## 2.1 The bank — 2.4 Timing of the game ## Source details **Canonical URL:** [2.1 The bank — 2.4 Timing of the game](https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021001-print-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2021/english/wpiea2021001-print-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2021/english/wpiea2021001-print-pdf.pdf.json) --- ### Overview of actors and incentives - Financial sector represented by a single bank managed by its owners (owners are residual claimants and not part of the public). - Funding: deposits of size 1, entirely costless government-insured deposits (no interest, never called). - Bank choices: - Initial asset risk profile  ∈ [0;1]. - If insolvent, whether to engage in a gamble for resurrection. - Bank owners obtain full upside of positive bank returns and none of the downside when insolvent (limited liability); bank profits do not enter public welfare. - Public bears expected cost of insolvency via lump-sum taxes funding deposit insurance. ### Bank’s risk profile — setup and tradeoffs - Deposit base size: 1. - Net return on bank assets r ∼ Uniform(mean , half-width w) with support [w, +w]. - Example:  = 0:02 and w = 0:05 imply r ranges from 3% to +7%. - Dependence on risk profile : - () = m  (equation (1)) - w() =  (equation (2)) -  ∈ [0;1] (equation (3)) - Parameter restrictions: - > > 0 (4) - + < 1/2 (5) - + > m > 0 (6) - Limited liability implies bank behaves like a call option owner: trades off higher variance (w increases with ) against lower mean ( decreases with ). - Bank expected profit with limited liability: - E[] = E[max{r;0}] =  + w ∫_0^{+w} r f(r) dr (equation (7)) - Uniform density: f(r) = 1/(2 ) (8) - Closed-form and corner solutions: - Lemma 1: bank chooses  = 1 if m < m̄;  = 0 otherwise (equation (9)) - Threshold m̄ = + 2 sqrt( ) (equation (10)) - Consequently model focuses on binary choice between good project ( = 0) and bad project ( = 1); bad project has lower mean and greater exploitation of safety net. ### Gambling for resurrection — mechanism and effect - Gamble represents zombie lending: reinvestment in failed project with negative expected return. - Gamble parameters: - Success probability g ∈ (0; 1/2]; success raises net return by ( + ) and suffices to bring bank back to positive profit; failure reduces net return by ( + ). - Expected return of gamble is negative because g ≤ 1/2. - Bank objective when gamble allowed: - E[] = E[r | r > 0] + E[g (r + + ) | r < 0] (equation (11)) - An insolvent bank always chooses to engage in the gamble. - Availability of gamble increases ex ante incentive to choose bad project: - Lemma 2: With gamble option, bank chooses  = 1 if m < m_b and  = 0 otherwise (equation (12)) - m_b = + 2 sqrt( (1g) + 1/4 ( + )^2 g^2/(1g)) (equation (13)) - m_b > m̄ (gamble raises threshold favoring bad project). ### Interaction with the public and information frictions - Public imperfectly informed; visible variable is realized bank solvency r while bank active. - Public receives a signal with revelation probability  ∈ (0;1): - With probability  the public learns true r while bank remains active. - If bank closed (deposit insurance paid), public always learns r. - Limits:  → 1 implies perfectly informed public;  → 0 implies public in the dark while bank open. - If public learns bank is insolvent, it imposes punitive cost x > 0 on the government (electoral loss, demonstrations). - Expected public cost when no gamble: E[min{r;0}]. - When gamble exists, expected loss to public becomes E[(1g)(r ) | r < 0]. - Public’s reaction function is known and anticipated by bank and government. ### Government: objectives, time inconsistency, and policy tool - Single policymaker represents public interests; government loss function: E[min{Bank returns;0} x] (text indicates E[minfBank returns;0g] x), with x measuring sensitivity to public punishment. - Define x(1) as "the extent to which the government is subject to pressure from a poorly informed public". - Binary regulatory tool l: - l = fAllowg: historical cost accounting (allows gambles). - l = fForbidg: mark-to-market accounting (forbids gambles by insolvent