## wp18123

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### Introduction: phenomenon and research questions
- Research question: Does financial tranquility always call for more stringent regulation/macroprudential policies over time?
- Empirical context and key observations:
  - Laeven and Valencia (2010): the median output loss of the recent financial crisis is 25 percent.
  - Reinhart and Rogoff (2009): in financial crises the unemployment rate increases by 7 percentage points and remains high for over four years on average.
  - IMF (2017a): “the longer booms last and the larger credit grows, the more dangerous they become.”
- Core mechanisms:
  - Risk-taking effect / input effect: prolonged tranquility builds investor confidence and leads to larger positions in risky assets and higher leverage.
  - Resilience effect / transmission effect: prolonged tranquility can update perceptions that the transmission from risk-taking to systemic crisis is weaker.
- Central policy puzzle: whether macroprudential policy should become more or less stringent during prolonged tranquility depends on the trade-off between the risk-taking/input effect and the resilience/transmission effect.

### Model framework (primitives and learning)
- Environment and assets:
  - Continuum of investors i ∈ [0, I], each endowed with 1 unit of investment good; two assets (safe and risky); discrete infinite time t = 1, 2, 3, ...
  - Payoffs: if no crisis safe asset pays μ_S; risky asset pays μ_R with μ_R > μ_S > 0. If crisis: safe pays τ_S μ_S; risky pays τ_R μ_R, with 0 ≤ τ_R ≤ τ_S ≤ 1.
- Crisis probability and states:
  - Two states: strong (“G”) or fragile (“B”). Crisis probability p_j(α_t) with p_G(α_t) < p_B(α_t), p′_j(α_t) > 0.
  - θ_t ≡ π_t p_G(α_t) + (1−π_t) p_B(α_t): unconditional perceived crisis probability.
- Bayesian learning and “complacency trap”:
  - Belief π_t ∈ [0,1] updated from π_{t−1} using realized crises C_{t−1} ∈ {0,1}.
  - π_t(C_{t−1}=0) > π_{t−1}; π_t(C_{t−1}=1) < π_{t−1}.
  - Proposition 1: a longer history of tranquility (no crisis) tends to build investor confidence; π_t strictly increases in the number of no-crisis periods.
  - Short-memory feature: investors revise down confidence only in the first period after a crisis; from the second post-crisis period on, absent a crisis in the previous period, investors become more confident again.

### Decentralized competitive equilibrium: characterization and comparative statics
- Investor optimization (rewritten):
  - max_{α_it ∈ [0,1]} [1 − θ(π_t, α_t)](μ_S + α_it Δ_0) + θ(π_t, α_t)(τ_S μ_S + α_it Δ_1)
  - Δ_0 ≡ μ_R − μ_S with Δ_0 > 0; Δ_1 ≡ τ_R μ_R − τ_S μ_S with Δ_1 < 0 (Assumption A1).
- First-order condition for interior solution:
  - [1 − θ(π_t, α_t)] Δ_0 + θ(π_t, α_t) Δ_1 = 0  (equation (2))
  - Equivalent: Δ_0 / (Δ_0 − Δ_1) − θ(π_t, α_t) = 0  (equation (3))
- Key implications:
  - In any interior competitive equilibrium θ_t is pinned to Δ_0/(Δ_0 − Δ_1) and thus is constant regardless of π_t.
  - Since p′_G(α) > 0, p′_B(α) > 0 and p_G < p_B, as π_t increases investors must raise α_t to keep θ_t constant.
  - Proposition 2: aggregate risky investment α_t in the competitive equilibrium strictly increases in investor confidence π_t.

### Constrained planner’s problem and externality
- Planner’s objective (aggregated):
  - max_{α_t ∈ [0,1]} [1 − θ(π_t, α_t)](μ_S + α_t Δ_0) + θ(π_t, α_t)(τ_S μ_S + α_t Δ_1)  (equation (4))
- Planner’s FOC (normalized):
  - Δ_0/(Δ_0 − Δ_1) − θ(π_t, α_t) − ξ(π_t, α_t) = 0  (equation (6))
  - ξ(π_t, α_t) ≡ θα(π_t, α_t)[α_t + (1−τ_S) μ_S/(Δ_0 − Δ_1)] captures the normalized externality.
- Proposition 3: aggregate risky investment α_t in the constrained planner’s equilibrium strictly increases in planner confidence π_t, under p′B(α) > p′G(α) > 0 and convexity assumptions.

### Comparison: decentralized vs constrained planner (inefficiency)
- Notation: α_Ct (competitive), α_Pt (constrained planner).
- Proposition 4 (Constrained inefficiency / excessive risk-taking):
  - α_Ct(π_t) > α_Pt(π_t) for any π_t: competitive equilibrium is constrained inefficient due to uninternalized negative externality ξ > 0.
- Interaction between learning and inefficiency:
  - Sufficiently convex condition (SCC) (as given): θαα > θα |θπ| / (|θαπ| − θα).
  - Lemma 1: SCC ⇔ slope of iso-vulnerability curve > slope of iso-externality curve.
  - Proposition 5:
    - As π_t increases, α_Ct increases faster than α_Pt iff SCC holds. Intuition decomposed into:
      - Round One (Resilience Effect, RE): higher π puts more weight on p_G, lowering θ and reducing externality, increasing α_P.
      - Round Two (Risk-Taking Effect, RTE): higher α raises θ and ξ, inducing planner to curb α_P. If RTE dominates RE (captured by SCC), α_P rises less than α_C.

