## Appendix A (wp18235) ## Source details **Canonical URL:** [Appendix A (wp18235)](https://www.imf.org/-/media/files/publications/wp/2018/wp18235.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2018/wp18235.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2018/wp18235.pdf.json) --- ### Introduction and context - Monetary and macroprudential policies are often assigned to different authorities, requiring coordination; central banks are usually at least part of macroprudential policy committees, but many macroprudential tools often reside at the bank regulator. - Monetary policy imposes a direct externality on financial stability via its effect on bank risk taking incentives (Maddaloni and Peydró, 2011; Jiménez et al., 2014; Dell’Ariccia, Laeven and Suarez, 2017). - Macroprudential policy affects real economic activity and inflation (Kim and Mehrotra, 2018; Richter, Schularick and Shim, 2018). - The separation of objectives between monetary and macroprudential authorities creates potential coordination problems; the central question addressed is whether introducing a financial-stability weight into the monetary authority’s objective ("leaning") facilitates coordination. ### Modeling approach and assumptions - Framework: a game between an unconstrained authority (monetary authority, MA) and a constrained authority (macroprudential regulator, REG). - Authorities minimize quadratic loss functions using their respective tools; tools impose externalities on the other authority’s main objective. - Constrained macroprudential policy is modeled to capture imprecision: examples include large, discrete adjustments (e.g., LTVs moving from 90% to 80%) arising from calibration difficulty, political resistance, infrequent adjustment, communication challenges, arbitrage, and political economy problems. - When both tools are unconstrained, a single Nash Equilibrium emerges that coincides with the social planner’s optimum. When one tool is constrained, multiple Nash Equilibria always arise. - Key formal elements and parameter roles: - MA minimizes Y (output gap) y; REG minimizes F (financial gap) f. - Tools: monetary rate r (neutral rD0) and macroprudential tool m (neutral mD0). Positive r or m implies tightening. - Shocks: "y and "f represent cyclical shocks. - Cross-impact assumptions: @y/@f < 1 and @f/@y < 1, implying @Y/@"y > @Y/@"f and @F/@"f > @F/@"y. - Dominance of tools: @Y/@r > @Y/@m and @F/@m > @F/@r. - Loss weights: MA: min_r [  Y^2 + (1) F^2 ] with  in (1/2,1); REG: min_m [ ! F^2 + (1!) Y^2 ] with ! in (1/2,1);  = 1 implies MA focuses only on Y;  < 1 implies MA leans. ### Definition of leaning - Leaning is defined as introducing in the unconstrained authority’s loss function a weight on "helping" the constrained authority by jointly targeting the constrained authority’s objective. - Leaning changes the similarity of authorities’ aims but does not imply identical objective weights. ### Main analytical findings - Multiplicity and conflict - With unconstrained tools (r,m ∈ R): there exists a social planner first-best (br, bm) setting Y^2 = 0 and F^2 = 0. Proposition 1: the simultaneous moves game has a unique Nash Equilibrium that attains the social optimum (Y^2 = 0 and F^2 = 0). - With constrained REG (m ∈ Z or m ∈ {m_ , m_C}): Proposition 2: the simultaneous moves game always yields multiple Nash Equilibria. - Multiple Nash Equilibria can include combinations such as tight macroprudential policy paired with lower policy rate and loose macroprudential policy paired with higher policy rate. - Three sufficient conditions for multiplicity to produce conflict (Proposition 3): 1. Linear separability of shocks ("y and "f as intercept shocks). 2. A marginal leaning weight:  = 1  with 0 <  < 1 (MA has a marginal weight on financial stability); and ! = 1 (REG focused on financial gap). 