bank). - Mark-to-market forces insolvent (r < 0) bank to recognize losses and shut down (prevents gambles); historical cost can veil insolvency and allow gambles. - Timing and time inconsistency: - Government legislates initially before bank chooses . - With probability  ∈ (0;1) the government has a second decision stage after r is realized; with probability 1 this stage does not exist. -  parameterizes agility/time consistency of legislation: high  = agile legislation but greater time inconsistency;  → 1 makes initial legislation moot;  → 0 initial decision binding. - Once r < 0 realizes, the government has incentive to allow gambles to avoid public imposition of x; ex ante regulation against gambles can be undermined if government can change policy later. - Regulatory tradeoffs: - Forbidding gambles ex ante reduces bank incentive to choose bad project but cannot directly target . - An ideal tool targets initial risk choice  (discussed in Section 5 extension); model focuses on l as imperfect realistic instrument. ### Game timing (seven stages) - Stage 1: Government determines l. - Stage 2: Bank chooses . - Stage 3: r is drawn. - If r > 0: game ends with payoffs r (bank), 0 (public), 0 (government). - If r < 0: game continues. - Stage 4: With probability  this stage occurs and government can again determine l. - Stage 5: If l = fForbidg, bank shuts down, public repaid deposits, government deposit insurance covers shortfall, public imposes x on government. If l = fAllowg, game continues. - Stage 6: With probability  the public learns r < 0 and imposes x on government. - Stage 7: Bank gambles. With probability g gamble succeeds, payoffs are (r + + ) (bank), 0 (public), government payoff either 0 or x depending on Stage 6. If gamble fails, payoffs are 0 (bank), (r ) (public), (r x) (government). ### Solution approach - Solve by backward induction: Stage 4, Stage 2, Stage 1 decision nodes analyzed in sequence. ### Key lemmas and conditions (baseline) - Lemma 3 (government at Stage 4): - Government sets l = fForbidg iff x(1 ) < r + ( + ) (1 g) / g (equation (14)). - Implication: Government forbids gambles when x(1 ) is smaller than a positive threshold. For x ! 0 government always forbids gambles at Stage 4. For  ! 1 government’s incentive to forbear disappears. - Lemma 4 (bank’s Stage 2 decision given Stage 1 policy): - If l = fAllowg at Stage 1, bank sets  = 1 if m < m_A and 0 otherwise (equation (15)). - If l = fForbidg at Stage 1, bank sets  = 1 if m < m_F and 0 otherwise (equation (16)). - Ordering: m_F < m_A and m_F < m < m_A. Expressions for m_A and m_F appear in (31) and (33). - Lemma 5 (government optimal policy at Stage 1): - Government chooses l = fAllowg if m < m_F and x(1 ) > 1/2 m + ( + ) (2 3g) / (2g) (equation (17)); otherwise l = fForbidg. - Equivalent (18): m < min { m_F ; 3 ( + ) + 2  x(1 ) + / g }. - Interpretation: Government allows gambles when political pressure from poorly informed public is large (x(1 ) large) or when m is too low. ### Proposition 1: non-monotonic interaction of time inconsistency  with policy - Statement:  interacts non-monotonically with government’s optimal policy. In (17), both @m_F/@ < 0 and @m_F/@ > 0 are possible. - @m_F/@ expression given in (38) decomposes into: - Term (39): strictly positive — less time consistency reduces tool effect on bank risk-taking. - Term (40): strictly negative — increased time inconsistency raises Stage 1 government’s incentive to tie its own hands. - Numerical examples: - Example 1: = 0:01, = 0:1,g= 0:4,= 0:5,x= 0:1,m= 0:03, at = 0:5 gives @m_F / @ =0:005 < 0. - Example 2: = 0:01, = 0:1,g= 0:4,= 0:75,x= 0:75,m= 0:03, at = 0:5 gives @m_F / @ = 0:047 > 0. - Intuition: -  ! 0 (tool time-consistent) maximizes government influence on bank behavior. - Time inconsistency creates conflict between Stage 1 government and Stage 4 future self; net effect on allowance of gambles depends on parameters. ### Extensions: conditioning policy on r and risk caps - Conditioning policy on bank solvency (l(r)): - Government chooses l(r) with l(r) = fAllowg if r ∈ S and fForbidg otherwise (equation (19)). - Proposition 2: For r < 0, l(r) = fAllowg iff r < x(1 ) ( + ) (1 g) / g; government plays mixed strategy when r ≥ 0 (equation (20)). - Interpretation: Government allows gambles only when bank is sufficiently insolvent; when x(1 ) ! 