### Parametric SCC example and sufficient conditions
- Parametric p_j specification:
  - p_G(α) = a_G + b_G · α^k
  - p_B(α) = a_B + b_B · α^k, with assumptions a_B ≥ a_G ≥ 0, b_B ≥ b_G > 0, a_i + b_i ≤ 1, k ≥ 1.
- Derived condition:
  - If k > 1 + (b_B − b_G)/(a_B − a_G) (under a_B > a_G and b_B > b_G), then SCC holds.
- Extremes:
  - If p′_G = p′_B (b_B = b_G): SCC holds for any k > 1; planner should raise tax when tranquility persists.
  - If a_B = a_G: SCC fails for any k; planner should optimally decrease tax when tranquility persists.

### Macroprudential policy instrument: capital income tax and optimal D*
- Capital income tax Dt levied per dollar of risky payoff; investor problem with tax (aggregate form):
  - max α_it ∈ [0,1] (1 − θ_t)[μ_S + α_it(Δ_0 − μ_R D_t)] + θ_t[τ_S μ_S + α_it(Δ_1 − τ_R μ_R D_t)]  (equation (8))
- Optimal tax that restores constrained efficiency:
  - D* = [Δ_0 + (Δ_1 − Δ_0) θ_P] / [μ_R + (τ_R μ_R − μ_R) θ_P]  (equation (10))
  - Rearranged: D* = 1 − μ_S [1 − (1 − τ_S) θ_P] / [μ_R [1 − (1 − τ_R) θ_P]]  (equation (11))
- Proposition 6: unique D* ∈ (0,1) exists for any belief π_t and can restore constrained efficiency.
- Proposition 7:
  - As π_t increases, the optimal tax D* needed to restore constrained efficiency will be higher if and only if SCC holds.
  - Macroprudential implication: if SCC holds, optimal policy should be countercyclical—raise capital income tax as market tranquility persists to curb excessive risk-taking.
  - Qualification: regulator must assess structural resilience (e.g., industrial composition) rather than only surface cyclical indicators; high credit growth driven by diversification may improve resilience and argue against tightening.

### Inefficient deregulation and regulation (regulator belief misspecification)
- If regulator overestimates |ξπ| and believes SCC fails when it actually holds:
  - As π rises by Δπ, the regulator may lower tax from D* to D_Reg(π + Δπ) = D* − ΔD, inducing inefficiently high α_Reg(π + Δπ).
- COROLLARY 1:
  - If regulator overestimates resilience and mistakenly believes SCC fails (|θ_π|/θ_α ≤ |ξ_π|/ξ_α), observing one more no-crisis period leads the regulator to lower the capital income tax and induce an inefficiently high aggregate risky position.
- COROLLARY 2:
  - If crisis damages the safe real sector more (lower τ_S) and/or the risky sector less (higher τ_R):
    - (i) market invests more in the risky sector;
    - (ii) planner is more likely to raise the tax when the market has been tranquil for longer.
  - Intuition: lower τ_S and/or higher τ_R make safe sector less attractive, increasing α_C responsiveness to π and prompting planner tax increases to contain externalities.

### Learning dynamics, Markov switching, and complacency trap
- Dynamics of learning (Corollary 3):
  - Investors do not revise down confidence until a crisis actually occurs.
  - A crisis reduces confidence only in the immediately next period; from the second post-crisis period onward, confidence rebuilds as long as previous period had no crisis—this creates a “complacency trap.”
  - Simulations (as reported in source) show sharp belief declines at crises and potential higher peaks in later cycles despite repeated crises.
- Markov switching extension:
  - With state persistence q ∈ (1/2, 1) and switching prob 1 − q, interim belief updates to π′_t(C_{t−1} = 0) = (2q − 1) π_t(C_{t−1} = 0) + (1 − q).
  - Since 2q − 1 > 0, π′_t is monotonically increasing in π_t; Proposition 1 and others continue to hold under Markov switching.

### Unconstrained (informed) planner vs constrained planner: information value
- Unconstrained/informed planner (full information) when true state is B:
  - Planner solves max over α_t [1 − p_B(α_t)](μ_S + α_t Δ_0) + p_B(α_t)(τ_S μ_S + α_t Δ_1)  (equation (12))
  - FOC: Δ_0/(Δ_0 − Δ_1) − p_B(α_t) − p′_B(α_t)[α_t + (1 − τ_S) μ_S Δ_0 − Δ_1] = 0  (equation (13))
- Proposition 8:
  - If true state is B, constrained planner’s α_Pt is strictly higher than unconstrained/informed planner’s α_IPt.
- Policy implication:
  - Information asymmetry causes the uninformed/constrained planner to choose inefficiently high risky investment relative to an informed planner; assigning macroprudential policy to independent agencies with technical expertise (better information) is desirable.
  - Note: α_IPt and the informed planner’s optimal tax do not depend on belief π_t and remain constant as market/planner confidence changes.

