formalized in financial markets by the Efficient Market Hypothesis, is that the competitive price system aggregates dispersed private information. In Walrasian general equilibrium models where traders receive private signals exogenously at zero cost (c = 0), Robert E. Lucas Jr. (1972), Jerry R. Green (1973), and Sanford J. Grossman (1976) showed that in a Rational Expectations Equilibrium (REE), uninformed traders condition their asset demand on the market-clearing price P. When the mapping θ ↦ P(θ) is invertible, the equilibrium price is a sufficient statistic for all dispersed private information: observing P tells an uninformed trader everything that informed traders know.

However, Grossman and Stiglitz (1980) identified a logical contradiction in this fully revealing Rational Expectations Equilibrium: when information is a public good transmitted freely through the market price P(θ), it triggers an information free-rider problem whenever information acquisition is endogenous and costly (c > 0).

Formal Statement of the Paradox (Non-Existence of Competitive Equilibrium Without Noise)

Consider a competitive financial market in which each trader i ∈ [0, 1] chooses at date t = 0 either to pay research cost c > 0 to observe the fundamental signal θ of a risky asset (Informed Traders, fraction λ ∈ [0, 1]) or to pay zero and observe only the market-clearing price P (Uninformed Traders, fraction 1 − λ). Suppose per-capita asset supply x̄ is a known constant (no supply noise), so P depends only on θ whenever λ > 0:

1. Case 1 — Positive Fraction Acquires Information (λ > 0): Because asset supply x̄ is known and informed traders condition demand on θ, the market-clearing price P(θ) is an invertible function of θ. Every uninformed trader observes P(θ), inverts it to learn θ for free, and earns the same gross trading return as an informed trader. Thus the net utility of an informed trader is lower by the research cost c > 0, so no trader chooses to become informed, contradicting λ > 0.
2. Case 2 — Zero Traders Acquire Information (λ = 0): If λ = 0, no trader knows θ, so the market-clearing price P₀ is independent of θ. A single atomistic trader i who pays c > 0 to learn θ while taking P₀ as given buys heavily when θ > E[θ] and shorts when θ < E[θ], earning an informational trading gain ΔU > c. Thus every trader prefers to become informed, contradicting λ = 0.

The Grossman–Stiglitz Paradox (Theorem 6, Grossman and Stiglitz, 1980): In a competitive financial market without supply noise (σ_x² = 0) where acquiring fundamental information carries a positive cost c > 0 (smaller than the autarky value of information), no competitive Rational Expectations Equilibrium exists.

Equilibrium requires supply noise that prevents uninformed traders from inverting P to extract θ, inspiring both the Noisy Rational Expectations Equilibrium (NREE) literature and the strategic microstructure models of Kyle (1985) and Glosten and Milgrom (1985).

Preferences, Asset Payoff Structure, and Supply Noise

To restore the existence of competitive equilibrium and solve for the endogenous degree of market informational efficiency, Grossman and Stiglitz (1980) embedded an unobservable random per-capita supply (liquidity / noise trading) shock x into a CARA-Normal asset pricing model:

• Traders and CARA Utility: A continuum of traders i ∈ [0, 1] share identical Constant Absolute Risk Aversion (CARA) utility over terminal wealth W_1i with risk-aversion coefficient a > 0:

• Safe and Risky Assets: Each trader is endowed with initial wealth W_0i and allocates funds between a risk-free asset with gross return R ≥ 1 (normalized so price is 1) and a risky asset trading at equilibrium price P per share. At date t = 1, each share of the risky asset pays a stochastic terminal dividend u:

where θ ~ N(E[θ], σ_θ²) is the observable fundamental component (which can be learned at date t = 0 by paying research cost c > 0) and ε ~ N(0, σ_ε²) is an unlearnable residual cash-flow shock independent of θ. Thus, the unconditional variance of the asset's terminal payoff is Var(u) = σ_θ² + σ_ε², while an informed trader who pays c and observes θ faces only the residual conditional variance Var(u | θ) = σ_ε².