3. Nonlinearity between targets and tools: second derivatives @^2Y/@r^2 and @^2F/@m^2 are nonzero (expected positive in practice). - Under these properties, MA and REG prefer different Nash Equilibria (genuine conflict of preferences). - Coordination cost and hump-shaped relation with leaning - Coordination cost C defined as the minimal transfer required to move authorities from one equilibrium to the other: - C = min { | [  Y^2_1 + (1) F^2_1 ] [  Y^2_2 + (1) F^2_2 ] | ; | [ ! F^2_1 + (1!) Y^2_1 ] [ ! F^2_2 + (1!) Y^2_2 ] | }. - Proposition 4: Coordination cost C is hump-shaped in the extent of leaning (1), with C = 0 at the corners: no leaning ( = 1) and sufficiently similar preferences ( = b ). - Intuition: - Starting at  = 1, introducing small leaning (reducing ) raises coordination cost because MA still prefers the equilibrium with lower output gap while leaning increases output gaps and reduces financial gaps within equilibria. - As  falls further, preferences converge and coordination cost eventually declines to zero at  = b . - Asymmetry remark: results for variation in  do not symmetrically carry over to ! because r and m are asymmetric (r is a precision tool; m is coarse). - Leaning can backfire on social welfare - Pareto criterion: a change in  is Pareto improving if both Y^2 and F^2 fall. - Y^2 is always lowest at  = 1 (inflation targeting) since r is adjusted to attain Y^2 = 0. - Within each Nash Equilibrium, leaning lowers F^2: @F^2_1/@ > 0 and @F^2_2/@ > 0. - However, leaning can increase the difficulty of coordinating between equilibria, raising the probability of landing on the worse equilibrium in terms of F^2. - With a uniform prior over equilibria, expected financial gap if uncoordinated: E[F^2] = (F^2_1 + F^2_2)/2. - Define "backfiring leaning" where E[F^2] increases with more leaning (a reduction in ): condition 1E[F^2]1 < 0 (as stated in text). - Proposition 5: If authorities fail to coordinate when C > 0, then there exists a _0 in (b , 1) such that a change from  = 1 to  ∈ (_0, 1) always Pareto worsens welfare: - 1E[F^2]1 < 0 and 1E[Y^2]1 < 0. - Note: uniform prior is convenient; qualitative results hold for any prior placing positive weight on each equilibrium. ### Numerical example and intuition (Appendix B selected results) - Linear functional example used: - y = "y - 1 r - 2 m + 1 f - f = "f - 1 r - 2 m + 2 y - Parameter constraints: 1; 2; 1; 2 > 0 and 1; 2 in (0,1). - Specific parameterization used in figures: -  = 1, ! = 0:75, "y = -5, "f = 0, 1 = 2, 2 = 0:5, 1 = 0:5, 2 = 2, 1 = 0:5, 2 = 0:5. - Unconstrained Nash Equilibrium: r_NE;m_NE = .-2:67;0:67/. - Constrained REG with m ∈ {0,1}: - REG sets m = 1 if r < r̄ and m = 0 if r > r̄, with r̄ = -2:45 (i.e., m_ = 1 if r < -2:45). - MA’s best responses: r_ = -2:89 if m = 1; r_ = -2:22 if m = 0. - Two Nash Equilibria under constrained REG: {. -2:89;1/ , . -2:22;0/ }. - Interpretation: tight macroprudential policy paired with lower policy rate, or loose macroprudential policy paired with higher policy rate, are both sustainable Nash equilibria. ### Alternative game forms and robustness - Stackelberg (first-mover) game - A fixed first-mover assignment yields coordination on the leader’s preferred equilibrium. - If MA first-mover → MA’s preferred Nash Equilibrium; if REG first-mover → REG’s preferred equilibrium. - Fixed Stackelberg assignment resolves multiplicity but does not ensure welfare dominance because which equilibrium is welfare-superior depends on shocks "y and "f and can vary over time. - Ensuring play toward the welfare-dominant equilibrium would require reassigning first-mover each period—impractical. - Dynamic (repeated) games - Repetition can support cooperative strategies (e.g., Tit-for-Tat) that sometimes avert bad equilibria, but when each authority strictly prefers a different equilibrium, repetition offers limited scope for improvement. - Repetition can produce de facto first-mover advantages if one authority’s tool has larger externalities; threats and punishments can