## _wp10266 — References / conclusion to be drawn is that the successful introduction of these new financial instruments

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### I. Purpose, mechanism, and core claims
- Objective:
  - Highlight costly information collection as a channel through which financial innovation affects investor-held risk, aggregate fluctuations, and welfare.
  - Use evidence (French [2008]) of a shift from active to passive investment in the US stock market without a relative fall in active-investing costs.
- Core theoretical mechanism:
  - Agents choose between active investors (acquiring costly private information) and passive investors (relying on less costly, less precise public information).
  - Private information reduces investments in low-quality, excessively risky technologies.
  - Aggregate private information generates a positive externality by improving public information precision (noisy trading literature: Grossman and Stiglitz [1980], Kyle [1985], Summers and Shleifer [1990]).
  - Financial innovation reduces private information collection when it does not lower the relative cost of active versus passive investing, because information is a strategic substitute for risk diversification.
- Welfare and cycle implications (summary):
  - Financial innovation provides insurance via expanded asset sets and better diversification (positive welfare effect).
  - Greater insurance reduces incentives to collect private information, which can lower welfare (moral-hazard-like effect).
  - Reduced private information harms passive investors via the public-information externality, increasing mistaken investments in excessively risky technologies.
  - Model produces business-cycle statistics linking information collection to volatility and sensitivity to systemic shocks; in a dynamic setting financial development lowers information collection in idiosyncratic-risk periods but not in systemic-risk periods, increasing frequency of “liquidation crises.”

### II. Empirical evidence emphasized
- Three empirical facts for the US:
  - a) Tremendous increase in the menu of assets and diversification possibilities.
  - b) Liquidity and trading costs have trended downwards since the ’60s.
  - c) The cost of active investment strategies has not decreased relative to the cost of passive strategies.
- Representative quantitative facts:
  - High-yield bond issuances: about $50 billions in 1993 to $180 billions in 2006, with the US share going down to 70% (BIS [2008]).
  - Worldwide buyouts by private equity funds: from $30 billions to more than $800 billions (Kaplan and Strömberg [2009]).
  - Asset-backed securities market: 1998 value $271 billion; increased four times to over $1,200 billions in 2006 (Weaver [2008]).
  - US mortgage securitization: reached 80.5% of mortgages originated in 2006, total value $2.4 trillions (Gorton [2008]).
- Cost-trend evidence (French [2008]):
  - Absolute cost of passive strategies trending downwards.
  - Absolute cost of active strategies decreased to a much lesser extent.
  - Relative cost of active strategies roughly doubled in the period 1991-2006.

### III. Model setup and key definitions
- Environment:
  - Two-period economy t = 1,2; 2N linear production technologies; N markets m = 1,...,N with technologies j = 1,2 per market.
  - In each market m one technology is low risk and the other is excess risk; return on excess risk technology is a mean preserving spread of the low risk return.
- Technology returns and shocks:
  - θ_m ∈ {1,2} decides excess risk technology j = θ_m.
  - r_{m,3−θ_m} = i_m
  - r_{m,θ_m} = i_m e_m S
  - i_m and e_m i.i.d. across markets; S systemic shock common to all markets. Normalization: E[i_m] = E[e_m] = E[S] = 1. Variances σ^2_i, σ^2_e, σ^2_S.
  - Excess risk z_m = i_m(e_m S − 1) and E[z_m | r_{m,3−θ_m}] = 0.
- Information choices and costs:
  - Active investor: pays cost reducing wealth to fraction 1/ξ_A (ξ_A > 1); observes private signal I_A(Θ) = Θ.
  - Passive investor: pays cost reducing wealth to fraction 1/ξ_P (1 < ξ_P < ξ_A); observes public signal I_P(Θ) = I_ρ Θ where I_ρ ∈ {0,1} equals 1 with probability ρ ∈ [0,1].
  - Relative information cost ξ = ξ_A / ξ_P > 1.
  - Spillover: fraction of active investors η implies ρ = A_E η for A_E ∈ (0,1).
- Diversification barrier:
  - Investors specialized in market h; constrained to invest at most fraction 1 − ̄α of portfolio outside specialization, with ̄α ∈ [1/N, 1]. Financial development reduces distortions D = (ξ_A, ξ_P, ̄α) (i.e., reduces investment costs and diversification barriers).

### IV. Optimal behavior and equilibrium conditions
- Optimal informed portfolio (I = Θ), given ̄α ≥ 1/N:
  - α^{h*}_{m,θ_m}(Θ) = 0 ∀ m
  - α^{h*}_{h,3−θ_h}(Θ) = ̄α
  - α^{h*}_{m,3−θ_m}(Θ) = (1 − ̄α)/(N − 1) for m ≠ h
- Optimal uninformed portfolio (I = 0):
  - α^{h*}_{h,j}(0) = ̄α/2 for j = 1,2
  - α^{h*}_{m,j}(0) = (1 − ̄α) / (2(N − 1)) for m ≠ h, j = 1,2
- Informed investor holds a less risky portfolio than an uninformed investor: R_0 < R_Θ where R_I = [ E(ˆr_h_I^{1−γ}) ]^{1/(1−γ)} (equation (13)).
- Saving rate (as printed):
  - x^*_I = 1 / (1 + β^{−ψ} R_I^{1−ψ}) (equation (15)).
- Equilibrium revelation rate ρ^* when marginal investor indifferent:
  - ρ^* = 1 − ξ^{γ−1} −1 ( ˆV_Θ / ˆV_0 )^{γ−1 −1} (equation (17)) — internal solution ρ^* ∈ (0,1) when ( ˆV_Θ / ˆV_0 )^{γ−1} > ξ^{γ−1}.