### Conclusion: contributions, policy lessons, and future research
- Contributions:
  - Shows how interaction of negative externality and Bayesian learning generates excessive risk-taking: longer tranquility raises investor confidence π_t, increasing α_t and crisis risk.
  - Characterizes when constrained inefficiency rises with confidence (SCC) and when optimal macroprudential policy is countercyclical.
  - Demonstrates that a capital income tax can restore constrained efficiency; its optimal level D* can be increasing in π_t if SCC holds.
- Policy lessons and caveats:
  - Optimal macroprudential policy is not necessarily countercyclical; it depends on trade-off between resilience effect and risk-taking effect.
  - Misassessment of resilience can lead to inefficient deregulation (loosen policy) or inefficient regulation (over-tightening), with adverse effects on financial activity and innovation.
  - Financial cycles are much longer than business cycles (about 15 years in U.S., U.K., and average G-7 per cited studies); policymakers should avoid mechanical cyclical responses.
- Suggested future research:
  - Provide micro foundations for crisis probability functions p_G(α) and p_B(α).
  - Empirically test model implications, e.g., convexity of nonbank households’ management fees in capital managed.

*Source: WP/18/123 — Does Financial Tranquility Call for Stringent Regulation? by Deepal Basak and Yunhui Zhao (May 2018).*

### Section 1

### wp18123 - Section 1

### Introduction: phenomenon and research questions
- Research question: Does financial tranquility always call for more stringent regulation/macroprudential policies over time?
- The paper links the Minsky hypothesis and the “volatility paradox” (Brunnermeier and Sannikov, 2014) to recent empirical evidence that financial crises tend to follow prolonged periods of financial stability and investor optimism (citations include Borio, 2012; Dell’Ariccia and others, 2012; Drehmann and others, 2012; Danielsson and others, 2016; Espinoza and others, 2017).
- Key observation from prior evidence:
  - Laeven and Valencia (2010): the median output loss of the recent financial crisis is 25 percent.
  - Reinhart and Rogoff (2009): in financial crises the unemployment rate increases by 7 percentage points and remains high for over four years on average.
  - IMF (2017a): “the longer booms last and the larger credit grows, the more dangerous they become.”
- Core mechanisms emphasized:
  - Risk-taking effect / input effect: prolonged tranquility builds investor confidence and leads to larger positions in risky assets and higher leverage.
  - Resilience effect / transmission effect: prolonged tranquility can update perceptions that the transmission from risk-taking to systemic crisis is weaker.
- Central policy puzzle: whether macroprudential policy should become more or less stringent during prolonged tranquility depends on the trade-off between the risk-taking/input effect and the resilience/transmission effect.

### Model framework (summary)
- Model primitives:
  - Simple portfolio choice model with a safe asset and a risky asset that may expose the financial sector to systemic crisis risk.
  - Investors are Bayesian learners about the extent to which risky asset exposures raise systemic-crisis probability; they learn from history of crises (or absence thereof).
- Learning features emphasized:
  - “Complacency trap”: investors increase risky investment whenever no crisis occurred in the previous period (“Cry only when death is staring one in the face”).
  - Short memory about crises: investors revise down confidence in the first period after a crisis only; from the second post-crisis period on, absent a crisis in the previous period, investors again become more confident (“Once on shore, pray no more”).
- Externality:
  - Each investor does not internalize how her/his risky position raises aggregate crisis probability, creating a negative externality and constrained inefficiency in the laissez-faire equilibrium.

### Main analytical results and conditions
- Inefficiency result:
  - The laissez-faire (decentralized) equilibrium is inefficient: equilibrium risky asset positions exceed those chosen by a constrained benevolent planner who also learns about the true systemic-crisis probability.
  - The paper derives a necessary and sufficient condition on the crisis probabilities under which rising investor confidence strengthens the negative externality, aggravating excessive risk-taking and increasing inefficiency.
- Policy instrument and corrective role:
  - A macroprudential regulator can reestablish efficiency at a point in time via a capital income tax set at an appropriate level (Pigouvian taxation). Footnote clarification: capital income tax is levied on gross investment income in the paper for convenience; results apply under the standard definition and to financial transaction taxes.
- Comparative statics on optimal regulation:
  - Under the same condition that strengthens the externality, the optimal tax rate rises as investor confidence (and market inefficiency) rises.
- Mechanism explaining optimal regulation’s cyclicality:
  - Persistence of tranquility leads the constrained planner to believe the transmission channel per unit risk is weaker (reducing perceived externality — resilience/transmission effect) but also leads to larger risky positions that increase externality directly (risk-taking/input effect).
  - Whether optimal macroprudential policy becomes more or less stringent depends on which effect dominates; thus optimal policy is not necessarily countercyclical.