0 condition never holds and government always forbids gambles. - Targeting bank risk with a risk cap  (risk-weighted capital requirement shorthand): - Government chooses  and l at Stage 1. - Frictions: - Regulatory arbitrage: with probability  ∈ (0,1) regulation binds ( binds ); with probability (1 ) arbitrage resets  to bank choice. - Adjustment cost: if bank chosen  >  and forced down, cost ( ) with  ∈ (0,1). - Lemma 6 (bank Stage 2 policy with risk cap): -  = 1 if m > b_m_A, 0 otherwise if l = fAllowg; -  = 1 if m > b_m_F, 0 otherwise if l = fForbidg (equation (21)). - Ordering: b_m_F < b_m_A, b_m_A < m_A, b_m_F < m_F. Limits: b_m_A ! m_A, b_m_F ! m_F as  ! 0. For  ! 1 bad project never chosen. - Lemma 7 (government Stage 1 with risk cap): - Government optimally sets  = 0, combined with l = fAllowg if m < b_m_F and x(1 ) > 1/2 m + ( + ) (2 3g) / (2g) (equation (22)); otherwise l = fForbidg. - Remark:  ! 1 implies  =  = 0 and the socially optimal outcome with payoffs (m; 0; 0). - Proposition 3: - Proposition 1 remains valid with replacements m_A → b_m_A and m_F → b_m_F. - Government uses tools  and l as complements: with risk cap government forbids gambles using l over a larger range of m than without risk cap. ### Policy implications and trade-offs - Main insight: Interaction of political pressure (public imperfect information) and time consistency of policy tools creates non-monotonic effects on regulatory incentives. - Key trade-offs: - More credible (time-consistent) tools strengthen policymaker ability to constrain bank incentives and can raise incentive to act up front. - Less credible tools increase urgency to be tough initially because policymaker anticipates future forbearance by its future self. - Policy-relevant implications: - Granting financial regulators more operational and financial independence can detach incentives from political cycles and reduce muddling through. - Improving public information or regulatory transparency (reducing x or increasing ) reduces political pressure and attractiveness of forbearance. - Combining direct tools targeting bank risk (risk caps) with regime choice on accounting/forbearance (l) can be complementary and yield closer-to-first-best outcomes, especially when risk cap is effective (high ). - Overarching message: Time consistency is a double-edged sword — more time-consistent policy tools can either increase or decrease the government’s incentive to protect the public, depending on parameter values and informational frictions. *Source: wpiea2021001-print-pdf — sections 2.1 and 2.4.* ### 2.1 The bank ### 2.1 The bank ### Overview of actors and incentives - Financial sector represented by a single bank managed by its owners; owners are residual claimants and are not part of the public. - Funding side is fixed: entirely costless government-insured deposits (no interest, never called). - Bank decisions center on asset-side choices: initial asset risk profile () and, if insolvent, whether to engage in a gamble for resurrection. - Bank profits do not enter public welfare; bank owners obtain the full upside of positive bank returns and none of the downside when insolvent. ### 2.1.1 The bank’s risk profile — setup and tradeoffs - Deposit base size: 1 (the bank invests deposits of size 1). - Net return on bank assets, r, is drawn from a uniform distribution with mean  and width w, defined so distribution ranges from w to +w. - Example:  = 0:02 and w = 0:05 imply r ranges from 3% to +7%. - Both  and w depend on risk profile : - () = m  (equation (1)) - w() =  (equation (2)) -  ∈ [0;1] (equation (3)) - Parameter restrictions: - > > 0 (4) - + < 1/2 (5) - + > m > 0 (6) - Interpretation: as  rises, width increases faster than mean declines ( > ), enabling possible preference for riskier profile; + < 1/2 ensures overall gross return nonnegative including gambles; + > m > 0 ensures fixed part of mean return is positive but not so large that safest profile is always chosen. - Due to limited