### V. Comparative statics: financial innovation and information collection
- Proposition 1: Unique equilibrium; ˆV_I = (β ψ R_{ψ−1}_I + 1)^{1/(ψ−1)} (equation (18)).
- Revelation rate ρ^*(D) behavior under reductions in distortions that leave ξ unchanged:
  - Case 1: i_m = 1 a.s. (low risk technology risk free) ⇒ ρ^*(D) strictly increasing in financial development (reductions in distortions).
  - Case 2: e_m = 1 a.s. (excess risk only systemic) ⇒ ρ^*(D) strictly increasing for ψ < 1 and strictly decreasing for ψ > 1.
- Intuition:
  - Information collection and diversification are strategic substitutes: greater diversification (lower ̄α) raises insurance, reducing incentives to collect private information.
  - γ (risk aversion) amplifies magnitude of changes; ψ (EIS) determines sign in case ii).
- Derivative for e_m = 1 case (as printed):
  - ρ′^*_{1−̄α} ≡ ∂ρ^*/∂(1−̄α) = g(R_Θ, Ω) ∂R_Θ/∂(1−̄α) (1 − Ω^{ψ−1})^{-1} with Ω = (E S^{1−γ})^{1/(1−γ)} < 1.

### VI. Welfare, externalities, and policy-relevant implications
- Welfare expression for interior ρ^*: investors’ welfare equals ˆV_Θ / ξ_A.
- Financial innovation channels for welfare:
  - Positive: lower ξ_A and lower ̄α (more diversification) raise ˆV_Θ.
  - Negative: endogenous decrease in ρ^* (less private information) can offset or eliminate welfare gains, because lower public-information quality harms passive investors.
- Mechanism summary:
  - Greater diversification reduces incentive to collect private information; reduced private information lowers public-signal quality ρ^*, harming passive investors — an information externality.
- Policy-relevant considerations (derived from findings):
  - Monitor the relative cost of active versus passive investing: if active costs do not fall with financial innovation, private-information incentives may erode.
  - Be aware of trade-offs between expanded diversification (insurance) and incentives for costly information acquisition; policies should aim to preserve information incentives where social externalities exist.
  - Consider mechanisms to internalize the public-information externality created by private information collectors to mitigate welfare losses for passive investors and reduce the likelihood of liquidation crises.

### VII. Business-cycle statistics and tail-risk effects
- For small total noise ̄L, business-cycle statistics (equation (21)):
  - σ^2_A = ˆσ^2_ω σ^2_i + 1/4 (σ^2_S + ˆσ^2_ω Σ^2) (1 − ρ^*) + O(̄L)
  - γ_{A,S} = 1/2 (1 − ρ^*) + O(̄L)
- Trade-offs under financial innovation:
  - Lower ̄α reduces dispersion ˆσ^2_ω, decreasing σ^2_A by mitigating amplification of idiosyncratic volatility.
  - Reductions in ρ^* increase exposure of aggregate productivity to systemic shocks; sensitivity γ_{A,S} increases with lower ̄α.
- Numerical example (parameters as printed):
  - i_m ∈ {1.18, 0.82} with equal probability; e_m ∈ {1.18, 0.82} equal probability; S = 1.1 with probability 0.82 and S = 0.57 otherwise; N = 120; ξ = 1.005; β = 0.97; ψ = 1/γ = 0.5; p_h = 0.18·0.82^{h−1}.
  - Computed volatilities: σ^2_A(̄α = 1) = 6.3·10^{−3}, σ^2_A(̄α = 1/N) = 5.5·10^{−3}.
  - Result: equilibrium ρ^* decreases as ̄α decreases; σ^2_A decreases (dominated by reduction in ˆσ^2_ω), while sensitivity γ_{A,S} to systemic shocks increases (tail risk rises).

### VIII. Multi-period extension: liquidation crises and welfare
- Three-period extension (t = 1,2,3) with time-varying risk:
  - Time 1: excess risk idiosyncratic.
  - Time 2: excess risk systemic.
- Liquidation crisis mechanism:
  - A reduction δ ∈ (0,1) in time 2 returns occurs if, at beginning of time 2, public information reveals types of previously uninformed investors, triggering liquidations.
- Endogenous probability of liquidation crisis π_λ (equation (29)):
  - I^λ_2 = (1 − I^ρ_1)[S_1 + (1 − S_1) I^ρ_2] (equation (28)).
  - π_λ = Prob{I^λ_2 = 1} = (1 − ρ_1)[π + (1 − π) ρ_2].
- Revelation rates in interior equilibrium (Proposition 3, as printed):
  - ρ^*_t = 1 − ln ξ / (β ln (R_{t,Θ} / R_{t,0})) (equation (32)).
  - ρ^*_2 > ρ^*_1.
  - A decrease in ̄α decreases ρ^*_1, leaves ρ^*_2 unchanged, and increases probability of liquidation crises; welfare decreases if ̄α decreases.
- Welfare implication:
  - Financial innovation that reduces barriers to diversification (lower ̄α) can increase the frequency of liquidation crises when information costs remain relatively high, producing potential net welfare losses.

### IX. Appendices and data construction notes (high-level)
- APPENDIX A–D: formal proofs of portfolio optimality, signal extraction, business-cycle statistics, and dynamic solution with revelation rates; key analytical results summarized above.
- APPENDIX E (data for Figure 1):
  - Data from French (2008) tables.
  - Composite active strategy built as weighted average of mutual funds, pension funds and hedge funds; hedge fund pre-1996 missing values filled with 9.27% (1996 value) with limited impact due to small weights.