### Policy implications and practical considerations
- Policymakers should not rely solely on surface cyclical indicators (for example, credit growth) but should closely examine deep structural changes in system resilience (for example, industrial composition and dependence on particular industries).
- The analysis highlights a regulatory information problem:
  - Inefficient deregulation: if regulators overestimate resilience, they may reduce stringency as tranquility persists, inducing risky buildups they would have opposed if they knew true resilience (some commentators argue this occurred in the United States in the 1990s and 2000s).
  - Inefficient regulation: if regulators underestimate resilience, they may repress beneficial financial activity and inhibit innovations that diversify risk.
- Institutional recommendation:
  - The importance of assigning the macroprudential policy function to independent agencies with technical expertise to better assess resilience and set appropriate policy.

### Linkages to literature and robustness
- The paper focuses on interaction between learning and externality rather than modeling a specific crisis mechanism; the crisis probability is exogenously specified as a function of aggregate risky positions.
- Results are intended as a reduced-form characterization and are consistent with more detailed models (examples cited include Biais and others, 2015; Gertler and Kiyotaki, 2015).
- Robustness: all results hold when the underlying state follows a Markov process (good states may become bad and vice versa); main difference is that agents may never learn the true state in that case (see Section V).

*Source: WP/18/123 — Does Financial Tranquility Call for Stringent Regulation? by Deepal Basak and Yunhui Zhao (May 2018).*

### Section 2

### wp18123 - Section 2

### Related literature and contribution
- Emphasizes uncertainty about the resilience of the financial system and the a priori unclear direction of optimal cyclical adjustment of macroprudential policy.
- Links to four literature strands:
  - Countercyclical macroprudential regulation: instruments include higher capital requirements, higher provisioning, higher borrowing cost, capital inflow taxation, lower caps on loan-to-value and debt-to-income ratios.
  - Learning: agents update beliefs about aggregate parameters of the financial industry; rising confidence can reduce managers’ risk-abatement effort.
  - Inefficient risk-taking from externalities: private agents fail to internalize systemic effects; learning interacts with these externalities in this paper.
  - Broader macroprudential policy literature: systemic risk taxation and measures of systemic risk.
- Distinguishes this paper from Bhattacharya and others (2015):
  - Focus: implications of learning over many periods for excessive risk-taking and optimal macroprudential stringency.
  - Approach: simpler framework to obtain analytical characterizations of welfare results and to highlight countervailing effects of rising investor risk-taking and increasing regulator confidence.
- Policy implication emphasized: desirability of assigning macroprudential policy to independent agencies with technical expertise that can better gauge true resilience of the financial system.

### Benchmark model — A. Decentralized problem: set up and learning
- Environment:
  - Continuum of investors indexed by i ∈ [0, I], each endowed with 1 unit of investment good.
  - Two assets: safe asset (real/conventional) and risky asset (financial/innovative).
  - Time is discrete and infinite: t = 1, 2, 3, ...
  - At beginning of period t, investor i chooses α_it ∈ [0,1] to invest in risky asset and 1−α_it in safe asset; agent consumes and exits at end of period t.
- Payoffs and crisis:
  - If no crisis: safe asset pays μ_S with probability 1; risky asset pays μ_R in expectation, with μ_R > μ_S > 0.
  - If crisis occurs: safe asset pays τ_S μ_S; risky asset pays τ_R μ_R, where 0 ≤ τ_R ≤ τ_S ≤ 1.
    - τ_S captures spillover from financial sector to real sector: τ_S = 1 implies no spillover; τ_S = τ_R implies maximal spillover.
- Financial sector states and crisis probability:
  - Two states: strong (“G”) or fragile (“B”).
  - Crisis probability at time t in state j ∈ {G, B} is p_j(α_t).
  - Assumptions on p_j(·):
    - p_j(α_t) ∈ [0,1], ∀α_t, ∀j
    - p_G(α_t) < p_B(α_t)
    - p′_j(α_t) > 0, ∀α, ∀j
- Beliefs and Bayesian updating:
  - π_t ∈ [0,1] is belief that the sector is strong at beginning of period t; agents inherit π_{t−1} and update using outcomes from period t−1.
  - C_{t−1} = 1 if crisis occurred at t−1, 0 otherwise.
  - Bayesian updating:
    - π_t(C_{t−1}=0) = π_{t−1}[1−p_G(α_{t−1})] / [π_{t−1}[1−p_G(α_{t−1})] + (1−π_{t−1})[1−p_B(α_{t−1})]] > π_{t−1}
    - π_t(C_{t−1}=1) = π_{t−1} p_G(α_{t−1}) / [π_{t−1} p_G(α_{t−1}) + (1−π_{t−1}) p_B(α_{t−1})] < π_{t−1}
- Proposition 1 (stated):
  - A longer history of tranquility (that is, no crisis) in the financial sector tends to build investor confidence that the sector is strong. That is, π_t strictly increases in the number of no-crisis periods.