liability, the risk-neutral bank behaves like a call option owner: accrues upside, bounded downside; trades off higher variance (w increases with ) against lower mean ( decreases with ). - Bank expected profit accounting for limited liability: - E[] = E[max{r;0}] =  + w ∫_0^{+w} r f(r) dr (equation (7)) - Uniform density: f(r) = 1/(2 ) (8) - Lemma 1 (closed-form solution for E[]) implies binary optimal choice: - Bank chooses  = 1 if m < m̄;  = 0 otherwise (equation (9)) - Threshold m̄ = + 2 sqrt( ) (equation (10)) - Economic interpretation: low m (less skin-in-the-game) makes variance valuable; high m reduces incentive to increase variance. - Consequently model focuses on binary choice between good project ( = 0) and bad project ( = 1); bad project is socially detrimental (lower mean, greater exploitation of safety net). ### 2.1.2 Gambling for resurrection — mechanism and effect - Gambling for resurrection modeled as negative expected return activity (zombie lending): reinvestment in failed project. - Gamble success probability g ∈ (0; 1/2]; success boosts net return by ( + ) and always suffices to bring bank back to positive profit; failure (probability 1g) reduces net return by additional ( + ). - Expected return on the gamble is negative because g ≤ 1/2. - Bank’s objective when gamble allowed: - E[] = E[r | r > 0] + E[g (r + + ) | r < 0] (equation (11)) - An insolvent bank always chooses to engage in the gamble (only sees upside). - Availability of gamble makes bank more inclined to choose bad project ex ante; threshold for choosing bad project increases (more values of m lead to bad project). - Lemma 2: With the gamble option (absent regulation), bank chooses  = 1 if m < m_b and  = 0 otherwise (equation (12)), where - m_b = + 2 sqrt( (1g) + 1/4 ( + )^2 g^2/(1g)) (equation (13)) - And m_b > m̄ (i.e., gamble raises threshold favoring bad project). ### Interaction with the public - Public bears cost of bank insolvency via lump-sum taxes funding deposit insurance. - Expected cost to public when no gamble option: E[min{r;0}]. - When gamble option exists, expected loss to public becomes (1g)(r ) when r < 0, so expected loss is E[(1g)(r ) | r < 0]. - Public is imperfectly informed about bank health; only visible (imperfect) variable is realized bank solvency r while bank remains active. - Public receives a signal with revelation probability  ∈ (0;1): with probability  the public learns true r while bank remains active; if bank closed (deposit insurance paid), public always learns r. - Limits:  → 1 implies perfectly informed public;  → 0 implies public completely in the dark while bank open. - Public prior: bank solvent (r ≥ 0) unless signal or closure reveals otherwise. - If public learns bank is insolvent, it imposes a punitive cost x > 0 on the government (electoral loss, mass demonstrations, etc.). - The public’s reaction is modeled as a known reaction function (not explicit optimization); bank and government anticipate this reaction. ### Government: objectives, time inconsistency, and policy tool - Single policymaker (government) represents public interests; does not value bank owners’ profits. - Government loss function: E[min{Bank returns;0} x] (text indicates E[minfBank returns;0g] x), where x measures sensitivity to public punishment. - Define x (1) as "the extent to which the government is subject to pressure from a poorly informed public". - Binary regulatory tool l: legislation about gambles for resurrection. - l = fAllowg represents historical cost accounting (allows gambles). - l = fForbidg represents mark-to-market accounting (forbids gambles by insolvent bank). - Mark-to-market forces insolvent (r < 0) bank to recognize losses and shut down (prevents gambles); historical cost accounting can veil insolvency and allow gambles. - Government legislates initially before bank chooses . With probability  ∈ (0;1) the government has a second decision stage after r is realized (can change policy); with probability 1 this second stage does not exist. - Interpretation:  parameterizes agility/time consistency of legislation. High  = agile legislation but greater time inconsistency;  → 1 makes initial