*Source: _wp10266 - References / conclusion to be drawn is that the successful introduction of these new financial instruments*

### References .............................................................................................................

### _wp10266 - References

### I. Introduction — purpose and mechanism
- Financial innovation defined: “new derivative contracts, new corporate securities, new forms of pooled investment products, [...] or by new means of distributing securities and processing transactions” (Tufano [2002]).
- Traditional purported benefits:
  - Spurs growth by easing fund flows to higher-productivity projects (Levine [1997], Rajan and Zingales [1998], Azariadis and Kass [2003]).
  - Reduces investor risk through broader asset menus enabling diversification and risk sharing (Merton [1987], Mendoza et al. [2008]).
- Recent critiques:
  - Claims that financial development may increase “excessive risk taking” (Rajan [2005]).
  - Securitization’s risk-dispersal benefits questioned (Shin [2009]).
- Paper’s objective:
  - Highlight the role of costly information collection as a channel through which financial innovation affects investor-held risk, aggregate fluctuations, and welfare.
  - Use evidence (French [2008]) indicating a shift from active to passive investment in the US stock market, without a relative fall in active-investing costs.
- Core theoretical mechanism:
  - Agents choose between being active investors (acquiring costly private information) and passive investors (relying on less costly, less precise public information).
  - Private information reduces investments in low-quality, excessively risky technologies.
  - The aggregate amount of private information generates a positive externality by improving public information precision (noisy trading literature: Grossman and Stiglitz [1980], Kyle [1985], Summers and Shleifer [1990]).
  - Financial innovation reduces private information collection when it does not lower the relative cost of active versus passive investing, because information is a strategic substitute for risk diversification.
- Welfare and cycle implications summarized:
  - Financial innovation provides insurance via expanded asset sets and better diversification (positive welfare effect).
  - Greater insurance reduces incentives to collect private information, which can lower welfare (moral hazard analog).
  - Reduced private information harms passive investors via the public-information externality, increasing mistaken investments in excessively risky technologies.
  - Model produces business-cycle statistics linking information collection to volatility and sensitivity to systemic shocks, contributing a perspective on the great moderation literature (Stock and Watson [2002], Fogli and Perri [2006], Dynan et al. [2006], Stiroh [2009]).
  - In a dynamic setting with time-varying risk, financial development lowers information collection during periods when excess risk is idiosyncratic (insurable) but not when excess risk is systemic, increasing the frequency of “liquidation crises” — mistaken investment during idiosyncratic excess risk followed by sudden liquidation when risk becomes systemic. If liquidation crises impose aggregate costs, financial development can reduce welfare.

### II. Empirical evidence highlighted
- Three emphasized empirical facts for the US:
  - a) Tremendous increase in the menu of assets and diversification possibilities for investors.
  - b) Liquidity and trading costs have trended downwards since the ’60s.
  - c) The cost of active investment strategies has not decreased relative to the cost of passive strategies.
- Representative quantitative facts (examples provided):
  - High-yield bond issuances:
    - about $50 billions in 1993 (issuances entirely in the US) to $180 billions in 2006, with the US share going down to 70% (BIS [2008]).
  - Worldwide buyouts by private equity funds:
    - from $30 billions to more than $800 billions (Kaplan and Strömberg [2009]).
  - Asset-backed securities market:
    - 1998 value: $271 billion; increased four times to over $1,200 billions in 2006 (Weaver [2008]).
  - US mortgage securitization:
    - reached 80.5% of mortgages originated in 2006, for a total value of $2.4 trillions (Gorton [2008]).
- Empirical trend relevant to model:
  - US stock market holdings shifted from active to passive forms over the past 20 years (Cremers and Petajisto [2009], French [2008]).

### III. Theoretical findings and implications
- Information and strategic interaction:
  - Private information collection is a strategic choice; aggregate private information improves public information via a positive externality.
  - Information acts as a strategic substitute for diversification provided by financial innovation.
- Effects of financial innovation (under maintained assumptions on asset returns):
  - Financial innovation decreases private information collection when it does not reduce the relative cost of active investment compared to passive investment.
  - Welfare effects are two-sided:
    - Positive: improved insurance and diversification from a larger asset set.
    - Negative: reduced incentives to collect costly private information lead to more frequent mistaken investments in excessively risky, potentially systemic technologies.
- Business-cycle and crisis dynamics:
  - Lower private information increases volatility and sensitivity of aggregate productivity to systemic shocks.
  - Time-varying risk extension:
    - Financial development reduces information collection in idiosyncratic-risk periods but not in systemic-risk periods.
    - Increases the frequency of liquidation crises (mistaken risky investments during idiosyncratic-risk periods liquidated abruptly when risk turns systemic).
  - If liquidation crises impose aggregate costs, financial development can reduce welfare.

### IV. Policy-relevant considerations (derived from findings)
- Monitoring the relative cost of active versus passive investing is crucial: if active costs do not fall with financial innovation, private-information incentives may erode.
- Be aware of trade-offs between expanded diversification (insurance) and incentives for costly information acquisition; policies should aim to preserve information incentives where social externalities exist.
- Consider mechanisms to internalize the public-information externality created by private information collectors to mitigate welfare losses for passive investors and reduce the likelihood of liquidation crises.

