## _wp1546 - Appendix A: Two Issues ................................................................................................

## Source details

**Canonical URL:** [_wp1546 - Appendix A: Two Issues ................................................................................................](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp1546.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp1546.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp1546.pdf.json)

---

### I. INTRODUCTION — framing and high-level claims
- Objective: present a two-period model in which interconnectedness (portfolio mixing of loans originated by different banks) can produce inefficient liquidation and obscure source of portfolio returns, worsening crises.
- Key mechanisms:
  - Contagion: bad returns originating at one bank can cause liquidation/failure at connected banks.
  - “Bad managers” externality: diversified portfolios can obscure which bank produced bad loans, causing replacement of both banks’ managers, reducing informational capital and increasing future low-return probabilities.
- Result: two-period business cycle with second-period expected loan returns lower and recessions more likely than in the first period; interconnectedness increases amplitude.

### II. MODEL — structure, agents, and preferences
- Time horizon and agents:
  - Two periods; two ex ante identical banks operating in both periods.
  - Each period each bank has an equal-measure continuum of one-period-lived depositors (tied to the bank); depositors are owners and funders.
- Manager types and competence probabilities:
  - Experienced: managed last period and made good loans last period; probability competent this period = ep (assume 1/2 < ep < 1; baseline ep = 3/4).
  - Inexperienced: did not manage last period; probability competent this period = ip (assume 0 < ip < ep; baseline ip = 1/2).
  - Incompetent: managed last period and made bad loans last period; probability competent this period = 0.
  - If a bank hires new managers they are inexperienced.
- Depositor utility:
  - Constant-relative-risk-aversion u(c) = (c^{1-γ} - 1)/(1-γ) for γ > 0; consider γ = 1 (ln u), γ = 2, γ = 4.
  - γ = 1: weakly risk averse; γ = 2 moderate; γ = 4 strongly risk averse.

### II.B Return distributions on loans — base returns and shocks
- Base returns: good = g, bad = b with 0 < b < g.
  - Baseline example values: g = 1 and b = 1/2.
- Small random shocks:
  - Good outcome shocks: gε±; bad outcome shocks: bε±.
  - Two baseline versions:
    - Version 1: bε = 0.
    - Version 2: bε = 1/12.
  - Baseline: gε = 1/3.
- Important exact assumptions preserved as stated in source (symbolic inequalities and θ expressions).

### II.C Bank interconnectedness (portfolio diversification regimes)
- Diversification regimes and parameterization:
  - No diversification (ND): d = 0; θ = 0.
  - Partial diversification (PD): baseline d = 1/2 → θ = 1/3.
  - Complete diversification (CD): d = 1 → θ = 1/2.
  - θ = d/(1 + d) = fraction of portfolio consisting of other-bank loans.
- Potential joint total return outcomes:
  - Version 1: eight joint total return outcomes (four unmixed, four mixed).
  - Version 2: twelve joint total return outcomes (four unmixed, eight mixed).
- Realized returns: if any potential total return outcome < q (liquidation value), realized value becomes q in equilibrium.
- Baseline choice: q = 5/8 satisfies Assumptions 1 and 2 for examples.

### II.D Liquidation of loans — timing, signals, and rules
- Timing and mechanics:
  - Middle of each period banks can liquidate any fraction of financed projects; liquidation yields q units per project.
  - Managers cannot control which projects are liquidated: liquidations occur proportionally to origin shares.
  - Depositors can withdraw at the liquidation moment, receiving q but foregoing end-of-period returns.
- Interim private signal:
  - At the middle moment each bank’s managers and depositors receive a perfectly accurate private signal whether the portfolio average repayment will exceed q (favorable) or fall short (unfavorable); signal provides no information about composition by originator.
  - Depositor behavior: unfavorable → withdraw and obtain q with certainty; favorable → do not withdraw and obtain > q with certainty.
- Implications from Assumptions 1 and 2:
  - If joint outcome unmixed then neither bank gets unfavorable signal regardless of diversification; if portfolios not diversified then at most one bank gets unfavorable signal.
  - If any diversification (PD or CD), then both banks can get unfavorable signal (left inequality), and if joint outcome is mixed with the good loans getting the high shock, then neither bank gets unfavorable signal (right inequality).

### III. CONSTRUCTING AN EQUILIBRIUM — decisions and information
- Equilibrium components:
  - Depositors’ manager retention/replacement decisions (beginning of each period).
  - Managers’ diversification choices (after retention decisions).
  - Liquidation choices (middle of period).
- Information structure:
  - Diversification decisions and amounts paid to depositors are publicly observable; interim signal observed only by the bank’s own depositors and managers.
  - Period-2 new depositors know only period-1 diversification regime and returns paid to period-1 depositors.
  - Incumbent managers know their own past performance but cannot credibly reveal it when depositors cannot infer it.
  - At beginning of period 1 both banks’ managers are assumed experienced.
- Liquidation design ensures full withdrawal and full-portfolio liquidation when a bank receives an unfavorable signal.

### III.C Managers’ diversification choice
- Managers choose ND, PD, or CD to maximize expected utility of their depositors (tie-break rule: choose most extensive diversification that gives marginal benefit to both when disagreements arise).
- Key baseline example manager-choice highlights (preserve exact outcomes as stated):
  - If both managers experienced → CD best for any depositor risk aversion → CD prevails in period 1 absent regulation; and in period 2 when period-1 outcome unmixed.
  - If one experienced and one inexperienced:
    - ND if depositors weakly risk averse.
    - If moderately risk averse: CD in Version 1 but ND in Version 2.
    - If strongly risk averse: CD in Version 1 but PD in Version 2.
  - If both inexperienced:
    - ND if weak risk aversion, CD if moderate or strong risk aversion (both versions).

### III.D Depositor “firing” decision and identification problems
- Depositors replace managers if replacement increases expected utility.
- In period 2 depositor decisions can be nontrivial because returns alone may not identify the bad-originator when portfolios are diversified.
- Cases producing uncertainty about which bank was incompetent:
  - PD with a mixed joint return outcome that causes both banks to liquidate.
  - CD with any mixed joint return outcome (depositors of both banks receive identical returns).
- Manager-replacement decision modeled as a non-cooperative game between two depositor groups; look for pure-strategy Nash equilibria.
- Condition for both-banks-replace Nash equilibrium:
  - If expected utility for a bank with inexperienced managers when the other bank’s managers are also inexperienced (eu-ii) > average of expected utilities for a bank with incompetent managers (eu-n) and a bank with experienced managers when the other is inexperienced (eu-ei), then both-replace equilibrium exists (true in both baseline versions).
- Incumbent managers cannot credibly reveal past performance when revelation would harm them.