• Unobserved Random Supply Noise (x): Total per-capita supply of the risky asset available in the market is a random variable x ~ N(E[x], σ_x²) with σ_x² > 0, independent of both θ and ε. Neither informed nor uninformed traders observe x directly; economically, x represents exogenous liquidity trades, endowment shocks, or lifecycle rebalancing by noise traders.

Portfolio Demand of Informed Traders (X_I)

Suppose a fraction λ ∈ [0, 1] of traders has paid the cost c > 0 to observe θ. For any informed trader I, terminal wealth when purchasing X_I shares of the risky asset at price P is:

Because u | θ ~ N(θ, σ_ε²), maximizing expected CARA utility E[−exp(−aW_1I) | θ, P] is equivalent to maximizing the mean-variance certainty equivalent E[W_1I | θ] − (a/2) Var(W_1I | θ), yielding the CARA optimal asset demand for an informed trader:

How Uninformed Traders Extract a Noisy Signal From the Market Price P_λ

What do the remaining 1 − λ uninformed traders know when they observe the market-clearing price P_λ(θ, x)? Although uninformed traders do not know θ or x separately, they understand the market-clearing condition:

Substituting the informed demand X_I(θ, P_λ) = (θ − RP_λ)/(aσ_ε²) into the market-clearing equation shows that the price P_λ depends on the two unobserved random variables (θ, x) through a single linear combination φ_λ(θ, x), which Grossman and Stiglitz term the normalized price signal:

When an uninformed trader sees a high market price P_λ (and hence a high price signal φ_λ), the uninformed trader cannot tell with certainty whether the price is high because informed traders observed good fundamental news (high θ) or because per-capita asset supply is scarce (low x < E[x]). The supply shock x camouflages the fundamental signal θ inside the market price P_λ.

Quantifying the Signal-to-Noise Ratio of the Price System

Because φ_λ(θ, x) = θ − (aσ_ε²/λ)(x − E[x]) is the sum of the true signal θ (variance σ_θ²) and the scaled supply-noise error −(aσ_ε²/λ)(x − E[x]) (variance (aσ_ε²/λ)² σ_x²), Grossman and Stiglitz defined the price-informativeness (precision) parameter w(λ) as the inverse variance of the supply noise in φ_λ:

Using standard normal projection rules, conditional on observing the market price P_λ (which is informationally equivalent to observing φ_λ), an uninformed trader's posterior expectation and posterior return variance of the risky payoff u = θ + ε are:

And the squared correlation coefficient R² between the equilibrium price signal φ_λ and the true fundamental θ—measuring how much private information the price system reveals to the public—is:

1. As the fraction of informed traders λ increases, informed trading volume dominates noise x, raising w(λ) and making the market price P_λ more informative (Var(u | P_λ) shrinks toward σ_ε²).
2. As supply noise σ_x² increases (or risk aversion a and residual risk σ_ε² increase), informed traders trade more cautiously per unit of θ and supply noise obscures their orders, lowering w(λ) and making the market price less informative.

Expected Utility Ratio and the Noise-Invariance Theorem

Before θ and x are realized, traders compare the unconditional expected utility E[V(W_1I)] of becoming an informed trader against the unconditional expected utility E[V(W_1U)] of remaining an uninformed trader. By evaluating the normal moment-generating function over the equilibrium price distribution, Grossman and Stiglitz derived an identity for the ratio of expected utilities of informed to uninformed traders at any given fraction λ ∈ [0, 1] (Grossman and Stiglitz, 1980, Equation 15):

For CARA utility V(W) = −exp(−aW) < 0, an interior Noisy Rational Expectations Equilibrium λ* ∈ (0, 1) requires that traders be indifferent ex ante between paying c to become informed and paying 0 to remain uninformed (E[V(W_1I)] / E[V(W_1U)] = 1):

Solving for the equilibrium price-informativeness statistic w* = w(λ*) gives:

Notice the Invariance of Equilibrium Price Informativeness to Supply Noise σ_x²: the supply-noise variance σ_x² does not appear on the right-hand side of the equation for w(λ*). If the variance of noise trading σ_x² doubles, at a fixed fraction λ of informed traders w(λ) initially falls by half, raising the return to private research above c. This attracts new entrants into the informed research sector (λ* increases by √2) until the larger group of informed traders offsets the higher supply noise, restoring w(λ*) = (λ*/a)² / (σ_ε² σ_x²) and ρ_θ,P² to their original values. Financial markets thus operate in an "equilibrium degree of disequilibrium" where prices reflect fundamental information up to the point where the marginal informational rent earned by active analysts covers the marginal cost c of research.

What is the Grossman–Stiglitz Paradox in plain language?

The Grossman–Stiglitz Paradox states that if stock prices always reflected 100% of all fundamental information, then no investor or security analyst could ever beat the market by spending time and money researching companies (c > 0), because anyone could get the same information for free simply by looking at the stock price. But if everyone stopped researching companies and relied passively on the stock price, then nobody would be gathering information, so stock prices would stop reflecting fundamentals. Therefore, markets can never be 100% informationally efficient when research is costly: prices must remain partially noisy so that active researchers earn enough excess return to pay for their research costs.

How do noise traders or random supply shocks (x) resolve the Grossman–Stiglitz Paradox?

When random supply or liquidity shocks x (often called noise trading following Fischer Black, 1986, and Kyle, 1985) hit the market, an uninformed trader who observes a price increase cannot tell whether the price rose because informed traders received good fundamental news θ or because noise traders happened to buy shares (low net supply x). This ambiguity prevents uninformed traders from learning θ for free from the market price P, preserving a positive informational advantage Var(u | P) > Var(u | θ) for informed traders who paid the research cost c.

What happens in the Grossman–Stiglitz model if the cost of acquiring information c falls due to better technology?

When the cost of acquiring information c decreases (for example, through faster electronic disclosures or computational analytics), exp(2*ac*) falls toward 1. To restore equilibrium indifference between informed and uninformed traders, more traders choose to become informed (λ* increases), driving up the price-informativeness statistic w(λ*) and raising the correlation ρ_θ,P² between the market price P and the true fundamental θ.

How does the Grossman–Stiglitz (1980) model relate to the Kyle (1985) and Glosten–Milgrom (1985) models?

All three models form the foundational triad of modern information economics in financial markets:

• Grossman and Stiglitz (1980) analyzed a competitive (price-taking) Noisy Rational Expectations Equilibrium with a continuum of small traders and endogenous costly information acquisition (c > 0).
• Kyle (1985) analyzed a strategic monopolistic insider who is large enough to move the market price and restricts order size X = β(θ − p₀) to hide behind noise traders' order flow u ~ N(0, σ_u²).
• Glosten and Milgrom (1985) analyzed sequential quote-driven trading where competitive market makers widen the bid–ask spread to protect themselves against adverse selection from informed traders.

1. Grossman, S. J., & Stiglitz, J. E. (1980). "On the Impossibility of Informationally Efficient Markets." American Economic Review, 70(3), 393–408. https://doi.org/10.7916/D8765R99
2. Grossman, S. J. (1976). "On the Efficiency of Competitive Stock Markets Where Trades Have Diverse Information." The Journal of Finance, 31(2), 573–585. https://doi.org/10.1111/j.1540-6261.1976.tb01907.x
3. Fama, E. F. (1970). "Efficient Capital Markets: A Review of Theory and Empirical Work." The Journal of Finance, 25(2), 383–417. https://doi.org/10.2307/2325486
4. Kyle, A. S. (1985). "Continuous Auctions and Insider Trading." Econometrica, 53(6), 1315–1335. https://doi.org/10.2307/1913210
5. Hellwig, M. F. (1980). "On the Aggregation of Information in Competitive Markets." Journal of Economic Theory, 22(3), 477–498. https://doi.org/10.1016/0022-0531(80)90056-390056-3)