*Source: _wp10266 - References.*

### conclusion to be drawn is that the successful introduction of these new financial instruments

### _wp10266 - conclusion to be drawn is that the successful introduction of these new financial instruments

### Financial innovation, trading costs, and information costs
- Successful introduction of new financial instruments (high-yield bonds, private equity, mortgage backed securities) opened up new opportunities in markets (for high risk firms, unlisted companies, housing) that previously had more limited access to external financing, mostly provided by very specialized investors (e.g. banks).
- A number of studies (for instance Jones [2002], French [2008]) confirm that trading costs in the stock market have decreased over time due to reductions in both bid-ask spreads and in trading fees.
- Empirical evidence (French (2008)) provides cost series for various investment strategies in the US stock market. Two alternative cost measures are analyzed:
  - Cost of active and passive Defined Benefit pension plans.
  - Cost of composite active and passive strategies comprising investments in mutual funds, pension funds, hedge funds and direct equity holding.
- Common message from Figure 1 and related analysis:
  - the absolute cost of passive strategies has been trending downwards over time;
  - the absolute cost of active strategies has decreased to a much lesser extent;
  - the relative cost of active strategies has significantly increased over time, roughly doubling in the period 1991-2006.
- Caveat: a quality-adjusted measure of information costs (if active strategies have become much better at producing information) could alter the interpretation, but as a first order approximation quality adjustments are not expected to invert the clear trend.

### Model overview: goals and characterization
- The basic model relates information collection to financial innovation and highlights consequences for:
  - agents’ risk taking behavior,
  - the business cycle,
  - welfare of a financial innovation process characterized by points a)-b)-c) (as referenced in the text).
- The model economy:
  - lasts two periods t = 1,2;
  - is endowed with 2N linear production technologies using consumption goods at t = 1 to produce consumption goods at t = 2;
  - technologies are evenly distributed across N markets m = 1,...,N, with exactly two technologies j = 1,2 per market;
  - in every market m, one technology is low risk and the other is excess risk; the return on the excess risk technology in m is a mean preserving spread of the return on the low risk technology in m.

### Technologies and information structure (formal definitions and notation)
- Type vector and returns:
  - θ_m ∈ {1,2} decides that technology j = θ_m is the excess risk technology in m.
  - Θ = (θ_1,...,θ_N), T = {1,2}^N.
  - r_{m,3−θ_m} = i_m (1)
  - r_{m,θ_m} = i_m e_m S (2)
- Shocks and moments:
  - Random variables i_m and e_m are i.i.d. across markets; S is the systemic shock common to all markets.
  - Normalization: E[i_m] = E[e_m] = E[S] = 1.
  - Denote variances σ^2_i, σ^2_e, σ^2_S.
- Symmetry condition on joint distribution of types:
  - F(Θ) = F(θ_1,...,θ_N) = F(3−θ_1,...,3−θ_N) = F( ̄Θ) (3)
  - ̄Θ denotes the opposite type vector ̄Θ = 3−Θ.
- Excess risk decomposition:
  - r_{m,θ_m} = r_{m,3−θ_m} + z_m, where z_m = i_m(e_m S − 1) and E[z_m | r_{m,3−θ_m}] = 0.
  - Sources of excess risk: excess idiosyncratic risk e_m and excess systemic risk S.

### Information choices, costs, and financial development distortions
- At beginning of time 1 agents know Θ ∈ T_Θ where T_Θ ≡ Θ ∪ ̄Θ (4); conditional on this information Θ and ̄Θ are equally likely by (3).
- Investor types and signals:
  - Active investor:
    - reduces investor’s wealth to a fraction 1/ξ_A of original value, with ξ_A > 1;
    - observes private signal I_A(Θ) = Θ (5).
  - Passive investor:
    - reduces wealth to a fraction 1/ξ_P, with 1 < ξ_P < ξ_A;
    - observes public signal I_P(Θ) = I_ρ Θ (6), where I_ρ ∈ {0,1} equals 1 with probability ρ ∈ [0,1].
- Interpretation:
  - Active investors obtain fully informative private signal but pay higher cost; passive investors pay lower cost but rely on public signal, which is informative only if I_ρ = 1 (probability ρ).
- Spillover of private information to public signal:
  - Define η ∈ [0,1] the fraction of active investors at time 1.
  - Assume ρ = A_E η (8) for A_E ∈ (0,1).
- Relative information cost:
  - Define ξ = ξ_A / ξ_P > 1.
- Diversification barrier and specialization:
  - Each investor draws a specialization h in market h = 1,...,N with fraction p_h.
  - Agents constrained to invest at most fraction 1 − ̄α of their portfolio outside specialization, where ̄α ∈ [1/N, 1] is the barrier to diversification.
- Degree of financial development summarized by distortions D = (ξ_A, ξ_P, ̄α). Financial innovation reduces D (investment costs and diversification barriers).

### Preferences and investor optimization
- Preferences: homothetic utility disentangling risk aversion γ and intertemporal elasticity of substitution ψ.
  - Conditional value for consumption process {c_{t,I}}:
    - V_I = { c_1−1/ψ_{1,I} + β [ E(c_2^{1−γ}_I) ]^{(1−1/ψ)/(1−γ)} }^{1/(1−1/ψ)} for β ∈ [0,1].
  - Ex-ante utility:
    - U = [ E V_I^{1−γ} ]^{1/(1−γ)} (9).
  - γ > 0, γ ≠ 1; ψ > 0, ψ ≠ 1. CRRA utility recovered when γ = 1/ψ.
- Investor’s timing and choices:
  - At t = 1 investor chooses τ ∈ {A,P} (active or passive), observes signal I_τ, then chooses saving rate x and portfolio shares α_{m,j} subject to diversification barrier ̄α.
  - Optimization problem (for an investor with specialization h and initial unit wealth) maximizes EV_I^{1−γ} with constraints:
    - c_{1,I} = (1 − x_I) / ξ_τ
    - c_2(Θ) = ˆr_h(Θ) x_I / ξ_τ
    - ˆr_h(Θ) = Σ_{(m,j)} α^h_{m,j} r_{m,j}(Θ)
    - α^h_{1,h} + α^h_{2,h} ≥ ̄α
    - Σ_{(m,j)} α^h_{m,j} = 1 (10)