### III.E Optimal regulation — government problem and policy instrument
- Government objective: maximize average ex ante expected utility of period-1 and period-2 depositors (depositors do not know bank assignment).
- Government can act only in period 1: options are do nothing or rule out one or two regimes (forbid CD or forbid CD and PD → force ND).
- Rationale: intertemporal externality — period-1 depositors ignore effects on future depositors; government values both periods.
- Government decision depends on depositor risk aversion and how period-1 diversification affects period-2 probabilities of manager-experience states.
- Government may prefer less diversification in period 1 to reduce external costs that raise period-2 recession probability and reduce informational capital.

### III.F Equilibrium sequence and period outcomes
- Period 1:
  - Depositors retain managers; managers choose diversification maximizing period-1 depositor utility given any regulation.
  - Interim signals determine liquidation decisions; returns paid at end of period.
- Period 2:
  - New depositors observe period-1 returns and diversification regime; they make manager retention/replacement decisions to maximize expected utility.
  - Managers then choose diversification to maximize period-2 depositor utility (no regulatory constraint).
- Key difference: period-2 may have states with one or two inexperienced managers; period-1 starts with both experienced.

### III.G Interpretations — recessions, crises, and business cycle
- Definitions in model:
  - Total loan return across two banks = total output for the period.
  - Recession: any period in which one base loan return outcome is bad.
  - Financial crisis: at least one bank liquidates its asset portfolio in a period; systemic crisis = both banks liquidate.
  - Severe recession: periods with double-liquidation (systemic crisis) — produces the lowest total loan return.
- Role of diversification:
  - Diversification can produce inefficient liquidation during recessions: good loans may be liquidated or bad loans may not be liquidated.
  - Systemic crises and severe recessions can occur only when portfolios are diversified.
- Built-in business cycle:
  - Period 1 begins with two experienced managers → lower recession probability.
  - A period-1 recession increases probability of period-2 recession via manager replacement dynamics.
  - Interconnectedness magnifies this effect: diversified period-1 portfolios make it more likely that period-1 bad outcomes cause joint manager replacement and thus higher period-2 recession probability and frequency of systemic crises.
- Government goal: regulate in period 1 (when outlook favorable) to improve period-2 prospects and raise average expected depositor utility.

### III.H Summary of the baseline example — key numerical features
- Two versions:
  - Version 1: bε = 0.
  - Version 2: bε = 1/12.
- Exact baseline numerical parameters:
  - g = 1
  - gε = 1/3
  - b = 1/2
  - q = 5/8
  - θ under PD = 1/3
  - ep = 3/4
  - ip = 1/2
  - mep = 2/3
- Noted exaggerations in baseline:
  - Period-1 recession probability set at 50 percent.
  - Output loss from a recession on order of 20 percent of non-recession output (b = 1/2 vs g = 1).
  - Large shocks: good shock = 1/3 of good return; bad shock (Version 2) = 1/6 of bad return.
  - These amplify both benefits and costs of regulation.

### Laissez-Faire (LF) equilibria — baseline (no regulation) outcomes
- Period-1 under LF:
  - Managers choose CD regardless of depositor risk aversion.
  - Expected output per depositor in period 1 is 11.5 percent below “normal” (normal = unity).
  - Probability of a recession in period 1 = 50 percent.
  - Average output in a recession = 22.9 percent below normal.
  - Half the recessions are accompanied by financial crises (liquidation); all these crises are systemic (both banks) because CD makes returns equal.
  - Inefficient liquidation occurs in all recessions:
    - In half the recessions, loans with returns < q are not liquidated.
    - In the other half, loans that would have yielded > q are liquidated.
- Period-2 under LF:
  - 50 percent probability period-2 outlook identical to period-1 (both managers retained experienced) — unmixed period-1 outcome.
  - 50 percent probability both banks replace managers → both inexperienced.
  - If both inexperienced, recession always occurs; thus period-2 recession probability = 75 percent.
  - Systemic crisis probability rises from 25 percent in period 1 to 37.5 percent in period 2.
- Risk-aversion effects:
  - If depositors weakly risk averse (γ = 1), when both inexperienced they choose ND, mitigating output loss from inefficient liquidation:
    - Recession output averages 20.1 percent below normal vs 22.9 percent under CD.
    - Average output in period 2 = 15.1 percent below normal (weak-risk-aversion case) vs 11.5 percent in period 1.
  - If depositors moderately or strongly risk averse, managers choose CD even when inexperienced → period-2 average output 17.2 percent below normal.
- Probability of a long (two-period) recession under LF = 50 percent (in this example, period-1 recession always followed by period-2 recession absent regulation).

### Equilibria under government regulation — selected baseline results
- If depositors weakly risk averse (γ = 1), government imposes ND in period 1 in both versions:
  - Rationale: period-1 depositors not greatly harmed by increased return variability from prohibiting diversification; gains to period-2 depositors from reduced probability of recession dominate.
  - Under ND in period 1:
    - Probability of a recession in period 1 unchanged at 50 percent.
    - Expected output in period 1 is only 9.4 percent below normal (improvement from LF 11.5 percent below normal).
    - Expected output in a recession = 18.8 percent below normal (less severe because liquidation efficient under ND).
    - ND eliminates systemic crises in period 1 (no possibility of both banks liquidating when portfolios not mixed).
  - Period-2 effects:
    - ND in period 1 replaces the 50 percent probability that both banks will have inexperienced managers with 50 percent probability that only one bank will have inexperienced managers.
    - When only one bank inexperienced, period-2 recession probability = 75 percent; unconditional period-2 recession probability falls to 62.5 percent (from 75 percent under LF).
    - Probability of a long recession falls from 50 percent to 37.5 percent.
- Regulation loosens the tight link between period-1 and period-2 recessions and lowers unconditional period-2 recession probability to 0.625 in the weak-risk-aversion case.

### Key quantitative findings on recessions, diversification, and regulation
- Under LF (full diversification), average period-1 recession reduces output by 22.9 percent; output averages 11.5 percent below normal in period 1.
- In Version 2 under similar circumstances, the period-1 figures are 23.6 percent and 11.8 percent, respectively.
- When diversification is prohibited in period 2 (after ND in period 1), the average output loss from recessions in period 2 would be 11.7 percent, compared to the LF figures of 15.1 or 17.2 percent (depending on depositors’ degree of risk aversion).
- Absent diversification in period 2, the increase in average output loss from period 1 to period 2 is 2.3 percentage points; under LF the increase would be 3.6 or 5.7 percentage points.
- When depositors are weakly risk averse and regulation induces ND in period 1, the probability of a long recession is 37.5 percent.
- With weakly risk averse depositors and regulation that prevents diversification by inexperienced managers in period 2, average output in recession is 20.4 percent below normal; average output (overall) is 12.8 percent below normal, compared to 15.1 percent without regulation.