shift outcomes, but welfare implications depend on shocks. - Merging the authorities - A merged authority optimizing welfare replicates the social planner (first-best). - If the merged authority inherits MA-only or REG-only preferences, outcomes parallel Stackelberg cases and welfare depends on shocks. - Unmodeled real-world complications: credibility, reputational effects, agency conflicts may hinder merger benefits. ### Policy implications and recommendations - Reframe evaluation of "leaning against the wind" to account for between-equilibrium coordination frictions in addition to within-equilibrium tradeoffs. - Small or intermediate degrees of leaning may raise coordination costs and produce welfare losses when multiple equilibria are present; a discrete coordination cost can outweigh marginal within-equilibrium benefits. - Enhancing the precision, adjustability, and credibility of macroprudential tools (thereby reducing constraint-induced multiplicity) could mitigate coordination problems. - Institutional arrangements that lower coordination costs or align preferences across authorities are important complements to any leaning strategy. - Policy translation depends on the scale of the hump in coordination cost as a function of leaning: - If the hump lies within 0–1% weight on financial stability in MA’s objective, between-equilibrium effects are negligible relative to within-equilibrium considerations. - If the hump covers a wider range (e.g., 0–20% weight), ignoring between-equilibrium effects could mislead policy. *Source: Appendix A of wp18235* ### Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ### Appendix A ### Introduction and context - Monetary and macroprudential policies are often assigned to different authorities, requiring coordination; central banks are usually at least part of macroprudential policy committees, but many macroprudential tools often reside at the bank regulator. - Monetary policy imposes a direct externality on financial stability via its effect on bank risk taking incentives (Maddaloni and Peydró, 2011; Jiménez et al., 2014; Dell’Ariccia, Laeven and Suarez, 2017). - Macroprudential policy affects real economic activity and inflation (Kim and Mehrotra, 2018; Richter, Schularick and Shim, 2018). - The separation of objectives between monetary and macroprudential authorities creates potential coordination problems; the central question addressed is whether introducing a financial-stability weight into the monetary authority’s objective ("leaning") facilitates coordination. ### Modeling approach and assumptions - Framework: a game between an unconstrained authority (monetary policy) and a constrained authority (macroprudential policy). - Authorities minimize quadratic loss functions using their respective tools; tools impose externalities on the other authority’s main objective. - Constrained macroprudential policy is modeled to capture imprecision: examples include large, discrete adjustments (e.g., LTVs moving from 90% to 80%) arising from calibration difficulty, political resistance, infrequent adjustment, communication challenges, arbitrage, and political economy problems. - When both tools are unconstrained, a single Nash Equilibrium emerges that coincides with the social planner’s optimum. When one tool is constrained, multiple Nash Equilibria always arise. ### Definition of leaning - Leaning is defined as introducing in the unconstrained authority’s loss function a weight on "helping" the constrained authority by jointly targeting the constrained authority’s objective. - Leaning changes the similarity of authorities’ aims but does not imply identical objective weights. ### Main analytical findings - Multiplicity of equilibria generates conflict: - Multiple Nash Equilibria exist (e.g., combinations of relatively loose monetary policy with tight macroprudential policy and vice versa). - Authorities can prefer different