### Optimal portfolios and implications of information
- Optimal portfolio shares given specialization h and signal I:
  - If I = Θ (informed), investor avoids excess risk technologies and spreads investment as evenly as possible across low risk technologies conditional on ̄α ≥ 1/N:
    - α^{h*}_{m,θ_m}(Θ) = 0 ∀ m
    - α^{h*}_{h,3−θ_h}(Θ) = ̄α
    - α^{h*}_{m,3−θ_m}(Θ) = (1 − ̄α)/(N − 1) for m ≠ h (11)
  - If I = 0 (uninformative), investments are spread as evenly as possible:
    - α^{h*}_{h,j}(0) = ̄α/2 for j = 1,2
    - α^{h*}_{m,j}(0) = (1 − ̄α) / (2(N − 1)) for m ≠ h, j = 1,2 (12)
- Portfolio returns:
  - For given Θ, portfolio return ˆr_h^Θ and ˆr_h^0 expressions:
    - ˆr_h^Θ = ̄α i_h + Σ_{m ≠ h} i_m
    - ˆr_h^0 = ̄α/2 (i_h + i_h e_h S) + (1 − ̄α)/(2(N − 1)) Σ_{m ≠ h} (i_m + i_m e_m S)
  - An informed investor holds a less risky portfolio than an uninformed investor:
    - r_h^0 = r_h^Θ + ε_h, where ε_h = ̄α/2 i_h (e_h S − 1) + (1 − ̄α)/(2(N − 1)) Σ_{m ≠ h} i_m (e_m S − 1) and E[ε_h | r_h^Θ] = 0.
  - Risk-adjusted return to portfolio with information I and specialization h:
    - R_I = [ E(ˆr_h_I^{1−γ}) ]^{1/(1−γ)} (13)
  - Jensen inequality implies R_0 < R_Θ.

### Saving rates and aggregate market shares
- Using homothetic preferences, value function can be rescaled:
  - V_I(τ,h) = (1/ξ_τ) ˆV_I where
    - ˆV_I = max_x { (1 − x)^{1−1/ψ} + β x^{1−1/ψ} R_I^{1−1/ψ} }^{1/(1−1/ψ)} (14)
  - First order condition yields optimal saving rate:
    - x^*_I = 1 / (1 + β^{−ψ} R_I^{1−ψ}) ? (text gives) x^*_I = 1 / (1 + β^{−ψ} R_I^{1−ψ}) but the stated formula in the text is:
    - x^*_I = 1 / (1 + β^{−ψ} R_I^{1−ψ}) (15)  (presented in the source as x^*_I = 1/(1+β^{−ψ} R^{1−ψ}_I) ; retain exact formula as printed)
- Aggregate market shares ω_m (fraction of aggregate investment to market m):
  - ω_m = ̄α p_m + (1 − ̄α)/(N − 1) (1 − p_m)
  - Special cases:
    - If ̄α = 1 (minimum financial development), ω_m = p_m.
    - If ̄α = 1/N (maximum financial development), ω_m = 1/N for all m.

### Microfoundation of public information and noisy investment
- Passive investors observe active investors’ investments plus noise. Noise interpreted as mistakes, measurement error, or liquidity investors’ random behavior.
- Noisy investment:
  - Total noisy investment ̄L distributed across markets with shares ω_m.
  - A random variable L with support [0, ̄L] determines total amount of noisy investment directed to excess risk technologies.
  - y(m,j) denotes sum of active and noisy investment in technology (m,j):
    - y(m,θ_m) = ω_m ξ_A x^*_Θ η + ω_m (̄L − L)
    - (text also writes y(m,3−θ_m) = ω_m L)
  - Assume ̄L < x^*_Θ / ξ_A.
- Passive investors infer types via Bayes rule. Define for a reference market m and j = 1,2:
  - L_j = y(m,j)/ω_m
  - Θ_j ∈ T_Θ the vector of types consistent with realization θ_m = j.
  - If (3) holds, Prob{Θ_j | y} = u(L_j) / [ u(L_j) + u(L_{3−j}) ] (16), where u(·) is the pdf of L.
- Uniform noise example (L ~ Uniform[0, ̄L]):
  - If L < η x^*_Θ / ξ_A then types are revealed to passive agents (Prob{Θ_{θ_m} | y} = 1).
  - If L ≥ η x^*_Θ / ξ_A then Prob{Θ_{θ_m} | y} = Prob{Θ_{3−θ_m} | y} = 1/2.
  - Revelation rate:
    - ρ = Prob{ L < η x^*_Θ / ξ_A } = η x^*_Θ / (̄L ξ_A)
  - Externality function (8) obtained by setting A_E = x^*_Θ / (̄L ξ_A).

### Equilibrium condition for the revelation rate
- In equilibrium with strictly positive revelation rate ρ^*, fraction of active investors η^* is strictly positive and the marginal investor is indifferent between being active and passive:
  - 1 / ξ_A^{1−γ} ˆV^{1−γ}_Θ = 1 / ξ_P^{1−γ} [ ρ^* ˆV^{1−γ}_Θ + (1 − ρ^*) ˆV^{1−γ}_0 ]
- Solving for ρ^* gives:
  - ρ^* = 1 − ξ^{γ−1} −1 ( ˆV_Θ / ˆV_0 )^{γ−1 −1} (17)
  - Internal solution ρ^* ∈ (0,1) obtained whenever ( ˆV_Θ / ˆV_0 )^{γ−1} > ξ^{γ−1}.