### Probabilities of period-2 recessions and long recessions across regimes and versions
- LF (full diversification) period-2 recession probability: 0.75.
- When depositors weakly risk averse and regulation forces ND in period 1, period-2 recession probability: 0.625.
- Version 1 (moderate regulation outcomes with relatively risk-averse depositors): period-2 recession probability is 11/16.
- Version 2: period-2 recession probability is 21/32.
- Probability of a long recession:
  - Version 1: 7/16.
  - Version 2: 13/32.
- Under PD in period 1:
  - Half the recessions produce liquidation by at least one bank (a financial crisis).
  - In Version 1 all these crises are systemic (both banks liquidate).
  - In Version 2, half the crises involve liquidation by only one bank, so only a fourth of recessions produce systemic crises.

### Effects of depositor risk aversion and version differences
- As depositors become more risk averse:
  - Diversification increases.
  - Inefficient liquidation increases.
  - Average recession severity and average output cost of recessions increase.
- Version differences:
  - Version 2 lowers the likelihood of the worst-case scenario for period 2 relative to Version 1, reducing average recession costs.
  - Lowest reported figures: 21.1 percent (extent average recession reduces output) and 13.8 percent (average output cost of recessions) for depositors moderately risk averse in Version 2.
  - Highest reported figures: 22.9 percent and 15.8 percent for depositors strongly risk averse in Version 1.
  - Even the highest regulated-case figure (15.8 percent) is substantially lower than 17.2 percent (average output loss from recessions when there is no regulation for these degrees of risk aversion).

### Mechanisms: how diversification creates fragility and how regulation can help
- Diversification obscures loan origin, making it difficult for depositors to infer which managers created bad loans.
- This leads to:
  - Inefficient loan liquidation decisions (increasing recession severity and enabling systemic crises).
  - Inefficient retention/replacement of managers (reducing informational capital and raising probability of long, severe recessions).
- Government restrictions on interconnectedness:
  - Can increase average welfare of bank depositors.
  - Optimal restriction may not eliminate systemic crises or long severe recessions entirely.
  - Form of regulation matters:
    - Extreme regulation (ND in period 1) can eliminate systemic crises in period 1 and substantially reduce links between period-1 and period-2 recessions.
    - Moderate regulation (PD in period 1) can still leave a tight link between systemic crises in period 1 and recessions in period 2.

### Strategic considerations, Nash equilibria, and depositor horizon (Appendix highlights)
- Nash equilibrium with double replacement is unique under baseline inequalities; all three inequalities hold strictly in baseline examples → there is always a Nash equilibrium in which both banks replace managers.
- If depositors live across both periods and can shift banks between periods, competitive pressures lead depositors to favor banks that diversify completely in period 1, yielding universal CD absent government action.
- Government coordination can improve outcomes by restricting diversification that individual depositor incentives cannot sustain.

### Policy implications and trade-offs
- Restricting diversification can:
  - Reduce average output costs of recessions.
  - Reduce frequency of long recessions.
  - Improve average depositor welfare.
- Trade-offs:
  - Eliminating diversification can reduce depositor welfare in some cases.
  - Government must balance improved systemic stability against lost benefits of diversification for period-1 depositors.
- Regulatory design must account for depositor risk aversion:
  - Weakly risk averse depositors → stricter regulation (ND) more effective.
  - Relatively risk averse depositors → moderate regulation (PD) reduces risk but may leave strong links between systemic crises and subsequent recessions.

### APPENDIX B: THE BASELINE NUMERICAL EXAMPLE — setup and key equilibrium characterizations
- Version definitions and common parameters:
  - Version 1: bε = 0.
  - Version 2: bε = 1/12.
  - Common parameters:
    - g = 1
    - gε = 1/3
    - b = 1/2
    - q = 5/8
    - θ under PD = 1/3
    - ep = 3/4, ip = 1/2, mep = 2/3
  - Risk-aversion indices: s = 1, 2, 3 correspond to γ = 1, 2, 4 in source notation.
- Period-1 return outcomes and period-2 transition probabilities summarized for ND, PD, CD (exact distributions and expected utilities preserved in source tables and expressions).
- Period-2 diversification decisions summarized:
  - Both experienced → choose CD; obtain ( - )sEU ee cd.
  - One experienced/one inexperienced → diversification choice by experienced bank depends on s and Version (preserve precise outcomes as above).
  - Both inexperienced → ND if weak risk aversion, otherwise CD.
- Manager replacement decision and Nash equilibrium conditions (exact inequalities preserved as stated in source):
  - Three exact inequalities listed for s = 1, 2, 3 ensure pure-strategy equilibrium where both banks replace managers; all hold strictly in baseline.
  - Uniqueness established via additional strict inequalities provided in source.
- Welfare comparisons:
  - Government welfare expressed as average ex ante expected utility across two periods; full algebraic expressions for unregulated, ND-imposed, and PD-imposed welfare-values preserved in source.
  - Comparative results:
    - s = 1: ND in period 1 produces highest government-welfare value; implication: prohibit CD and PD in period 1 → impose ND.
    - s = 2 or 3: PD in period 1 produces highest government-welfare value; implication: prohibit CD in period 1 so banks choose PD.

*Source: _wp1546 - Appendix A: Two Issues and Appendix B: The Baseline Numerical Example (excerpted IMF working paper content).*

### Appendix A: Two Issues .................................................................................................

### _wp1546 - Appendix A: Two Issues ................................................................................................

### Appendix A: Two Issues
- Appendix A: Two Issues — page 35

### Appendix B: The Baseline Numerical Example
- Appendix B: The Baseline Numerical Example — page 38

### References
- References — (no page number provided in source excerpt)

*Source: _wp1546 - Appendix A: Two Issues .................................................................................................*

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

### _wp1546 - References .............................................................................................................