legislation moot (government can change later);  → 0 eliminates time inconsistency (initial decision binding). - Time inconsistency: once r < 0 realizes, government has incentive to allow the bank to gamble to avoid public imposition of x; hence ex ante regulation against gambles is undermined if government can change policy later. - Regulatory policy tradeoffs: - Forbidding gambles ex ante reduces bank incentive to choose bad project but is imperfect because it cannot directly target . - An ideal regulatory tool would target the initial risk choice  so the bank would always choose  = 0; such a tool could overcome the model tradeoffs (discussed in an extension in Section 5). - The model focuses on mark-to-market vs historical cost accounting as an imperfect but realistic regulatory instrument; alternative imperfect tools discussed in Sections 4 and 5. *Source: wpiea2021001-print-pdf - 2.1 The bank.* ### 2.4 Timing of the game ### 2.4 Timing of the game ### Game timing and stages - The game consists of seven stages (Figure 1). Uncertain outcomes are represented by striped boxes. - Formal stages: - Stage 1: The government determines l. - Stage 2: The bank chooses . - Stage 3: r is drawn. If r > 0: the game ends with payoffs r, 0, 0 for, respectively, the bank, the public and the government. If r < 0: the game continues. - Stage 4: With probability  this stage occurs, and the government can once more determine l. - Stage 5: If l = fForbidg, then the bank shuts down, the public is repaid its deposits, with the government’s deposit insurance covering the shortfall, while the public imposes x on the government. If l = fAllowg, then the game continues. - Stage 6: With probability  the public learns that r < 0, and imposes x on the government. - Stage 7: The bank gambles. With probability g the gamble succeeds, then payoffs are (r + + ), 0 for, respectively, the bank and the public, while the government’s payoff is either 0 or x, depending on the outcome of Stage 6. If the gamble fails, then payoffs are 0, (r ), (r x) for, respectively, the bank, the public and the government. ### Solution approach - Decision stages solved by backward induction: Stage 4, Stage 2, and Stage 1 are the decision nodes analyzed in sequence. ### Baseline model: key lemmas and conditions - Lemma 3: If the government gains the opportunity to legislate at Stage 4, its policy is to set l = fForbidg if and only if x(1 ) < r + ( + ) (1 g) / g (equation (14)). - Implication: Government forbids gambles when x(1 ) is smaller than a positive threshold. For x ! 0 the government always forbids gambles at Stage 4. For  ! 1 the government’s incentive to forbear disappears. - Lemma 4: The bank’s Stage 2 decision depends on the government’s Stage 1 decision: - If l = fAllowg at Stage 1, then bank sets  = 1 if m < m_A and 0 otherwise (equation (15)). - If l = fForbidg at Stage 1, then bank sets  = 1 if m < m_F and 0 otherwise (equation (16)). - m_F < m_A and m_F < m < m_A. Expressions for m_A and m_F appear in (31) and (33). - Lemma 5: At Stage 1, the government’s optimal policy is l = fAllowg if m < m_F and x(1 ) > 1/2 m + ( + ) (2 3g) / (2g) (equation (17)); otherwise l = fForbidg. - Equivalent expression (18): m < min { m_F ; 3 ( + ) + 2  x(1 ) + / g }. - Interpretation: Government allows gambles when political pressure from a poorly informed public is large (x(1 ) large) or when m is too low (bank has little skin in the game). Thus worst bank incentives to risk coincide with strongest government incentives to allow gambles. ### Proposition 1: time inconsistency interacts non-monotonically with policy - Statement: The time inconsistency of the government’s tool, , interacts non-monotonically with the government’s optimal policy. In (17), both @m_F/@ < 0 and @m_F/@ > 0 are possible. - The expression for @m_F/@ (shown in (38)) splits into: - Term (39): strictly positive — a less time consistent tool has less effect on the bank’s risk-taking incentives. - Term (40): strictly negative — increased time inconsistency raises the Stage 1 government’s incentive to tie its own hands. - Numerical analysis shows either effect can dominate depending on parameters. - Intuition and implications: - When  ! 