### Benchmark model — A.2 Characterization of the competitive equilibrium
- Investor optimization (period t, investor i):
  - Objective (as given in text):
    π_t{(1−p_G)[(1−α_it)μ_S + α_it μ_R] + p_G[(1−α_it)τ_S μ_S + α_it τ_R μ_R]} +  
    (1−π_t){(1−p_B)[(1−α_it)μ_S + α_it μ_R] + p_B[(1−α_it)τ_S μ_S + α_it τ_R μ_R]}
- Notation and assumptions:
  - Δ_0 ≡ μ_R − μ_S (risky asset’s excess payoff conditional on no-crisis at t−1). Assume Δ_0 > 0.
  - Δ_1 ≡ τ_R μ_R − τ_S μ_S (excess payoff conditional on crisis at t−1). Assumption A1: Δ_0 > 0, and Δ_1 < 0 (i.e., τ_R μ_R < τ_S μ_S).
  - θ_t ≡ θ(π_t, α_t) ≡ π_t p_G(α_t) + (1−π_t) p_B(α_t): unconditional probability of crisis at period t (investors’ perceived vulnerability).
- Rewritten investor problem:
  - max_{α_it ∈ [0,1]} [1 − θ(π_t, α_t)](μ_S + α_it Δ_0) + θ(π_t, α_t)(τ_S μ_S + α_it Δ_1)   (equation (1))
- First-order condition (FOC) for interior solution:
  - [1 − θ(π_t, α_t)] Δ_0 + θ(π_t, α_t) Δ_1 = 0   (equation (2))
  - Interpretation: no-arbitrage condition—expected excess payoff of risky asset relative to safe asset is zero in any interior equilibrium.
  - Equivalent rearrangement:
    - Δ_0 / (Δ_0 − Δ_1) − θ(π_t, α_t) = 0   (equation (3))
- Existence condition for interior equilibrium:
  - For a given π_t, an interior equilibrium exists if θ(π_t, α_t) crosses Δ_0/(Δ_0 − Δ_1) as α_t varies.
  - Sufficient condition for unique interior equilibrium (given continuity and monotonicity of p_G and p_B): p_B(0) < Δ_0/(Δ_0 − Δ_1) and p_G(1) > Δ_0/(Δ_0 − Δ_1).
  - If conditions fail, agents could learn to a corner solution (not analyzed here).
- Comparative statics and equilibrium dynamics:
  - In any interior competitive equilibrium, θ_t is constant regardless of belief π_t, because FOC pins θ_t to Δ_0/(Δ_0 − Δ_1).
  - Since θ_t = π_t p_G(α_t) + (1−π_t) p_B(α_t) and p_G < p_B with positive derivatives p′_G(α) > 0, p′_B(α) > 0:
    - As π_t increases, investors must increase α_t to keep θ_t constant.
    - Therefore, α_t strictly increases in π_t under competitive equilibrium.
- Proposition 2 (stated):
  - The aggregate risky investment α_t in the competitive equilibrium strictly increases in investor confidence π_t, provided p′_G(α) > 0 and p′_B(α) > 0.
- Intuition:
  - Decision on α_t is likened to choosing the speed of a car while driving in the fog: rising confidence (longer tranquility) induces higher risky investment even though true fragility may remain. 

*Source: wp18123 - Section 2*

### Section 3

### Section 3

### B. Constrained Planner’s Problem
- Planner faces same information constraint as decentralized investors; planner updates belief πt through realized crises.
- Planner’s per-period problem (aggregating across i) can be written as:
  - max αt ∈ [0,1] [1−θ(πt,αt)](μS+αtΔ0) + θ(πt,αt)(τSμS+αtΔ1)  (equation (4))
- First-order condition (FOC) for planner:
  - [1−θ(πt,αt)]Δ0 + θ(πt,αt)Δ1 − θα(πt,αt)[(μS+αtΔ0) − (τSμS+αtΔ1)] = 0  (equation (5))
- Normalized FOC (divide by Δ0−Δ1):
  - Δ0/(Δ0−Δ1) − θ(πt,αt) − ξ(πt,αt) = 0  (equation (6))
  - where ξ(πt,αt) ≡ θα(πt,αt)[αt + (1−τS)μS/(Δ0−Δ1)] captures the normalized externality.
- Interpretation:
  - Planner internalizes that higher αt raises crisis probability θt and lowers excess payoff and welfare — a negative externality absent in decentralized equilibrium.
  - Externality consists of: (i) probability term θα(πt,αt) (increase in crisis probability due to α); (ii) payoff term (μS+αtΔ0) − (τSμS+αtΔ1) (reduction in total payoff when crisis occurs).
- Proposition 3:
  - The aggregate risky investment αt in the constrained planner’s equilibrium strictly increases in the planner’s confidence πt, provided p′B(α)> p′G(α)>0, p′′B(α)>0, and p′′G(α)>0.

### C. Comparison between the Decentralized and Constrained Planner’s Equilibria
- Notation:
  - Competitive equilibrium: αCt
  - Constrained planner’s equilibrium: αPt
- Proposition 4 (Constrained inefficiency / excessive risk-taking):
  - The competitive equilibrium is constrained inefficient and characterized by excessive risk-taking: αCt(πt) > αPt(πt) for any πt.
  - Proof outline: Competitive equilibrium solves F(πt,αCt)=0; planner solves Φ(πt,αPt)=F(πt,αPt)−ξ(πt,αPt)=0 with ξ(πt,αPt)>0; since F is strictly decreasing in αt, αP t < αC t.