*Source: _wp10266 - conclusion to be drawn is that the successful introduction of these new financial instruments*

### 1.  investment policiesx

### 1.  investment policiesx

### Equilibrium characterization
- Proposition 1: There exists one and only one equilibrium.
- Values ˆV_I are given by
  - ˆV_I = (β ψ R_{ψ−1}_I + 1)^{1/(ψ−1)} (equation (18)) for I ∈ {0, Θ}.
- For interior solutions, the revelation rate ρ^* is
  - ρ^* = 1 − ξ^{γ−1 \over γ−1} ( (β ψ R_{ψ−1}_Θ + 1) / (β ψ R_{ψ−1}_0 + 1) )^{(γ−1)/(ψ−1)} − 1 (equation (19)).
- The equilibrium revelation rate ρ^* is a function of the exogenous vector of distortions D, written ρ^*(D).
- ρ^* depends on the relative information cost ξ (not on ξ_A and ξ_B separately) because it equalizes the value of being active and passive.

### Comparative statics: how financial innovation affects ρ^*(D)
- The analysis considers exogenous reductions in ξ_A, ξ_B and ̄α leaving ξ unchanged; focus on interior equilibria ρ^* > 0.
- Two cases analyzed:
  1. i_m = 1 (low risk technology is risk free; excess risk is both idiosyncratic e_m and systemic S):
     - Proposition 2.i: If i_m = 1 a.s. then ρ^*(D) is a strictly increasing function (i.e., information collection increases with reductions in distortions that leave ξ unchanged).
  2. e_m = 1 (all technologies have an idiosyncratic component i_m and excess risk is only systemic S):
     - Proposition 2.ii: If e_m = 1 a.s. then ρ^*(D) is strictly increasing for ψ < 1 and strictly decreasing for ψ > 1.
- Welfare strictly decreases in the distortions D.

- Intuition via ratios:
  - ζ_x = x_Θ / x_0 (saving rate of informed relative to uninformed).
  - ζ_R = R_Θ / R_0 (risk-adjusted return of informed relative to uninformed).
  - Case i) (i_m = 1): informed investors face no risk (R_Θ = 1); financial development boosts R_0, decreasing ζ_R and favoring passive investors relative to active.
  - Case ii) (e_m = 1): ζ_R = Ω^{−1} where Ω = (E S^{1−γ})^{1/(1−γ)} < 1, so ζ_R unaffected by diversification; the channel is changes in ζ_x.
    - If ψ < 1 then ζ_x < 1 and financial innovation favors uninformed/passive investors (ρ^* decreases).
    - If ψ > 1 the opposite holds, but ψ > 1 considered less relevant empirically.
- Roles of parameters:
  - ψ (elasticity of intertemporal substitution) determines sign of change in ρ^* for e_m = 1.
  - γ (relative risk aversion) influences magnitude of change in ρ^*; higher γ amplifies ∂R_Θ/∂(1−̄α).

- Derivative expression (for e_m = 1):
  - ρ′^*_{1−̄α} ≡ ∂ρ^*/∂(1−̄α) = g(R_Θ, Ω) ∂R_Θ/∂(1−̄α) (1 − Ω^{ψ−1})^{-1}
  - g(R_Θ, Ω) is strictly positive; ∂R_Θ/∂(1−̄α) > 0; sign determined by ψ via (1 − Ω^{ψ−1}).

### Welfare and the information externality
- For interior ρ^*, investors’ welfare equals ˆV_Θ / ξ_A.
- Welfare gains from financial innovation come from:
  - lower costs ξ_A and
  - a lower barrier ̄α to diversification which raises R_Θ in (18).
- However, welfare gains can be offset or eliminated by an endogenous decrease in ρ^* (less information collection).
  - Example: in case a) (i_m = 1), a decrease in ̄α does not change ˆV_Θ, so keeping ξ_A and ξ_B constant, welfare is unaffected by a strict decrease in ̄α because lower public-information quality ρ^* exactly offsets diversification gains for passive investors.
- Mechanism: information collection and diversification are strategic substitutes.
  - Greater diversification (lower ̄α) makes investors feel more insured and reduces incentive to collect costly private information.
  - Lower private information collection reduces public signal quality ρ^*, harming passive investors and potentially offsetting diversification benefits.
- Extended model (Section IV) shows financial development can increase probability of "liquidation crises" when periods of idiosyncratic excess risk are followed by systemic excess risk, potentially reducing welfare.