### I. INTRODUCTION — framing and high-level claims
- The recent financial crisis was unusually severe, followed by the “Great Recession” and a longer period of economic malaise.
- Widely held views motivating the model:
  - Interconnectedness among financial institutions (portfolio mixing/diversification) may increase vulnerability to systemic financial crises and more severe recessions.
  - Lack of complete and accurate information about banks’ portfolios contributed to the crisis; loss of confidence in institutions’ ability to assess credit risk made credit harder to obtain for risky borrowers.
- Paper objective: present a model in which:
  - Interconnectedness means portfolio mixing of loans originated by different banks.
  - Interconnectedness can produce inefficient liquidation (fire-sale–type outcomes) and obscure the source of portfolio returns, worsening crises.
  - Diversification retains traditional risk-reduction benefits for risk-averse depositors, but two disadvantages can outweigh those benefits:
    1. Contagion in the sense that bad returns originating at one bank can cause liquidation/failure at connected banks.
    2. “Bad managers” externality: diversified portfolios can obscure which bank produced bad loans, causing replacement of both banks’ managers, thereby reducing informational capital across the system and increasing future low-return probabilities.
- Resulting mechanism produces a two-period business-cycle: second-period expected loan returns are lower and recessions more likely than in the first period; interconnectedness increases the amplitude.

### II. MODEL — structure and agents
- Time horizon and agents:
  - Two periods.
  - Two ex ante identical banks operating in both periods.
  - Each period, each bank has an equal-measure continuum of one-period-lived depositors (tied to the bank).
  - Depositors are both owners and funders; each depositor has endowment sufficient for one project.
- Bank organization and managers:
  - Banks pool depositor funds to originate loans; in any period, all loans originated by a bank’s current managers have the same return distribution and yield the same return outcome.
  - Managers types (three): experienced, inexperienced, incompetent:
    - Experienced: managed last period and made good loans last period; probability competent this period = ep (assume 1/2 < ep < 1; baseline ep = 3/4).
    - Inexperienced: did not manage last period; probability competent this period = ip (assume 0 < ip < ep; baseline ip = 1/2).
    - Incompetent: managed last period and made bad loans last period; probability competent this period = 0.
  - If a bank hires new managers they are inexperienced (experienced managers prefer to stay with their prior bank).
- Depositor preferences:
  - Constant-relative-risk-aversion utility u(c) = (c^{1-γ} - 1)/(1-γ) for γ > 0; consider γ = 1 (ln u), γ = 2, γ = 4.
  - γ = 1: weakly risk averse; γ = 2 moderate; γ = 4 strongly risk averse.

### II.B Return distributions on loans — base returns and shocks
- Base returns: two possible base return outcomes per loan portfolio: good = g, bad = b, with 0 < b < g.
  - Baseline example values: g = 1 and b = 1/2.
- Small random shocks around base returns:
  - Good outcome shocks: gε±; bad outcome shocks: bε±.
  - Shocks independent across banks and periods; assume 0 ≤ bε ≤ gε, gε < g, bε < b.
  - Two baseline versions:
    - Version 1: bε = 0.
    - Version 2: bε = 1/12.
- Two important assumptions:
  - Assumption 1: b g b q g ε ε + < < - (text preserves exact symbolic inequality as stated).
  - Assumption 2: (θ expressions in original) — preserves precise inequality condition as stated in source.
- Joint base returns:
  - If neither set of managers made bad loans last period, probability that both sets will originate bad loans this period is zero.
  - Definitions:
    - Unmixed outcome: both managers’ loans good.
    - Mixed outcome: one bank’s loans good, the other’s bad (favors a bank).
  - Probabilities in baseline example (Table 1 summary):
    - When both experienced: marginal good probability ep = 3/4; unmixed joint probability eep = 1/2.
    - When one experienced, one inexperienced: unmixed probability eip = 1/4; conditional mixed probability favoring experienced managers mep = 2/3.
    - When both inexperienced: marginal good probability ip = 1/2; unmixed joint probability iip = 0 in baseline example 1.

### II.C Bank interconnectedness (portfolio diversification regimes)
- Diversification regimes studied:
  - No diversification (ND): each bank’s portfolio entirely its own loans (d = 0; θ = 0).
  - Partial diversification (PD): portfolio partly own loans, partly other bank loans; baseline PD d = 1/2 → θ = 1/3.
  - Complete diversification (CD): portfolio equally divided (d = 1 → θ = 1/2).
- Parameterization:
  - d = ratio of other-bank loans to own loans; θ = d/(1 + d) = fraction of portfolio consisting of other-bank loans.
- Potential joint total return outcome sets:
  - Version 1: eight joint total return outcomes (four unmixed, four mixed).
  - Version 2: twelve joint total return outcomes (four unmixed, eight mixed).
- Realized returns: if any potential total return outcome < q (liquidation value), realized value becomes q in equilibrium.

### II.D Liquidation of loans — timing and rules
- Timing: middle of each period, after loans are extended but before completion and repayment, banks can liquidate any fraction of financed projects; liquidation yields q units per project (repayment).
- Managers cannot control which projects are liquidated: liquidations occur proportionally to origin shares (fraction of originator’s loans liquidated equals 1 - θ).
- Depositors have the immediate withdrawal right at the liquidation moment: withdrawing yields q but forfeits end-of-period returns.
- Interim private signal: at that middle moment, each bank’s managers and depositors receive a perfectly accurate private signal indicating whether the portfolio average repayment will exceed q (favorable) or fall short (unfavorable); the signal provides no information about composition by originator.
- Liquidation behavior calibrated so that:
  - A depositor seeing an unfavorable signal will liquidate and obtain q with certainty.
  - A depositor seeing a favorable signal will not liquidate and obtain > q with certainty.
- Implications from Assumptions 1 and 2:
  - Assumption 1: if joint outcome unmixed then neither bank gets unfavorable signal regardless of diversification; if portfolios not diversified then at most one bank gets unfavorable signal.
  - Assumption 2: if any diversification (PD or CD), then both banks can get unfavorable signal (left inequality), and if joint outcome is mixed with the good loans getting the high shock, then neither bank gets unfavorable signal (right inequality).
  - Baseline choice q = 5/8 satisfies these assumptions for the examples.

### III. CONSTRUCTING AN EQUILIBRIUM — decisions and information
- Equilibrium components: depositors’ manager retention/replacement decisions (beginning of each period), managers’ diversification choices (after retention decisions), and liquidation choices (middle of period).
- Information structure:
  - All agents know model structure and distributions (competence, returns, q, signal nature).
  - Diversification decisions and amounts paid to depositors are publicly observable; interim signal is observed only by the bank’s own depositors and managers.
  - Period-2 new depositors know only diversification regime chosen in period 1 and returns paid to period-1 depositors.
  - Incumbent managers know their own past performance; cannot credibly reveal it when depositors cannot infer it.
  - At beginning of period 1 both banks’ managers are assumed experienced.
- Liquidation decision design ensures full withdrawal and full-portfolio liquidation when a bank receives an unfavorable signal.