equilibria; sufficient conditions for divergent preferences include tool effectiveness depending on distance from neutral policy. - Coordination cost and its relation to leaning: - Coordination cost is defined as the minimum utility transfer that would convince the least resistant authority to coordinate on another equilibrium. - The relation between leaning and coordination cost is hump-shaped: - At corners (no leaning or sufficiently large leaning), one equilibrium Pareto dominates and coordination is costless. - At intermediate degrees of leaning, authorities still prefer different equilibria and the payoff gaps between equilibria are large, raising coordination costs. - Leaning can backfire: - Introducing a leaning weight can make coordination more difficult and can perversely hurt the constrained authority the leaning was intended to help. - With a uniform prior over multiple equilibria, a small leaning weight always backfires, producing a Pareto welfare loss (worse performance on both objectives). - The established tradeoff is incomplete: - Within a single Nash Equilibrium, leaning reduces financial imbalances at the cost of reduced output-gap stabilization (the established tradeoff). - Among multiple equilibria, the ability to coordinate across equilibria matters; leaning can harden the constrained authority’s position and make unfavorable equilibria harder to avoid. ### Relation to existing literature - The paper’s mechanism differs from other backfire results (e.g., Galí (2014) where rate hikes can spur rational bubbles); here backfiring arises from multiplicity of equilibria and coordination frictions. - Calibrated DSGE and empirical studies often assume unique equilibria and focus on within-equilibrium tradeoffs; findings on the benefits of leaning vary (Svensson; IMF (2015) find small benefits; Filardo and Rungcharoenkitkul (2016) find large benefits). - Several DSGE papers yield differing policy implications: some find separation optimal (Collard et al., 2017), others find leaning welfare-improving (Bodenstein, Guerrieri and LaBriola, 2016; Carrillo et al., 2017; Van der Ghote, 2018). - Rules-based versus discretion debate: Laureys and Meeks (2018) show coordinated discretion can outperform rules when coordination succeeds. - First-mover advantage literature yields mixed results: De Paoli and Paustian (2017) favor macroprudential first-mover; Cecchetti and Kohler (2014) sometimes find first-mover inferior to coordination. ### Conceptual example and intuition - A simple 2x2 payoff example illustrates the coordination problem with two Nash equilibria where authorities disagree on which equilibrium is preferable: - Payoffs (Monetary, Macroprudential) - Loose Monetary / Loose Macroprudential: Low, Low - Loose Monetary / Tight Macroprudential: High, Medium - Tight Monetary / Loose Macroprudential: Medium, High - Tight Monetary / Tight Macroprudential: Low, Low - The constrained-tool setup generically produces multiple equilibria because a single constrained tool cannot always be finely calibrated to simultaneously satisfy both authorities’ objectives. ### Implications for policy design - Policymakers should account for between-equilibrium coordination frictions when evaluating leaning as a policy option; focusing solely on within-equilibrium tradeoffs can be misleading. - Small or intermediate degrees of leaning may raise coordination costs and produce welfare losses when multiple equilibria are present. - Enhancing the precision, adjustability, and credibility of macroprudential tools (thereby reducing constraint-induced multiplicity) could mitigate coordination problems. - Institutional arrangements that lower coordination costs or align preferences across authorities are important complements to any leaning strategy. *Source: Appendix A of wp18235* ### Section VII discusses alternative game forms. Section VIII concludes. ### wp18235 - Section VII discusses alternative