0 (tool perfectly time-consistent), government influence on bank behavior is maximized; the government can commit to forbidding or allowing gambles. - Time inconsistency creates a conflict between the Stage 1 government and its future self (Stage 4). The possibility of a more lenient future government can make the Stage 1 government more or less inclined to forbid gambles. - Net result: The range of m for which the government allows gambles can either expand or contract when  increases. More time consistent tools are not universally welfare-improving; time consistency can "backfire" under some parameter values. ### Extension: conditioning policy on bank solvency - Setup: - Government chooses a policy mapping l(r) with l(r) = fAllowg if r ∈ S and fForbidg if r ∉ S (equation (19)). Government maximizes expected payoff over choice of S. - Stage 1 becomes choosing l(r); Stage 4 is cut; stages reindexed accordingly. - To prevent public inference of solvency from observed policy, government can play mixed strategies over ranges of r where indifferent. - Proposition 2: When government conditions legislation on future bank returns, its optimal policy: - If r < 0, set l(r) = fAllowg if r < x(1 ) ( + ) (1 g) / g, and fForbidg otherwise (equation (20)). - Government plays a mixed strategy when r ≥ 0. - Interpretation: - Government allows gambles only when the bank is sufficiently insolvent. - When x(1 ) ! 0 (public perfectly informed  ! 1 or x ! 0), condition in (20) never holds and government always forbids gambles. - Greater insolvency (more negative r) increases the upside up to 0 that partially accrues to the public and government, making forbearance more attractive under political pressure. ### Extension: targeting the bank’s risk profile with a risk cap - New policy instrument: risk cap  (risk-weighted capital requirement shorthand). Timing modified so government chooses  and l at Stage 1. - Two frictions: - Regulatory arbitrage: with probability  ∈ (0,1) the regulation successfully binds bank risk ( binds ); with probability (1 ) arbitrage succeeds and  is reset to bank choice. - Adjustment cost: if bank chosen  >  and subsequently forced down, cost is ( ) with  ∈ (0,1). - Revised timing (Stages 1–8) and payoffs include adjustment costs ( ) where applicable. - Lemma 6: Bank’s Stage 2 optimal policy: -  = 1 if m > b_m_A, 0 otherwise if l = fAllowg; -  = 1 if m > b_m_F, 0 otherwise if l = fForbidg (equation (21)). - Ordering: b_m_F < b_m_A, b_m_A < m_A, b_m_F < m_F. Limits: b_m_A ! m_A, b_m_F ! m_F as  ! 0. For  ! 1 bad project never chosen. - Lemma 7: At Stage 1, government optimally sets  = 0, combined with l = fAllowg if m < b_m_F and x(1 ) > 1/2 m + ( + ) (2 3g) / (2g) (equation (22)); otherwise l = fForbidg. - Remark 1:  ! 1 implies  =  = 0 and the socially optimal outcome with payoffs (m; 0; 0) for, respectively, the bank, the public and the government. - Proposition 3: - Proposition 1 remains valid in this extension: both @b_m_F/@ < 0 and @b_m_F/@ > 0 are possible. - Government uses tools  and l as complements rather than substitutes: with the risk cap the government forbids gambles using l over a larger range of m than without the risk cap. - Intuition: - A more binding risk cap reduces frequency of bank failure and thereby reduces Stage 1 government’s incentive to tolerate future gambles to hide insolvency. Hence government becomes more inclined to forbid gambles (mark-to-market) when it has an effective risk cap. ### Conclusion: policy implications and trade-offs - Main insight: Interaction of political pressure (public’s imperfect information) and time consistency of policy tools creates non-monotonic effects on regulatory incentives. - Key trade-offs: - A more credible tool strengthens the policy maker’s ability to constrain bank incentives and can raise the incentive to act up front. - A less credible tool increases the urgency to be tough initially because the policy maker anticipates future forbearance by its future self. - Policy-relevant implications: - Granting financial regulators more operational and financial independence can detach their incentives