### Business cycle implications and volatility
- Multifactor productivity A(Θ, I_P) defined as weighted average portfolio return divided by total investment (equation (20)).
- Define:
  - Σ^2 = Var(r_{m,θ_m}) − σ^2_S − σ^2_i (volatility of excess risk technologies minus volatilities of i and S).
  - ˆσ^2_ω = ∑_m ω_m^2 (dispersion of market shares).
- For small total noise ̄L, business cycle statistics (Appendix C) are:
  - σ^2_A = ˆσ^2_ω σ^2_i + 1/4 (σ^2_S + ˆσ^2_ω Σ^2) (1 − ρ^*) + O(̄L)
  - γ_{A,S} = 1/2 (1 − ρ^*) + O(̄L)  (equation (21))
- Financial innovation accompanied by a reduction in ρ^* has ambiguous effects on aggregate volatility σ^2_A:
  - Lower ̄α always reduces dispersion ˆσ^2_ω of market sizes, lowering σ^2_A by reducing amplification of idiosyncratic volatility.
  - In the limit ̄α = 1/N and N = ∞, ˆσ^2_ω = 0 and only systemic volatility σ^2_S remains.
  - Reductions in ρ^* increase exposure of aggregate productivity to systemic shocks.
- Sensitivity γ_{A,S} always increases in 1 − ̄α (i.e., smaller barriers to diversification raise sensitivity to systemic shocks via lower revelation rate).
- Numerical example (Figure 3 parameters and computed volatilities):
  - Parameters: i_m ∈ {1.18, 0.82} with equal probability, e_m ∈ {1.18, 0.82} with equal probability; S = 1.1 with probability 0.82 and S = 0.57 otherwise; N = 120; ξ = 1.005; β = 0.97; ψ = 1/γ = 0.5; p_h = 0.18·0.82^{h−1}.
  - Computed volatilities: σ^2_A(̄α = 1) = 6.3·10^{−3}, σ^2_A(̄α = 1/N) = 5.5·10^{−3}.
  - Result: equilibrium ρ^* decreases as ̄α decreases; σ^2_A decreases (dominated by reduction in ˆσ^2_ω), while sensitivity γ_{A,S} to asymmetric systematic shocks increases (tail risk rises).

### Multi-period model: information cycles and liquidation crises
- Model extension: three periods t = 1, 2, 3; information and investment choices at beginnings of t = 1 and t = 2.
- Time-varying risk structure:
  - At time 1: excess risk is idiosyncratic.
  - At time 2: excess risk is all systemic.
- Liquidation crisis: a reduction δ in time 2 returns of all technologies occurs if, at beginning of time 2, public information reveals types of previously uninformed investors, triggering liquidations.
  - δ ∈ (0,1) interpreted as capital adjustment costs or productivity costs of banking crises.
- Probability of liquidation crisis π_λ is endogenous and depends on revelation rates ρ_t which relate to D.
  - Indicator I^λ_2 = (1 − I^ρ_1)[S_1 + (1 − S_1) I^ρ_2] (equation (28)).
  - π_λ = Prob{I^λ_2 = 1} = (1 − ρ_1)[π + (1 − π) ρ_2] (equation (29)).
  - Necessary condition for crisis at time 2: public information at time 1 not revealing (I^ρ_1 = 0).
- Returns setup (three-period extension):
  - r_{m,3−θ_m,1} = 1 (equation (24))
  - r_{m,3−θ_m,2} = 1 − I^λ_2 δ (equation (25))
  - r_{m,θ_m,1} = 1 + S_1 (e_m − 1) (equation (26))
  - r_{m,θ_m,2} = (1 − I^λ_2) S_2 (equation (27))
  - S_1 ∈ {0,1} with Prob{S_1 = 1} = π; S_2 and e_m i.i.d., mean 1, e_m, S_2 ≠ 1 a.s., independent of S_1 and I^P_t.
- Revelation rates in interior equilibrium (Proposition 3):
  - ρ^*_t = 1 − ln ξ / (β ln (R_{t,Θ} / R_{t,0})) (equation (32)).
  - ρ^*_2 > ρ^*_1.
  - A decrease in ̄α decreases ρ^*_1, leaves ρ^*_2 unchanged, and increases probability of liquidation crises; welfare decreases if ̄α decreases.
- Mechanism and welfare:
  - Financial innovation (lower ̄α) reduces ρ^*_1 because time 1 excess risk is idiosyncratic; ρ^*_2 unchanged because time 2 excess risk is systemic.
  - Lower ρ^*_1 increases likelihood of uninformed investments in excess-risk technologies at t = 1, raising π_λ.
  - Increased π_λ raises probability active investors face reduced returns 1 − δ at time 2, lowering expected utility and welfare.
  - Therefore, absent a reduction in relative cost of information, financial innovation can decrease welfare by increasing likelihood of liquidation crises.

### Conclusions
- Empirical evidence suggests financial innovation increases diversification opportunities and reduces investment costs, but does not reduce relative cost of information.
- The model shows financial innovation tends to reduce private information collection and, via information externalities, the quality of public information.
- Implications:
  - Aggregate productivity becomes more exposed to systemic shocks even when overall volatility decreases.
  - Financial innovation and endogenous information collection can increase occurrence of liquidity/liquidation crises when risk is time varying.
  - If liquidation crises impose aggregate costs, financial innovation can reduce welfare despite lower costs and greater diversification.
- The paper highlights the need for further research on interactions between financial innovation and endogenous information collection.

*Italic: Source — _wp10266 - 1.  investment policiesx*

### REFERENCES

### REFERENCES

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### Appendix inventory (content summaries)
- APPENDIX A: Optimal portfolio shares and risk-adjusted returns
  - Proof structure in five steps derives optimal portfolio shares α^*_{m,j}(I).
  - Key intermediate results and constructions:
    - For I = 0, portfolio transformations (ν_h_{m,j} to ̃α_h_{m,j}) show mean-preserving spreads and use Jensen’s inequality: E[(q_{̃α})^{1−γ}]^{1/(1−γ)} ≥ E[(q_ν)^{1−γ}]^{1/(1−γ)}.
    - For I = 0, further dominance arguments replace ̃α by α with α_{h,m,m} = ̃α_{h,m,m} for m ≠ h and α_{h,m,j} = ̄α/2 for m = h.
    - For I = Θ, portfolios ν are dominated by α with α_{h,m,3−θ_m} = ν_{h,m,1}+ν_{h,m,2} and α_{h,m,θ_m}=0.
  - Main conclusion:
    - For I ∈ {Θ,0}, the constructed portfolio return q_α(̄α) = ˆr_h^I(̄α), with ˆr_h^I(̄α) = (1−̄α)∑_{m≠h} r_m + ̄α r_h.
    - Writing ˆr_h^I(̄α) = ˆr_h^I(1/N) + φ z with φ = (N ̄α − 1)/N and z = r_h − (1/(N−1))∑_{m≠h} r_m, Jensen’s inequality implies: as ̄α decreases the risk adjusted return of the investors’ portfolios increase.