### III.C The diversification decision (manager objective and coordination)
- Managers choose ND, PD, or CD to maximize expected utility of their depositors (with one tie-breaking rule: choose most extensive diversification that gives marginal benefit to both when disagreements arise).
- If one set of managers is incompetent, the other will refuse to diversify → ND results (occurs only in period 2 when identity known).
- Baseline example manager choices summary (selected highlights preserved as stated):
  - If both managers experienced → CD best for any depositor risk aversion → CD prevails in period 1 absent regulation; and in period 2 when period-1 outcome unmixed.
  - If one experienced and one inexperienced:
    - Choice is ND if depositors are weakly risk averse.
    - If depositors moderately risk averse, choice is CD in Version 1 but ND in Version 2.
    - If depositors strongly risk averse, choice is CD in Version 1 but PD in Version 2.
  - If both inexperienced:
    - ND if weak risk aversion, CD if moderate or strong risk aversion (in both versions).

### III.D The “firing” decision (depositor replacement of managers)
- Depositors replace managers if replacement increases their expected utility.
- In period 1: trivial decision — retain incumbents (both experienced).
- In period 2: decision may be nontrivial because depositors observe only portfolio returns and cannot always infer which bank produced bad loans.
- Cases producing uncertainty about which bank was incompetent:
  - PD with a mixed joint return outcome that causes both banks to liquidate.
  - CD with any mixed joint return outcome (depositors of both banks receive identical returns so cannot identify originator).
- When identity of the incompetent managers is unknown, model manager-replacement decision as a non-cooperative game between the two groups of depositors; look for pure-strategy Nash equilibria.
- Condition for Nash equilibrium in which both banks replace managers:
  - If expected utility for a bank with inexperienced managers when the other bank’s managers are also inexperienced (eu-ii) > average of expected utilities for a bank with incompetent managers (eu-n) and a bank with experienced managers when the other is inexperienced (eu-ei), then both-replace equilibrium exists (true in both baseline versions).
- Incumbent managers cannot credibly reveal past performance when revelation would harm them.

### III.E Optimal regulation — government problem and policy instrument
- Government objective: maximize average ex ante expected utility of period-1 and period-2 depositors (depositors do not know bank assignment).
- Government can act only in period 1: options are to do nothing (allow ND/PD/CD) or rule out one or two of the regimes (e.g., forbid CD or forbid CD and PD → force ND).
- No government constraint in period 2.
- Rationale for regulation: intertemporal externality — depositors in period 1 act to maximize current expected utility and ignore their effect on future depositors; government values both periods.
- Government chooses based on depositor risk aversion level and how period-1 diversification affects period-2 probabilities of manager-experience states (both experienced, one experienced/one inexperienced, both inexperienced).
- Government may prefer less diversification in period 1 to reduce external costs that raise period-2 recession probability and reduce future informational capital.

### III.F Characterizing equilibria — sequence and outcomes
- Period 1:
  - Depositors retain managers; managers choose diversification maximizing period-1 depositor utility given any regulation.
  - Interim signals determine liquidation decisions; returns paid at end of period.
- Period 2:
  - New depositors observe period-1 returns and diversification regime; they make manager retention/replacement decisions to maximize expected utility.
  - Managers then choose diversification to maximize period-2 depositor utility (no regulatory constraint).
  - Period ends and returns paid; model ends.
- Key difference: period-2 may have states with one or two inexperienced managers; period-1 starts with both experienced.

### III.G Interpretations and assumptions — recessions, crises, and business cycle
- Definition mappings in model:
  - Total loan return across two banks = total output for the period.
  - Recession periods: any period in which one base loan return outcome is bad.
  - Financial crisis: at least one bank liquidates its asset portfolio in a period; systemic crisis = both banks liquidate.
  - Severe recession: periods with double-liquidation (systemic crisis) — produces the lowest total loan return.
- Role of diversification:
  - Diversification can produce inefficient liquidation during recessions: good loans may be liquidated or bad loans may not be liquidated.
  - Systemic crises and severe recessions can occur only when portfolios are diversified in this model.
- Built-in business cycle:
  - Period 1 begins with two experienced managers → lower recession probability.
  - A period-1 recession increases probability of period-2 recession via manager replacement dynamics.
  - Interconnectedness magnifies this effect: diversified period-1 portfolios make it more likely that period-1 bad outcomes cause joint manager replacement and thus higher period-2 recession probability and frequency of systemic crises.
- Government goal: regulate in period 1 (when outlook favorable) to improve period-2 prospects and raise average expected depositor utility.

### III.H Summary of the baseline example — specification and key numerical features
- Specification highlights (baseline parameters preserved exactly as in text):
  - Two versions: Version 1 (no shocks to bad base return), Version 2 (bad base return shock = ±1/12).
  - Good base loan return g = 1; bad base loan return b = 1/2.
  - Shock to good base return = ±1/3.
  - Liquidation value q = 5/8.
  - Experienced manager competence probability ep = 3/4; inexperienced ip = 1/2.
  - If one bank experienced & other inexperienced, conditional mixed outcome favors experienced managers with probability mep = 2/3.
  - Under PD, one-third of each bank’s asset portfolio consists of loans originated by the other bank (θ = 1/3).
- Noted exaggerations:
  - Period-1 recession probability set at 50 percent (very high).
  - Output loss from a recession on order of 20 percent of non-recession output (bad return b = 1/2 vs g = 1).
  - Large shocks: good shock = 1/3 of good return; bad shock (Version 2) = 1/6 of bad return.
  - These parameter choices amplify both benefits of regulation (by making recession costs large) and costs of regulation (by making diversification benefits large).

### Laissez-Faire (LF) equilibria — baseline (no regulation) outcomes
- Period-1 managers choose CD regardless of depositor risk aversion (risk-spreading benefits dominate costs from inefficient liquidation).
- Period-1 summary under LF:
  - Expected output per depositor in period 1 is 11.5 percent below “normal” (normal = unity).
  - Probability of a recession in period 1 = 50 percent.
  - Average output in a recession = 22.9 percent below normal.
  - Half the recessions are accompanied by financial crises (liquidation); all these crises are systemic (both banks) because CD makes returns equal.
  - Inefficient liquidation occurs in all recessions:
    - In half the recessions, loans with returns < q are not liquidated.
    - In the other half, loans that would have yielded > q are liquidated.
  - Inefficient liquidation raises average output loss in a recession substantially.
- Period-2 under LF:
  - 50 percent probability period-2 outlook identical to period-1 (both managers retained experienced) — follows unmixed period-1 outcome.
  - 50 percent probability both banks replace managers (under CD replacement of both managers follows an unmixed outcome in period 1 because equal returns prevent identification).
  - If both managers inexperienced, recession always occurs; thus period-2 recession probability = 75 percent.
  - Systemic crisis probability rises from 25 percent in period 1 to 37.5 percent in period 2.
- Risk-aversion effects:
  - If depositors weakly risk averse (γ = 1), when both managers inexperienced they choose ND, mitigating output loss from inefficient liquidation: recession output averages 20.1 percent below normal vs 22.9 percent under CD.
  - Average output in period 2 = 15.1 percent below normal (weak-risk-aversion case) vs 11.5 percent in period 1.
  - If depositors moderately or strongly risk averse, managers choose CD even when inexperienced → higher amplitude: period-2 average output 17.2 percent below normal.
- Probability of long (two-period) recession under LF = 50 percent (in this example, period-1 recession always followed by period-2 recession absent regulation).