game forms. Section VIII concludes. ### II. GAME WITH UNCONSTRAINED TOOLS - Actors: MA (monetary authority) and REG (macroprudential regulator). - Objectives: - MA minimizes Y (output gap), denoted y. - REG minimizes F (financial gap), denoted f. - Tools and neutrality: - Monetary policy rate r, neutral rate rD0. - Macroprudential tool m, neutral stance mD0. - Positive (negative) r or m implies tightening (loosening) relative to neutral. - Shocks: "y and "f represent cyclical shocks (values of y and f if rD0 and mD0). - System of relations: - y = "y C ;m;r;f./C! - f = "f C ;m;r;y./C! - Equivalently: - Y = "y C ;"f C ;m;r! - F = "f C ;"y C ;m;r! - Cross-impact assumptions: - @y/@f < 1 and @f/@y < 1, implying @Y/@"y > @Y/@"f and @F/@"f > @F/@"y. - Loss minimization problems: - MA: min_r [  Y^2 + (1) F^2 ] with  in (1/2,1) - REG: min_m [ ! F^2 + (1!) Y^2 ] with ! in (1/2,1) - Dominance of tools: @Y/@r > @Y/@m and @F/@m > @F/@r. -  = 1 implies MA focuses only on Y;  < 1 implies MA leans (weights financial gap). - Benchmark (unconstrained tools: r,m ∈ R): - There exists a social planner first-best (b r, b m) that sets Y^2 = 0 and F^2 = 0. - Proposition 1: The simultaneous moves game has a unique Nash Equilibrium which attains the social optimum (Y^2 = 0 and F^2 = 0). - Intuition: Authorities share the same global loss-minimizing point; dominant-tool structure ensures convergence to the social optimum despite externalities. - Numerical example notes: - Example equilibrium .r;m/ = .2:67;0:67/ (specific-form in Appendix B). - Figures illustrate a single crossing where Y^2 = F^2 = 0 and play converges to the social optimum. ### III. CONSTRAINED TOOLS - Constraint considered: m ∈ Z (macroprudential tool is coarse; integer-valued). - REG’s optimization becomes: min_{m = ...; -2; -1; 0; 1; 2; ...} [ ! F^2 + (1!) Y^2 ]. - Proposition 2: With m ∈ Z, the simultaneous moves game always yields multiple Nash Equilibria. - Numerical-example characterization: - REG’s optimal rule can be threshold-form: set m = 1 when r < 2:45 and m = 0 when r > 2:45 (Appendix B quantification). - Mutual reinforcement: if m = 1 then MA's optimal r < 2:45; if m = 0 then MA's optimal r > 2:45 — both combinations are sustainable Nash Equilibria. ### IV. CONFLICT AMONG THE AUTHORITIES - Multiple equilibria do not automatically imply conflict; conflict arises when authorities strictly prefer different equilibria. - Notation for comparing two equilibria: - Equilibrium 1: (Y_1, F_1) with squared values Y^2_1, F^2_1. - Equilibrium 2: (Y_2, F_2) with squared values Y^2_2, F^2_2. - Potential conflict when: - Y^2_1 > Y^2_2 and F^2_1 < F^2_2, or vice versa. - Need for a more refined constraint than m ∈ Z to analyze symmetric deviations around b m: - Let m_ and m_C be feasible values below and above b m, with m_C b m = b m m_ (symmetric constraint). - REG’s choice set m ∈ { m_, m_C } yields optimization min_{m ∈ {m_, m_C}} [ ! F^2 + (1!) Y^2 ]. - Three sufficient conditions for multiplicity to produce conflict: 1. Linear separability of shocks: "y and "f are intercept shocks. 2. A marginal leaning weight:  = 1  with 0 <  < 1 (MA has a marginal weight on financial stability); and ! = 1 (REG focused on financial gap). 3. Nonlinearity between targets and tools: second derivatives @^2Y/@r^2 and @^2F/@m^2 are nonzero (expected positive in practice due to weakening marginal impacts away from neutral). - Proposition 3: Under these three properties, MA and REG prefer different Nash Equilibria (i.e., genuine conflict of preferences). ### V. COORDINATION COST - Define coordination cost C as the minimal joint transfer required to move authorities from one equilibrium to the other: - C = min { | [  Y^2_1 + (1) F^2_1 ] [  Y^2_2 + (1) F^2_2 ] | ; | [ ! F^2_1 + (1!) Y^2_1 ] [ ! F^2_2 + (1!) Y^2_2 ] | }. - (Exact expression in text: equation (9).) - Impact of leaning (variation in ) on coordination cost: - Proposition 4: Coordination