C.2 Interaction between Learning and Inefficiency
- Sufficiently convex condition (SCC):
  - θ(πt,αt) is sufficiently convex in αt for any πt if and only if θαα(πt,αt) > θα(πt,αt) |θπ(πt,αt)| / (| θαπ(πt,αt) | − θα(πt,αt)).  (SCC definition as given)
- Definitions for analysis:
  - Iso-vulnerability curve (IV): locus (π,α) along which θ(π,α) is constant: π pG(α) + (1−π) pB(α) = θ̄ ∈ [0,1].
  - Iso-externality curve (IE): locus (π,α) along which normalized externality ξ(π,α) is constant: θα(πt,αt) 1/X (αt) = ξ̄ > 0.
- Lemma 1:
  - The SCC is satisfied iff the slope of the iso-vulnerability curve is larger than that of the iso-externality curve.
- Proposition 5:
  - As investors become more optimistic (πt increases), the competitive equilibrium investment αCt increases faster than the constrained efficient equilibrium αPt if and only if the SCC is satisfied.
  - Interpretation via "two-round" decomposition:
    - Round One (Resilience Effect, RE): Increase in π raises weight on pG, lowering perceived vulnerability θ and inducing both market and planner to increase α. Planner also perceives greater resilience (through p′G weight), decreasing externality ξ and further increasing αP relative to αC by |αPξ|/|ξπ| Δπ.
    - Round Two (Risk-Taking Effect, RTE): Increase in α raises θ and hence ξ, inducing planner to decrease αP relative to αC by |αPξ| ξα Δα.
    - Whether αP increases less than αC depends on whether RTE dominates RE. SCC captures this comparison: if θ is sufficiently convex in α, RTE dominates RE, so αP increases less than αC.
  - Mathematical condition comparing slopes:
    - dαCt/dπt = −θπ/θα
    - dαPt/dπt = −(θπ + ξπ)/(θα + ξα)
    - dαCt/dπt > dαPt/dπt iff −θπ/θα > −(θπ+ξπ)/(θα+ξα) ⇔ −θπ/θα > −ξπ/ξα (equation (7)), which is equivalent to SCC by Lemma 1.

C.3 An Economic Example for SCC
- Fire-sale example:
  - Larger α strengthens fire-sale effects after bad shock; when α small, fire sales have small effect; when α large, fire sales significantly depress asset values and can trigger systemic crisis. Convex asset-management costs for nonbank households imply recovery value falls more with larger α, making runs more likely — consistent with model.
- Empirical test implication: test whether nonbank households’ management fee is convex in amount of capital managed.

### III. Competitive Equilibrium with Tax
A. Decentralized Equilibrium with Tax
- Capital income tax scheme: investor pays Dt dollar for every dollar of payoff from risky asset.
- Investor’s problem with tax (aggregated form):
  - max αit ∈ [0,1] (1−θt)[μS + αit(Δ0 − μR Dt)] + θt[τS μS + αit(Δ1 − τR μR Dt)]  (equation (8))
- FOC (taking pG and pB as given):
  - Δ0 − μR Dt + [(Δ1 − τR μR Dt) − (Δ0 − μR Dt)] θt = 0  (equation (9))

B. Optimal Tax and Macroprudential Policy Implications
- Define αP as social planner’s risky position, with associated θP ≡ πt pG(αP) + (1−πt) pB(αP) ∈ (0,1) — the target θ.
- Plugging θP into competitive FOC yields unique optimal tax rate:
  - D* = [Δ0 + (Δ1 − Δ0) θP] / [μR + (τR μR − μR) θP]  (equation (10))
  - Rearranged form:
    - D* = 1 − μS [1 − (1−τS) θP] / [μR [1 − (1−τR) θP]]  (equation (11))
- Proposition 6:
  - For any belief πt, there exists a unique optimal tax rate D* ∈ (0,1) given by equation (11), which can restore constrained efficiency of competitive equilibrium.
- Proposition 7:
  - As investors and constrained planner become more confident (πt increases), the optimal tax rate D* needed to restore constrained efficiency will be higher if and only if the SCC is satisfied.
- Macroprudential policy implication:
  - If and only if SCC holds, the optimal macroprudential policy should be countercyclical: as market tranquility lasts longer (πt rises), policymaker should raise capital income tax to curb excessive risk-taking and lower systemic risk.
  - Qualification: SCC requires θ(αt,πt) sufficiently convex in αt; regulator must assess structural resilience changes, not just cyclical indicators.
  - Example policy nuance: high credit growth driven by diversification (e.g., lending to non-oil industries) may improve resilience; tightening may not be desirable despite cyclical signals.
  - Financial cycles are much longer than business cycles (about 15 years in U.S., U.K., and average G-7 per cited studies), so policymakers should avoid mechanically loosening/tightening based solely on cyclical indicators.

C. Inefficient Deregulation/Regulation and Further Comparative Statics
- Consider a regulator who, like constrained planner, learns and cares about welfare but mechanically overestimates |ξπ| such that:
  - |θπ(πt,αt)| / θα(πt,αt) ≤ |ξπ(πt,αt)| / ξα(πt,αt) for some πt and αt (i.e., regulator believes SCC fails though SCC actually holds).
- Regulator makes decisions based on overestimated |ξπ|.
- Comparative statics example:
  - Starting from belief π with optimal tax D*, as π increases to π + Δπ, the regulator’s chosen tax will decrease to DReg(π + Δπ) = D* − ΔD (i.e., the regulator may mistakenly loosen policy).