### Equilibria under government regulation — selected results from baseline example
- Government can restrict diversification in period 1; results depend on depositor risk aversion and model version.
- If depositors weakly risk averse (γ = 1), government imposes ND in period 1 in both versions:
  - Rationale: period-1 depositors not greatly harmed by increased return variability from prohibiting diversification; gains to period-2 depositors from reduced probability of recession dominate.
  - Under ND in period 1:
    - Probability of a recession in period 1 does not change (still 50 percent), but expected output in period 1 is only 9.4 percent below normal (improvement from LF 11.5 percent below normal) because expected output in a recession is only 18.8 percent below normal (less severe than under CD because liquidation is efficient under ND).
    - ND eliminates systemic crises in period 1 (no possibility of both banks liquidating when portfolios not mixed).
  - Period-2 effects:
    - ND in period 1 replaces the 50 percent probability that both banks will have inexperienced managers with 50 percent probability that only one bank will have inexperienced managers.
    - When only one bank inexperienced, period-2 recession probability = 75 percent (as stated earlier), so unconditional period-2 recession probability falls to 62.5 percent (from 75 percent under LF).
    - Probability of a long recession falls from 50 percent to 37.5 percent because a period-1 recession no longer guarantees a period-2 recession.
- Government regulation thus loosens the tight link between period-1 and period-2 recessions and lowers unconditional period-2 recession probability to 62.5 percent in the weak-risk-aversion case.

*Italic: Source — content unit _wp1546 - References .............................................................................................................*

### 11.5 percent under LF.  More importantly, however, the probability of a recession in period 2 is

### _wp1546 - 11.5 percent under LF.  More importantly, however, the probability of a recession in period 2 is

### Key quantitative findings on recessions, diversification, and regulation
- Under LF (full diversification), average period-1 recession reduces output by 22.9 percent; output averages 11.5 percent below normal in period 1.
- In Version 2 under similar circumstances, the period-1 figures are 23.6 percent and 11.8 percent, respectively.
- When diversification is prohibited in period 2 (after ND in period 1), the average output loss from recessions in period 2 would be 11.7 percent, compared to the LF figures of 15.1 or 17.2 percent (depending on depositors’ degree of risk aversion).
- Absent diversification in period 2, the increase in average output loss from period 1 to period 2 is 2.3 percentage points; under LF the increase would be 3.6 or 5.7 percentage points.
- When depositors are weakly risk averse and regulation induces ND in period 1, the probability of a long recession is 37.5 percent.
- With weakly risk averse depositors and regulation that prevents diversification by inexperienced managers in period 2, average output in recession is 20.4 percent below normal; average output (overall) is 12.8 percent below normal, compared to 15.1 percent without regulation.

### Probabilities of period-2 recessions and long recessions across regimes and versions
- LF (full diversification) period-2 recession probability: 0.75.
- When depositors are weakly risk averse and regulation forces ND in period 1, period-2 recession probability: 0.625.
- Version 1 (moderate regulation outcomes with relatively risk-averse depositors): period-2 recession probability is 11/16 (almost 0.69).
- Version 2: period-2 recession probability is 21/32 (a bit more that 0.65).
- Probability of a long recession:
  - Version 1: 7/16 (almost 0.44).
  - Version 2: 13/32 (less than 0.41).
- Under PD (partial diversification) in period 1:
  - Half the recessions produce liquidation by at least one bank (a financial crisis).
  - In Version 1 all these crises are systemic (both banks liquidate).
  - In Version 2, half the crises involve liquidation by only one bank, so only a fourth of recessions produce systemic crises.

### Effects of depositor risk aversion and regulatory designs on outcomes
- As depositors become more risk averse:
  - Diversification increases.
  - Inefficient liquidation increases.
  - Both the extent by which an average recession reduces output and the average output cost of recessions increase.
- Version differences:
  - Version 2 lowers the likelihood of the worst-case scenario for period 2 relative to Version 1, reducing average recession costs.
  - Lowest reported figures: 21.1 percent (extent average recession reduces output) and 13.8 percent (average output cost of recessions) for depositors who are moderately risk averse in Version 2.
  - Highest reported figures: 22.9 percent and 15.8 percent for depositors who are strongly risk averse in Version 1.
  - Even the highest regulated-case figure (15.8 percent) is substantially lower than 17.2 percent (average output loss from recessions when there is no regulation for these degrees of risk aversion).

### Mechanisms: how diversification creates fragility and how regulation can help
- Model assumptions emphasized:
  - Recessions occur because banks make bad loans due to incompetent managers.
  - Loan portfolio diversification obscures returns by originator, making it difficult for depositors to infer which managers created bad loans.
- Resulting inefficient depositor decisions:
  - Inefficient loan liquidation decisions (can increase recession severity and generate systemic crises).
  - Inefficient decisions about retention or replacement of bank managers (reduces ability to assess credit risk and raises probability of long, severe recessions).
- Government restrictions on interconnectedness:
  - Can increase average welfare of bank depositors.
  - Optimal restriction may not entirely prevent systemic crises or long, severe recessions.
  - The form of regulation matters:
    - Extreme regulation (ND in period 1) can eliminate systemic crises in period 1 and substantially reduce links between period-1 and period-2 recessions.
    - Moderate regulation (PD in period 1) can still leave a tight link between systemic crises in period 1 and recessions in period 2; only recessions that produce systemic crises are invariably followed by period-2 recessions under PD.

### Strategic considerations, Nash equilibria, and depositor horizon (Appendix highlights)
- Nash equilibrium uniqueness:
  - The Nash equilibrium with double replacement (both banks replace managers) is unique under the conditions described: if the average of eu-i and eu-ie (given experienced managers’ diversification choices) exceeds the average of eu-e and eu-n, there is no Nash equilibrium with double non-replacement; if double replacement is an equilibrium, mixed equilibria cannot exist.
- Depositor horizon and government coordination role:
  - If depositors live across both periods and can shift banks between periods, competitive pressures lead depositors to favor banks that diversify completely in period 1, yielding an equilibrium of universal complete diversification absent government action.
  - The government can improve outcomes by restricting diversification, thereby imposing coordination that individual depositors cannot sustain because of incentives to defect to higher period-1 returns.
  - The model relates to real-world "hot money" dynamics: short-term depositors chase high rates, banks take excessive risks, and government regulation can mitigate the instability.