cost C is hump-shaped in the extent of leaning (1), with C = 0 at the corners: no leaning ( = 1) and sufficiently similar preferences ( = b ). - Explanation: - Starting at  = 1, introducing small leaning (reducing ) raises coordination cost because MA still prefers the equilibrium with lower output gap while leaning increases output gaps and reduces financial gaps within equilibria. - As  falls further, preferences converge and coordination cost eventually declines to zero at  = b . - Asymmetry remark: - There is no symmetric qualitative statement for ! because tools r and m are asymmetric (r is a precision tool; m is coarse). Thus results for variation in  do not carry over to !. ### VI. SOCIAL WELFARE IMPLICATIONS - Welfare comparison is done via Pareto improvements: a change in  is Pareto improving if both Y^2 and F^2 fall. - Observations: - Y^2 is always lowest at  = 1 (inflation targeting), since r is adjusted to attain Y^2 = 0. - Therefore welfare comparison centers on F^2: if  < 1 implies higher F^2 than at  = 1, then  < 1 Pareto worsens welfare. - Leaning and social welfare: - Within each Nash Equilibrium, leaning lowers F^2: @F^2_1/@ > 0 and @F^2_2/@ > 0 (leaning reduces F^2 for given r or m^*). - However, leaning can increase the difficulty of coordinating between equilibria, raising the probability of landing on the worse equilibrium (in terms of F^2). - With a uniform prior over equilibria, expected financial gap if uncoordinated: - E[F^2] = (F^2_1 + F^2_2)/2. - Define "backfiring leaning" as the case where E[F^2] increases with more leaning (a reduction in ). - Backfiring leaning condition: 1E[F^2]1 < 0 (as stated in text). - Proposition 5: If authorities fail to coordinate when C > 0, then there exists a _0 in (b , 1) such that a change from  = 1 to  ∈ (_0, 1) always Pareto worsens welfare: - 1E[F^2]1 < 0 and 1E[Y^2]1 < 0. - Note: Uniform prior is a convenient assumption; qualitative results hold for any prior placing positive weight on each equilibrium. *Source: wp18235 - Section VII discusses alternative game forms. Section VIII concludes.* ### Section II showed that under unconstrained tools the two separate authorities could replicate ### wp18235 - Section II showed that under unconstrained tools the two separate authorities could replicate ### Alternative game forms — overview - The paper focuses on one-shot simultaneous-moves games between two authorities (MA and REG) and highlights multiplicity of equilibria under constrained tools. - Extensions considered: Stackelberg (first-mover), dynamic (repeated) games, and merging the authorities. Outcomes are characterized as straightforward variations of the one-shot tradeoffs. ### A. Stackelberg game (first-mover) - A first-mover assignment leads to coordination on the preferred equilibrium of the Stackelberg leader. - If MA is first mover: MA sets r at a value that, anticipating REG’s response, replicates MA’s preferred Nash Equilibrium. - If REG is first mover: REG sets m so that MA’s subsequent response leads to REG’s preferred equilibrium. - Fixed Stackelberg assignment resolves multiplicity of equilibria but does not resolve the coordination problem since: - Either MA or REG’s preferred equilibrium may be socially optimal; which one depends on parameters and shocks. - A general welfare function W(Y2,F2) is introduced without specifying its exact form; it is (decreasing) in output and financial gaps. - From (3) and (4): @Y/@"y > @Y/@"f > 0 and @F/@"f > @F/@"y > 0, so shocks "y and "f affect Y2 and F2 differently. - Implication: which equilibrium, (Y2 1,F2 1) or (Y2 2,F2 2), is better for welfare can vary with shocks "y and "f. - Time variation: the welfare-preferred equilibrium can change over time with economic conditions. - Ensuring play toward the welfare-dominant equilibrium would require an overarching authority reassigning first-mover each period — not realistic. ### B. Dynamic games (repeated