*Source: wp18123 - Section 3*

### Section 4

### wp18123 - Section 4

### Inefficient regulation and learning: corollaries and intuition
- Because the SCC actually holds, Proposition 7 implies the constrained-planner optimal capital income tax D_P(π+∆π) is larger than D^∗, and the regulator-chosen tax D_Reg(π+∆π) satisfies:
  - D_Reg(π+∆π) < D^∗ < D_P(π+∆π)
  - Hence α_Reg(π+∆π) > α_P(π+∆π)
- COROLLARY 1:
  - If the regulator overestimates the resilience and mistakenly believes the SCC fails (|θ_π|/θ_α ≤ |ξ_π|/ξ_α): as the regulator observes one more no-crisis period and becomes more confident, she will lower the capital income tax and induce an inefficiently high aggregate risky investment position.
- The flip side—inefficiently tight regulation:
  - If the regulator underestimates resilience and mistakenly believes the SCC holds (|θ_π|/θ_α > |ξ_π|/ξ_α): as the regulator observes one more no-crisis period and the market becomes more confident, she will tighten regulation and induce an inefficiently low aggregate investment in the industry.
- COROLLARY 2:
  - If the crisis damages the safe real sector more (lower τ_S) and/or the risky financial sector less (higher τ_R), then:
    - (i) the market will invest more in the risky sector.
    - (ii) the planner will be more likely to raise the tax when the market has been tranquil for a longer time.
  - Intuition: lower τ_S and/or higher τ_R make the safe sector less attractive, increasing market incentives to invest in risky sector; α_C becomes more responsive to confidence π, so the planner raises taxes to contain larger externalities as tranquility persists.

### Parametric examples (p_G(α) and p_B(α) specification)
- Specification:
  - p_G(α) = a_G + b_G · α^k
  - p_B(α) = a_B + b_B · α^k
- Assumptions:
  - (1) a_B ≥ a_G ≥ 0, b_B ≥ b_G > 0
  - (2) a_i + b_i ≤ 1 for any i = B, G
  - (3) k ≥ 1
- Derived expressions:
  - θ = (π a_G + (1−π) a_B) + (π b_G + (1−π) b_B) α^k
  - −θ_π = (a_B − a_G) + α^k (b_B − b_G)
  - θ_α = k α^(k−1) (π b_G + (1−π) b_B)
  - −θ_πα = k α^(k−1) (b_B − b_G)
  - θ_αα = k(k−1) α^(k−2) (π b_G + (1−π) b_B)
  - Slope of IV: −θ_π/θ_α = [α(b_B − b_G) + (a_B − a_G) α^(1−k)] / [k(π b_G + (1−π) b_B)]
  - ξ(π_t, α_t) = θ_α(π_t, α_t) [α_t + (1−τ_S) μ_S ∆_0 − ∆_1] / 1_X (notation preserved from source)
  - Slope of IE: −ξ_π/ξ_α = α(b_B − b_G) / [((k−1) + αX)(π b_G + (1−π) b_B)]
- Condition SCC (slope of IV > slope of IE) reduces under strict inequalities a_B > a_G and b_B > b_G:
  - If k > 1 + (b_B − b_G)/(a_B − a_G), then condition SCC holds; otherwise SCC may not hold.
- Two extreme cases:
  - If p′_G = p′_B (i.e., b_B = b_G): RHS becomes 1 − αX < 1, so SCC holds for any k > 1. Therefore the planner should optimally increase the tax when market tranquility persists.
  - If a_B = a_G: RHS becomes ∞, so SCC does not hold regardless of k. Therefore the planner should optimally decrease the tax when market tranquility persists.

### Dynamics of learning and Markov switching
- COROLLARY 3 (dynamics of investors’ learning):
  - First, investors will not revise down their confidence until a crisis has actually occurred.
  - Second, a crisis affects investor confidence only in the immediately next period; confidence will improve from the second post-crisis period onward as long as there is no crisis in the previous period.
- Proof sketch and properties:
  - If first crisis occurs in Period t_C1, then at t = t_C1 + 1, π_{t_C1+1} < π_{t_C1}.
  - For t_C1 + 2 ≤ t ≤ t_C2 (t_C2 being second-crisis period), posterior belief π_t increases period-by-period as long as C_{t−1} = 0; hence the immediate one-period confidence drop followed by gradual re-build—termed a "complacency trap."
  - Simulations (Figures 5 and 6 in source) show sharp declines corresponding to crises; peak belief in later cycles can exceed earlier peaks despite repeated crises.
- Markov switching extension:
  - If underlying state follows Markov transition with persistence q ∈ (1/2, 1) and switching probability 1 − q, interim belief π_t updated to final belief π′_t as:
    - π′_t(C_{t−1} = 0) = q π_t(C_{t−1} = 0) + (1−q)[1 − π_t(C_{t−1} = 0)] = (2q − 1) π_t(C_{t−1} = 0) + (1 − q)
  - Since q > 1/2 implies 2q − 1 > 0, π′_t under Markov switching is monotonically increasing in π_t under fixed true states. Therefore Proposition 1 and all other propositions continue to hold under Markov switching.
  - Simulations (Figure 7 in source) illustrate belief and risky investment dynamics for decentralized market and social planner under Markov switching.