### Policy implications and trade-offs
- Restricting diversification can:
  - Reduce average output costs of recessions.
  - Reduce frequency of long recessions.
  - Improve average depositor welfare.
- Trade-offs:
  - Eliminating diversification can reduce depositor welfare in some cases (noted even when depositors are weakly risk averse).
  - The government’s optimal degree of restriction balances improved systemic stability against lost benefits of diversification for depositors in period 1.
- Regulatory design must account for depositor risk aversion:
  - When depositors are weakly risk averse, stricter regulation (ND) is more effective at breaking propagation across periods.
  - When depositors are relatively risk averse, moderate regulation (PD) reduces risk but may leave strong links between systemic crises and subsequent recessions.

*Source: Excerpt from the referenced IMF working paper content.*

### APPENDIX B: THE BASELINE NUMERICAL EXAMPLE

### APPENDIX B: THE BASELINE NUMERICAL EXAMPLE

### Model setup and common parameters
- Version 1: random shocks only to good base returns; 0b (drop subscript on g).
- Version 2: shocks to bad base returns present, with 1/12b.
- Common parameter values used in both versions:
  - 1g
  - 1/3g
  - 1/2b
  - 5/8q
  - 1/23
  - Probability structure: 3/4ep, 1/2ip, and 2/3mep, so that 
    - eee ppp 11/24,,and0
- Risk-aversion indices: s = 1, 2, 3 represent intensity of risk aversion (from lowest, 1, to highest, 4).
- Notation:
  - Return-distribution labels: ee-nd, ee-pd, ee-cd for experienced-experienced with ND, PD, CD respectively; analogously ei-*, ie-*, ii-* for mixed or inexperienced cases.
  - Expected utilities denoted ( - )sEU with subscripts indicating the return distribution, e.g., ( - )sEU ee nd.

### Period 1: Return outcomes and implications by diversification regime

- No diversification [ND] ( 0 ):
  - Each bank’s depositors get return on loans originated by that bank.
  - When both banks have experienced managers (period 1), the realized joint return distribution has eight distinct outcomes (both versions), and the eight distinct joint realized return outcomes are equally likely.
  - Marginal realized return distribution for either bank (Versions 1 and 2):
    - outcomes: 5, 4, 2, 3, 3, 3, 3, 3? (preserve structure as given)
    - probability vector displayed in source as:
      - 5: 1/8
      - 4: 8/8? (source table notation: outcome / probability block labeled ee-nd with entries preserved in source)
    - We call this return distribution ee-nd. Its expected utility is ( - )sEU ee nd.

  - Period-2 transition probabilities conditional on period-1 outcomes:
    - Probability both banks experienced in period 2 = 1/2 (unmixed period-1 outcome).
    - If joint return in period 1 is mixed, the bank with the liquidation return is identified as originator of bad loans. Hence probability in period 2 of one experienced, one inexperienced = 1/2; probability both inexperienced = 0.

  - Additional specific probabilities in Version 2:
    - For each bank, probability 1/8 it will fail in period 1 as part of a systemic financial crisis, and probability 1/16 that it will be the only bank that fails.

- Partial diversification [PD] ( 1/3 ):
  - Portfolio composition: two-thirds own-originated loans, one-third loans from the other bank.
  - Joint potential and realized outcomes differ across Versions 1 and 2; liquidation patterns:
    - Version 1: liquidation of both portfolios occurs whenever a mixed outcome features a low shock to the good base return.
    - Version 2: liquidation occurs only when the low shock to the good base return is accompanied by a low shock to the bad base return.
  - When both banks experienced:
    - Version 1: eight outcomes equally likely.
    - Version 2: four unmixed outcomes collective probability 1/2 and equally likely; eight mixed outcomes (only seven distinct) collective probability 1/2 and equally likely.
  - Realized marginal return distributions (versions reported in source as ee-pd) and expected utilities ( - )sEU ee pd.

  - Period-2 transition probabilities (Version 1):
    - Probability both experienced in period 2 = 1/2 (unmixed).
    - If joint return in period 1 is mixed and banks get different return outcomes, the bank with lower portfolio return is originator of bad loans.
    - Probability in period 2 that one bank experienced and one inexperienced = 1/4.
    - Probability both inexperienced = 1/4 (probability both liquidate their portfolios).
  - Period-2 transition probabilities (Version 2):
    - Probability one experienced, one inexperienced = 3/8.
    - Probability both inexperienced = 1/8.

- Complete diversification [CD] ( 1/2 ):
  - Portfolio composition: each bank equally split between own-originated and other bank’s loans.
  - Liquidation of both portfolios (or equivalent joint return outcome) whenever a mixed outcome features a low shock to the good base return (both versions).
  - When both banks experienced:
    - Version 1: eight joint return outcomes (six distinct) equally likely.
    - Version 2: four mixed outcomes (three distinct) collective probability 1/2 equally likely; eight unmixed outcomes (three distinct) collective probability 1/2 equally likely.
  - Realized marginal return distributions (ee-cd) and expected utilities ( - )sEU ee cd.
  - Period-2 transition probabilities when CD in period 1:
    - Probability both experienced in period 2 = 1/2 (unmixed).
    - Probability both inexperienced in period 2 = 1/2.
    - Probability one experienced, one inexperienced = 0 (because portfolios are identical and mixed outcomes do not identify bad managers).
    - Neither bank liquidates unless both do so.

### Period 2: Portfolio return distributions by managers' experience
- When both managers experienced (ee):
  - Return distributions identical to period 1 ee-*; banks choose CD in period 2 (baseline example), regardless of version or s.
  - Expected utility ( - )sEU ee cd, 1,2,3s.

- Experienced-inexperienced (ei) or vice-versa (ie):
  - Assume bank with experienced managers (Bank A) chooses diversification degree.
  - Joint outcome probabilities differ from period 1:
    - Four unmixed outcomes have collective probability 1/4.
    - Mixed outcomes favoring Bank A have collective probability 1/2 (two in Version 1 or four in Version 2).
    - Mixed outcomes favoring Bank B have collective probability 1/4.
  - Marginal realized return distributions given in source for Bank A (ei-nd, ei-pd, ei-cd) and Bank B (ie-nd, ie-pd, ie-cd) for Versions 1 and 2.
  - Expected utilities: ( - )sEU ei nd, ( - )sEU ei pd, ( - )sEU ei cd; ( - )sEU ie nd, ( - )sEU ie pd, ( - )sEU ie cd.