from political cycles and reduce muddling through. - Improving public information or regulatory transparency (reducing x or increasing ) reduces political pressure and the attractiveness of forbearance. - Combining direct tools targeting bank risk (risk caps) with regime choice on accounting/forbearance (l) can be complementary and yield closer-to-first-best outcomes, especially when the risk cap is effective (high ). - Overarching message: Time consistency is a double-edged sword — more time-consistent policy tools can either increase or decrease the government’s incentive to protect the public, depending on parameter values and informational frictions. *Source: 2.4 Timing of the game — wpiea2021001-print-pdf* ### References ### wpiea2021001-print-pdf - References ### References - Acharya, V. 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Equation for m: - m = (m+ ) 2 4 (equation (24)), giving solution in (10) and a second excluded solution m = + + 2 p . - Lemma 2 (bank optimization with distributional cutoff): - Bank’s optimization in (25); expected profit in (26): - E[] = m if = 0 - E[] = (m+ ) 2 g(m ) 2 +2g( + )( + m) 4 if = 1 - Optimal solution: = 1 when m < m (m given by (13)), and = 0 when m > m; m > m found numerically from parameter constraints. - Lemma 3 (government decision at Stage 4): - Government expected payoff when l=fAllowg: - (1g) (r( + )x) +g(0x(1) (0)) = (1g) (r )x(1g+g) (equation (27)) - When l=fForbidg government payoff is rx (equation (28)). - Government forbids the gamble when inequality in (29) holds, solved to (14). - Lemma 4 (bank policy thresholds with legislative action l): - Bank solves maximization in (30); integrals given f(r) from (8). - Thresholds: - When l=fAllowg: bank chooses = 1 if m < m_A and = 0 if m > m_A, where - m_A = + q A 2 g 1g (equation (31)) - A expression given in (32). - When l=fForbidg: bank chooses = 1 if m < m_F and = 0 if m > m_F, where - m_F = + q F 2 g 1g (equation (33)) - F expression given in (34). - Ordering: m < m_F < m_A < m. - Lemma 5 (government Stage 1 payoff structure and condition for l=fAllowg dominance): - Payoff matrix depends on m relative to m_A and m_F; condition (35) simplifies to - x(1) > 1 2 m + ( + ) (23g) 2g (equation (36)) - Also written as - m < 3 ( + ) + 2  x(1) + g (equation (37)); combined with m < m_F gives condition in (18). - Proposition 1 (comparative statics of m_F with respect to): - @m_F / @ expression given in (38), decomposed into positive term (39) and negative term (40). - Numerical examples demonstrating @m_F / @ can be negative or positive: - Example 1: = 0:01, = 0:1,g= 0:4,= 0:5,x= 0:1,m= 0:03, at = 0:5 gives @m_F / @ =0:005 < 0. - Example 2: = 0:01, = 0:1,g= 0:4,= 0:75,x= 0:75,m= 0:03, at = 0:5 gives @m_F / @ = 0:047 > 0. - Proposition 2 (government indifference and mixed strategy): - Government expected payoff conditional on r is (27) for r ∈ S and (28) for r ∈ 2S; solving rx < (1g) (r )x(1g+g) gives expression in (20). - Government uses a mixed strategy over policy for r ≥ 0 with weights satisfying E[r | l=fAllowg] ≥ 0 and E[r | l=fForbidg] ≥ 0. - Lemma 6 (effect of arbitrage failure probability  on bank thresholds): - With term (m) added, convexity preserved; bank still chooses between = 0 and 1. - Modified optimization in (41) implies b m_A < m_A and b m_F < m_F. - Limits: - b m_A → m_A and b m_F → m_F as  → 0. - As  → 1, bank always chooses = 0. - Lemma 7 (government choice of risk cap  at Stage 1 under binding conditions): - When  not binding government indifferent; when binding, government chooses  = 0. - For legislative choice l, steps mirror Lemma 5; when bank arbitrage fails and  binds, gambles cannot occur and l is irrelevant. - Proposition 3 (effects of arbitrage constraint on earlier results): - Proposition 1 remains valid with replacements m_A → b m_A and m_F → b m_F. - Since b m_F < m_F, government chooses l=fAllowg for a smaller subset of m > 0 and l=fForbidg for a larger subset compared to baseline (Lemma 7 vs Lemma 5). *Appendix: proofs reference equations and thresholds as presented in the source PDF.* --- _Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021001-print-pdf.pdf_