### Unconstrained (informed) planner vs constrained planner
- Motivation:
  - Macrprudential policymakers (central banks or dedicated agencies) may have more information about the true state via stress tests and indicators. Modeled here as an unconstrained planner with full information.
- Case: true state is B (fragile)
  - θ(π_t, α_t) = p_B(α_t)
  - ξ(α_t, π_t) = p′_B(α)[α_t + (1−τ_S) μ_S ∆_0 − ∆_1]
  - Unconstrained planner at period t maximizes over α_t in [0,1]:
    - [1 − p_B(α_t)](μ_S + α_t ∆_0) + p_B(α_t)(τ_S μ_S + α_t ∆_1)   (equation (12))
  - FOC for unconstrained planner (normalized):
    - ∆_0/(∆_0 − ∆_1) − p_B(α_t) − p′_B(α_t)[α_t + (1−τ_S) μ_S ∆_0 − ∆_1] = 0   (equation (13))
- PROPOSITION 8:
  - If true state is B (fragile), aggregate risky investment in constrained planner’s equilibrium α_Pt is strictly higher than in unconstrained/informed planner’s equilibrium α_IPt.
  - Proof idea: p_B(α) > θ(π, α) and p′_B(α) > θ_α(π, α) for any α, hence the unconstrained planner sets a lower α to satisfy FOC.
- Policy implication:
  - Information asymmetry causes the uninformed/constrained planner to choose inefficiently high risky investment (though still more efficient than decentralized investors). Therefore assigning macroprudential policy to independent agencies with technical expertise is desirable.
  - Note (source): since α_IPt does not depend on belief π_t, both α_IPt and the unconstrained planner’s optimal tax remain constant as market and constrained planner confidence change.

### Conclusion: contributions and policy lessons
- Empirical motivation: financial crises tend to follow prolonged periods of stability and investor optimism.
- Contributions:
  - Demonstrates excessive risk-taking arising from interaction of negative externality and learning: longer histories of tranquility increase investor confidence and aggregate risky positions, raising crisis probability.
  - Provides framework to assess macroprudential regulation efficiency: characterizes conditions when constrained inefficiency is increasing in investor confidence and when optimal macroprudential policy is countercyclical.
  - Evaluates a capital income tax as macroprudential tool and shows optimal tax should be higher as market tranquility persists under the same conditions.
- Policy caveats:
  - Optimal macroprudential policies are not always countercyclical; they depend on trade-off between resilience effect and risk-taking effect.
  - Misassessment of trade-offs can lead to inefficient regulation—either excessive loosening or unnecessary repression of financial activity.
- Suggested future research:
  - Provide micro foundations for crisis probability functions p_G(α) and p_B(α).
  - Empirically test model implications, possibly following approaches indicated in Subsection II.C.3 of the source.

*Source: wp18123 - Section 4*

### Section 5

### wp18123 - Section 5

### Derivation of dθ_P/dπ and interpretation of effects
- Recall that θ_t ≡ π_t p_G(α_t) + (1−π_t) p_B(α_t). The derivative of θ_P with respect to π is given by:
  - dθ_P/dπ = θ_π(α_t,π_t) + θ_α(α_t,π_t) dα_P/dπ  (17)
- Components:
  - θ_π(α_t,π_t) = p_G(α_P) − p_B(α_P), which is negative. This term captures the direct effect of π on θ_P: as investors and the constrained planner become more optimistic that the financial industry is strong (so the crisis is less likely), the unconditional crisis probability θ_P perceived by them tends to be lower.
  - θ_α(α_t,π_t) dα_P/dπ is positive. This term captures the indirect effect of π on θ_P: a more optimistic belief induces investors and the constrained planner to increase their positions in the risky asset (α becomes higher), which in turn raises the unconditional crisis probability θ.
- Rearranging (17) yields:
  - dθ_P/dπ = θ_α(α_t,π_t)[ θ_π(α_t,π_t)/θ_α(α_t,π_t) + dα_P/dπ ]
  - = θ_α(α_t,π_t)( dα_P/dπ − dα_C/dπ )
- Since θ_α(α_t,π_t) > 0, the sign of dθ_P/dπ is determined by the bracketed difference.

### Condition for dθ_P/dπ < 0 and relation to SCC
- dθ_P/dπ < 0 if and only if dα_C/dπ > dα_P/dπ.
- Interpretation:
  - The unconditional crisis probability perceived by investors and the constrained planner falls with greater optimism (dθ_P/dπ < 0) exactly when the change in the constrained planner's risky position with respect to π (dα_C/dπ) exceeds that of the planner (dα_P/dπ).
  - This condition is equivalent to the SCC being satisfied (by Proposition 5).
- Conclusion: Q.E.D.

*Source: wp18123 - Section 5*

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_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18123.pdf_