play) - Repetition can allow strategies (e.g., Tit-for-Tat) that sometimes avert bad equilibria, but when each authority has a single preferred equilibrium, scope for improvement is limited. - Authorities disagree on which equilibrium is bad; repetition does not change preferences. - One authority may gain an upper hand if its tool has larger externalities (e.g., MA’s tool affects F2 more than REG’s tool affects Y2). - Example strategy: MA threatens outcomes particularly damaging to REG; REG may yield. - Repetition may resolve multiplicity by effectively assigning a Stackelberg-like advantage but does not guarantee welfare improvement because which equilibrium is welfare-superior depends on shocks. ### C. Merging the authorities - Merging MA and REG into a single authority with two tools: - If the merged authority optimizes welfare, it replicates the social planner. - If the merged authority has the preferences of MA or REG only, outcome parallels Stackelberg case; welfare depends on shocks. - Unmodeled real-world considerations: - Credibility and reputational effects: when central bank conducts both policies, a financial crisis attributed to insufficient prudential policy can harm monetary policy credibility. - Agency considerations and conflicts of interest (Eijffinger, 2001) may complicate merger decisions. ### Conclusions — core findings and policy implications - Paper reframes the "leaning against the wind" debate from within-equilibrium cost-benefit to between-equilibrium coordination tradeoffs. - Core model components: quadratic loss functions, externalities of tools, one tool constrained → multiplicity of equilibria. - Adding the empirical notion that tool impacts weaken the farther they are from neutral generates divergent equilibrium preferences between authorities. - The disagreement is captured by an expression measuring the distance between authorities’ preferences over equilibria; comparative statics relate this to the weight the unconstrained authority places on the constrained authority’s objective ("leaning"). - Main quantitative/qualitative result: - The distance between authorities’ preferences over equilibria first rises in leaning and then falls (a hump-shaped relation). - As the unconstrained player becomes more similar to the constrained player, its ability to meet its target is impeded, making it more averse to switching equilibria. - Small positive weights on leaning can produce a discrete coordination cost (creating a strict preference and hence a coordination problem) while only marginally reducing financial instability within equilibria: “A discrete cost always outweighs a marginal benefit at first.” - Policy translation depends on the scale of the hump: - If hump lies within 0-1% weight on financial stability in monetary policy optimization, between-equilibrium effects are negligible compared to within-equilibrium literature. - If hump ranges, e.g., 0-20% weight, ignoring between-equilibrium effects could mislead policy. - Modeling frontiers: - Potential to merge this game-theoretic approach with calibrated macro models (reference to Gertler, Kiyotaki and Prestipino (2017)), though multiplicity here is endogenous to policy rather than agent sunspots. ### Appendix A — key proof insights and propositions (select highlights) - Proposition 1: - Y2 = 0 ^ F2 = 0 is a Nash Equilibrium. - Uniqueness under unconstrained tools follows from inequalities: @Y/@r > @Y/@m and @F/@m > @F/@r, implying monotonic, continuous, unbounded reaction functions and a single crossing. - Proposition 2: - Consider unconstrained social optimum (br,bm) and feasible closest values m_ (below) and m^ (above). - Multiple Nash equilibria can arise when comparative inequalities in Y2 and F2 under these constrained choices hold (formal inequalities preserved in source). - Proposition 3: - For linearly separable shocks: - Y("y,"f;m,r) = 2_y("y,"f) + eY(m,r) - F("y,"f;m,r) = 