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Asset Pricing & Information Economics

Grossman-Stiglitz Equilibrium Informed Trader Formula

Solve Sanford Grossman & Joseph Stiglitz’s (1980) equilibrium informed-trader share lambda* and prove why costly fundamental research requires noisy market prices.

Prove the Grossman-Stiglitz (1980) paradox: 100% price efficiency collapses when research is costly!

Sanford Grossman and Joseph Stiglitz (1980) resolved a foundational paradox in Strong/Semi-Strong Efficient Market theory: if market prices reflected 100% of costly fundamental research for free, NO rational investor would pay research cost c_info > 0 — leaving prices completely uninformative! In their CARA-Normal noisy rational expectations model (risk aversion a > 0, learnable signal variance sigma_theta^2, unlearnable residual variance sigma_eps^2, supply noise variance sigma_x^2 > 0), equilibrium requires indifference EXP(2 * a * c_info) = Var(u | P) / sigma_eps^2, yielding exact Equilibrium Informed Share lambda* = MIN(1, ((a * sigma_eps^2 * SQRT(sigma_x^2)) / SQRT(sigma_theta^2)) * SQRT((g_max - k_exp) / k_exp)) where g_max = sigma_theta^2 / sigma_eps^2 and k_exp = EXP(2 * a * c_info) - 1!

Verify on CARA_a = 1 (B2), Signal_Var_sigTheta2 = 4 (B3), Residual_Var_sigEps2 = 1 (B4 -> g_max = 4), Supply_Noise_sigX2 = 0.25 (B5), and Research_Cost_c = 0.25 (B6): k_exp = EXP(0.5) - 1 = 0.6487, Conditional Uninformed Variance Var(u | P) = 1.6487, Equilibrium Informed Trader Fraction is lambda* = 0.5682 (56.82%), and Price Informativeness R^2_P = 1 - (Var(u | P) - sigma_eps^2) / sigma_theta^2 = 0.8378 (83.78% of signal revealed — leaving an exact 16.22% alpha wedge to compensate research!). Now set Supply_Noise_sigX2 = 0 with c = 0.25 to see the Grossman-Stiglitz Paradox trigger explicitly!

Grossman-Stiglitz Metric (a=1, sig_th^2=4, sig_ep^2=1, sig_x^2=0.25, c=0.25)FormulaVerified Output
Uninformed Conditional Variance Var(u | P)sigma_eps^2 * EXP(2 * a * c)1.6487
Equilibrium Informed Share (lambda*)CARA-Normal Indifference Root0.5682 (56.82% Informed)
Price Signal Informativeness (R^2_price)1 - (Var(u|P) - sigma_eps^2)/sigma_theta^20.8378 (83.78% Revealed)
Zero Noise + Positive Cost (sig_x^2 = 0)Grossman-Stiglitz ImpossibilityNo Equilibrium Paradox

Grossman-Stiglitz Equilibrium Informed Trader Fraction lambda* Formula

Branch cleanly on c_info = 0 (100% informed), sigma_x^2 = 0 with c_info > 0 (Grossman-Stiglitz Paradox), and k_exp >= g_max (0% informed when research is too expensive).

=LET(a_ra, B2, st_2, B3, se_2, B4, sx_2, B5, c_in, B6, IF(OR(a_ra <= 0, st_2 <= 0, se_2 <= 0, sx_2 < 0, c_in < 0), "Invalid GS Input", IF(c_in = 0, 1, IF(sx_2 = 0, "Grossman-Stiglitz Paradox: No Equilibrium (Zero Noise + Costly Info)", LET(g_mx, st_2 / se_2, k_ex, EXP(2 * a_ra * c_in) - 1, IF(k_ex >= g_mx, 0, ROUND(MIN(1, ((a_ra * se_2 * SQRT(sx_2)) / SQRT(st_2)) * SQRT((g_mx - k_ex) / k_ex)), 4)))))))

Equilibrium Price Signal Informativeness R^2_price Formula

Observe the striking comparative static: Var(u | P) = MIN(sigma_theta^2 + sigma_eps^2, sigma_eps^2 * EXP(2 * a * c)) depends ONLY on research cost c, risk aversion a, and residual variance — doubling noise trading sigma_x^2 draws in proportionally more researchers lambda*, leaving price informativeness unchanged!

=LET(a_ra, B2, st_2, B3, se_2, B4, sx_2, B5, c_in, B6, IF(OR(a_ra <= 0, st_2 <= 0, se_2 <= 0, sx_2 < 0, c_in < 0), "Invalid GS Input", IF(AND(sx_2 = 0, c_in > 0), "Grossman-Stiglitz Paradox: No Equilibrium (Zero Noise + Costly Info)", LET(k_ex, EXP(2 * a_ra * c_in) - 1, ROUND(MAX(0, 1 - (se_2 * k_ex) / st_2), 4)))))

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