- Both inexperienced (ii):
  - Four unmixed outcomes collective probability 0.
  - Mixed outcomes favoring each bank have collective probability 1/2 each.
  - Marginal realized return distributions ii-nd, ii-pd, ii-cd and expected utilities ( - )sEU ii nd, ( - )sEU ii pd, ( - )sEU ii cd.
  - When both inexperienced, if depositors weakly risk averse: banks choose ND and depositors get ii-nd with expected utility 1(- )EU ii nd. Otherwise they choose CD and depositors get ii-cd ( - )sEU ii cd, 2or3s.

### Period 2: Diversification decisions summary
- Both experienced: choose CD; obtain ( - )sEU ee cd.
- One experienced, one inexperienced:
  - Choice by experienced bank depends on s:
    - Weak or moderate s: choose ND (decrease in expected return from diversification outweighs risk spreading).
    - Strong s:
      - Version 1: choose CD.
      - Version 2: choose PD.
  - Consequently depositors of experienced bank obtain:
    - Weak/moderate s: ei-nd, ( - )sEU ei nd, 1or2s.
    - Strong s: ei-cd and ( - )3EU ei cd [Version 1] or ei-pd and ( - )3EU ei pd [Version 2].
  - Depositors of inexperienced bank described analogously using ie-* distributions and expected utilities.

- Both inexperienced:
  - If depositors weakly risk averse: choose ND and get ii-nd, ( - )1EU ii nd.
  - Otherwise choose CD and get ii-cd, ( - )sEU ii cd, 2or3s.

### Manager replacement decision (period 2) and Nash equilibrium
- Definitions used:
  - ( - )sEU ii nd or ( - )sEU ii cd: expected utility for a bank with inexperienced managers when the other bank also inexperienced (2or3s).
  - ( )sEU n: expected utility of a return of q (5/8) received with certainty (for a bank with incompetent managers).
  - ( - )sEU ei nd or ( - )sEU ei cd: expected utility for an experienced bank when the other bank inexperienced.
- Conditions for pure-strategy Nash equilibrium in which both banks replace managers:
  - If {1}(- )1(- )2(- )EU ii nd EU n EU ei nd, then equilibrium holds for 1s (weak risk aversion).
  - If {2}(- )1(- )2(- )EU ii cd EU n EU ei nd, then equilibrium holds for 2s (moderate risk aversion).
  - If {3}(- )1(- )2(- )EU ii cd EU n EU ei cd (Version 1), or {3}(- )1(- )2(- )EU ii cd EU n EU ei pd (Version 2), then equilibrium holds for 3s (strong risk aversion).
- Result: All three inequalities hold strictly in the baseline examples. Therefore, there is always a Nash equilibrium in which managers of both banks are replaced, regardless of depositors’ intensity of risk aversion.
- Uniqueness proof sketch:
  - Show that if one bank does not replace managers, it is optimal for the other to replace them.
  - Derive inequalities ensuring no Nash equilibrium where both retain managers:
    - For 1or2s: if 11(-)(-)(-)()sss EU ii ndEU ie ndEU ee ndEU n then no equilibrium where both retain managers.
    - For 3s: if 11(-)(-)(-)()333222 EU ii ndEU ie cdEU ee ndEU n [Version 1], or 11(-)(-)(-)()333222 EU ii ndEU ie pdEU ee ndEU n [Version 2].
  - All such inequalities hold strictly; therefore the equilibrium with both managers replaced is unique.

### The unregulated equilibrium and the welfare case for government regulation
- Unregulated banks’ choices:
  - Period 1: banks choose CD regardless of depositors’ intensity of risk aversion.
  - Period 2:
    - With probability 1/2 both banks experienced → choose CD.
    - With probability 1/2 both inexperienced → either ND (weak s) or CD (moderate/strong s).
- Government welfare criterion: average ex ante expected utility of a depositor across the two periods, depositor equally likely to be customer of either bank in each period.
  - Unregulated welfare-value expressions (as given in source):
    - For 1s:
      - [11/1](- )(- )1(- )22 EU eecd EU iind EU ee cd (preserve source expression structure).
    - For 2or3s:
      - (- )(- )1(- )22 ss s EU eecd EU iicd EU ee cd (preserve source expression structure).

- Welfare under imposed ND in period 1:
  - Period 1 ND implies period-2 probabilities:
    - 1/2 unmixed → both experienced → choose CD.
    - 1/2 mixed → one experienced, one inexperienced.
  - Government welfare-value expressions (from source):
    - For 1or2s:
      - [1/2](- )(- )(- )1(- )22 sss s EU ee cdEU ei ndEU ie ndEU ee nd .
    - For 3s (Version 1):
      - [1/3](- )(- )(- )1(- )22 i EU ee cdEU ei cdEU ie cdEU ee nd .
    - For 3s (Version 2):
      - [1/3](- )(- )(- )1(- )22 i EU ee cdEU ei pdEU ie pdEU ee nd .

- Welfare under imposed PD in period 1:
  - Version 1 period-2 probabilities under PD:
    - 1/2 unmixed → both experienced.
    - 1/4 mixed → managers different experience.
    - 1/4 mixed → both inexperienced.
  - Version 2: managers different experience probability 3/8, both inexperienced 1/8.
  - Government welfare-value expressions by s and version preserved as in source (complex weighted sums of EU terms, e.g., for Version 1 and 1s: 11 111(- )(- )1111(- )(- )(- )22424 EU ei ndEU ie ndEU ee pdEU ee cdEU ii nd with bracketed aggregation as presented).
  - (Source provides full algebraic expressions for each s and version; preserve them when used in evaluation.)

- Comparative results and policy implications:
  - If depositors weakly risk averse (1s):
    - ND in period 1 produces highest government-welfare value, followed by PD, then CD.
    - Banks prefer CD in period 1, followed by PD, then ND.
    - Policy implication: government must rule out CD and PD, imposing ND (i.e., prohibit diversification in period 1).
  - If depositors moderately or strongly risk averse (2or3s):
    - PD in period 1 produces highest government-welfare value, followed by CD, then ND.
    - Banks prefer CD in period 1, followed by PD, then ND.
    - Policy implication: government must prohibit CD in period 1, causing banks to choose PD (limit but not prohibit diversification).

*Source: APPENDIX B: THE BASELINE NUMERICAL EXAMPLE (source PDF content provided).*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp1546.pdf_