2_f("y,"f) + eF(m,r) - Y2 = 2 2_y + eY^2 + 22_y eY - F2 = 2 2_f + eF^2 + 22_f eF - At social optimum (br,bm): 2_y + eY = 0 and 2_f + eF = 0, so first derivatives zero, second derivatives > 0, third derivatives ambiguous. - Third derivatives near social optimum: - @^3 Y2/@r^3 D 6 @eY/@r @^2 eY/@r^2 - @^3 F2/@m^3 D 6 @eF/@m @^2 eF/@m^2 - Signs of @^2 eY/@r^2 and @^2 eF/@m^2 determine whether Y2 and F2 tilt toward conflict as in (15) or (16). - Proposition 4: - Combining convexity/concavity of MA’s loss across equilibria yields a hump-shaped or hump-with-plateau relation between coordination cost C and leaning parameter . - Proposition 5: - Starting from C = 0 at  = 1, decreasing  implies C > 0 and a rise in expected F2 from min{F2 1,F2 2} to (F2 1 + F2 2)/2; 0 may equal b or be larger. ### Appendix B — analytical example (selected numeric results) - Linear functional example: - y = "y - 1 r - 2 m + 1 f - f = "f - 1 r - 2 m + 2 y - Parameter constraints: 1; 2; 1; 2 > 0 and 1; 2 in (0,1). - Reformulated Y and F given in (21) and (22). - Nash Equilibrium with unconstrained tools (closed form): - r_NE = ( 2 "f - 2 "y)/( 2(1 - 1/2)) - m_NE = ( 1 "f - 1 "y)/( 1(2 - 2/1)) - Under Proposition 1 parameterization, .br,bm/ = r_NE;m_NE. - Constrained macroprudential tool example (m ∈ {1,0}): - REG sets m = 1 if r < r̄ and m = 0 if r > r̄, with a threshold r̄ determined by parameters. - Specific parameterization used in figures: -  = 1, ! = 0:75, "y = -5, "f = 0, 1 = 2, 2 = 0:5, 1 = 0:5, 2 = 2, 1 = 0:5, 2 = 0:5. - Unconstrained Nash Equilibrium: r_NE;m_NE = .-2:67;0:67/. - REG threshold for m = 1: r̄ = -2:45 (i.e., m_ = 1 if r < -2:45) — equation (29). - MA’s reaction functions under constraints: - r_ = -2:89 if m = 1 - r_ = -2:22 if m = 0 — equation (30). - Two Nash Equilibria under constrained REG: - r_NE;m_NE = {. -2:89;1/ , . -2:22;0/ } — equation (31). - Interpretation: tight macroprudential policy paired with lower policy rate, or loose policy paired with higher policy rate, are both sustainable Nash equilibria. *Italic: Source — wp18235 (pdf section excerpts provided).* ### REFERENCES ### REFERENCES ### Monetary policy, financial stability, and "leaning against the wind" - Adrian, Tobias, and Nellie Liang, 2018, Monetary policy, financial conditions, and financial stability, International Journal of Central Banking 14(1), 73-131. - Agur, Itai, and Maria Demertzis, 2013, "Leaning against the wind" and the timing of monetary policy, Journal of International Money and Finance 35, 179-194. - Agur, Itai and Maria Demertzis, 2018, Will macroprudential policy counteract monetary policy’s effects on financial stability, Bruegel Working Paper 2018/1. - Borio, Claudio, and William White, 2004, Whither monetary policy and financial stability? 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Econometrica, 82(2), 463-505. - Maddaloni, Angela and José-Luis Peydró, 2011, Bank risk-taking, securitization, supervision, and low interest-rates: Evidence from the Euro Area and US lending standards, Review of Financial Studies 24, 2121-65. - Gerdrup, Karsten R., Frank Hansen, Tord Krogh, and Junior Maih, 2017, Leaning against the wind when credit bites back, International Journal of Central Banking 13(3), 287-320. - Heider, Florian, Farzad Saidi, and Glenn Schepens, 2017, Life below zero: Bank lending under negative policy rates, mimeo. - Hong, Gee Hee, and John Kandrac, 2018, Pushed past the limit? How Japanese banks reacted to negative rates. IMF Working Paper (forthcoming). - Kim, Soyoung, and Aaron Mehrotra, 2018, Effects of monetary and macroprudential policies - evidence from inflation targeting economies, Journal of Money, Credit and Banking 50(5), 967-992. - Richter, Björn, Moritz Schularick, and Ilhyock Shim, 2018, The macroeconomic effects of macroprudential policy, BIS Working Paper No. 740. *Source: wp18235 - REFERENCES* --- _Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18235.pdf_