- APPENDIX B: Signal extraction for passive investors
  - For market m and j = 1,2 define L_j = y(m,j)/ω_m. Realized noise L ∈ {L_1,L_2}.
  - Association: if L = L_j then θ_m = j′ when y(m′,j′)/ω_m = L_j.
  - Using Bayes’ rule for continuous variables and relation Θ_j = ̄Θ_{3−j} (equation A33):
    - Prob{Θ_j | y} = u(L_j) / (u(L_j) + u(L_{3−j})),
      where u is the probability density function of L.

- APPENDIX C: Business cycle statistics and monotonicity of ˆσ^2_ω in ̄α
  - Definitions:
    - A_Θ = ∑_m ω_m i_m
    - A_0 = 1/2 ∑_m ω_m i_m (1 + e_m S)
  - Conditional on informative signal I = Θ all agents invest in low risk technologies so A = A_Θ.
  - If I = 0 passive investors split portfolios equally and derive:
    - A = A_0 + η^* x^*_Θ (A_Θ − A_0) / ((1−η^*) ξ x^*_0 + η^* x^*_Θ), with η^* = ρ^*/A_E.
  - Unconditional covariance Cov(A,S):
    - Cov(A,S) = E[Cov(A,S|I=0)] + E[Cov(A,S|I=Θ)] = E[Cov(A,S|I=0)]
    - = E[Cov(A_0,S|I=0)] + O(̄L) = (1−ρ^*) Cov(A_0,S) = 1/2 (1−ρ^*) σ^2_S
  - Unconditional variance σ^2_A:
    - σ^2_A = E[Var(A|I=0)] + E[Var(A_Θ|I=Θ)]
    - = (1−ρ^*) Var(A_0) + ρ^* Var(A_Θ) + O(̄L)
  - Dispersion and monotonicity:
    - ω_m ≤ 1/N ⇔ p_m ≤ 1/N
    - ∂ω_m / ∂(1−̄α) ≥ 0 ⇔ p_m ≤ 1/N
    - Hence ∂ω_m / ∂(1−̄α) ≥ 0 ⇔ ω_m ≤ 1/N.
    - σ^2_ω = (1/N) ∑_m (ω_m − 1/N)^2 = ̃σ^2_ω (N−1)/N^2.
    - Since ˆσ^2_ω = N σ^2_ω + 1/N, ˆσ^2_ω decreases in 1−̄α (i.e., ˆσ^2_ω is decreasing in 1−̄α).

- APPENDIX D: Solution to the extended model (revelation rates and dynamic values)
  - Revelation rates ρ_t solved by backward induction.
  - At time 2, conditional on I ρ_1 + S_1 = 0, define V_{2,I}(w):
    - V_{2,I}(w) = max_x exp{(1−β) ln(1−x) w + β E ln x w ˆr_{h,2,I} }.
    - First order condition gives x^* = β.
    - V_{2,I}(w) = w (1−β)^{1−β} β^{β} (R_{2,I})^{β}.
  - Revelation rate ρ^*_2 found by equalizing expected continuation values:
    - E ln V_{2,Θ}(1/ξ_A) = ρ^*_2 E ln V_{2,Θ}(1/ξ_P) + (1−ρ^*_2) V_{2,0}(1/ξ_P).
    - Revelation at time 2 causes liquidation cost δ on all technologies when I ρ_2 = 1.
  - At time 1, given continuation value U_2 per unit wealth:
    - V_{1,I}(w) = max_x exp{(1−β) ln(1−x) w + β E ln x w ˆr_{h,1,I} U_2 } with x = β.
    - Solution: V_{1,I}(w) = w (1−β)^{1−β} β^{β} (R_{1,I})^{β} E U_2^{β}.
    - Revelation rate ρ^*_1 solves:
      - ln V_{1,Θ}(w/ξ_A) = ρ^*_1 ln V_{1,Θ}(w/ξ_P) + (1−ρ^*_1) ln V_{1,0}(w/ξ_P).

- APPENDIX E: Construction of series in Figure 1 (data details)
  - All data taken from the Tables in French (2008).
  - Top panel: active and passive costs for DB plans reported in Table III.
  - Bottom panel: cost of the composite passive strategy from Table VII, column “Total Cost”.
  - Composite active strategy constructed as weighted average of mutual funds, pension funds and hedge funds:
    - Weight for mutual funds from market share for Open-end funds in Table I.
    - Note: possible overestimate because mutual funds shifted over time toward passive strategies; nonetheless in 2006 as much as 88% of the Open-end funds are defined as active (Table II).
    - Weight of pension funds = sum of shares of DB plans and DC plans in Table I.
    - Weight for hedge funds from the share in Table I.
  - Cost components:
    - Cost of Open-end funds from Table II, column “Total”.
    - Costs of pension funds = cost of active DB plans in Table III.
    - Cost of hedge funds = entry in Table IV under “Hedge Fund plus Fund of Funds”; missing pre-1996 values filled with 9.27% (the 1996 value).
    - Hedge funds have weights smaller than 2.3% for the years prior to 1996, so the 9.27% fill has no significant impact on the cost index estimation.

*Content derived from the PDF file "_wp10266 - REFERENCES" (appendices A–E and reference list).*

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