## wpiea2025166-source-pdf ## Source details **Canonical URL:** [wpiea2025166-source-pdf](https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025166-source-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2025/english/wpiea2025166-source-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2025/english/wpiea2025166-source-pdf.pdf.json) --- ### Purpose and overview - Objective: develop a tractable macro-banking theory that simultaneously and endogenously incorporates banks’ core real-life function — endogenous (inside) money creation by liquidity, maturity and credit transformations — subject to regulatory and demand constraints. - Motivation: fill a gap in the literature where no existing tractable equilibrium macro-banking model addresses all five tightly and nonlinearly interlinked aspects (money creation, solvency, liquidity, maturity risks, regulatory and demand constraints). - Use cases: monetary and macroprudential policy analysis, counterfactual simulations (e.g., introduction of a central bank digital currency (CBDC)), forecasting, and tail-risk identification for central banks. ### Key model features - Endogenous inside-money creation via banks’ balance-sheet expansions (liquidity, maturity, credit transformations). - Simultaneous treatment of: - Solvency risk (capital adequacy). - Liquidity risk (including dependence on central bank collateral eligibility policy). - Maturity transformation. - Regulatory constraints (capital adequacy requirements, liquidity requirements, central bank collateral eligibility policy, reserve requirements). - Demand constraints (macroeconomic conditions and nonfinancial sector reactions to interest rate adjustments). - Nonlinear dynamics capable of generating: - Excessive leverage during loose monetary policy. - Credit crunches or sharp deleveraging even without large shocks if initial conditions are fragile. - Asymmetric speed of impacts from monetary tightening versus easing. - Incorporates interactions between monetary policy, macroprudential policy, and fiscal policy in driving broad money and price-level dynamics. ### Main implications and mechanisms - Liquidity regulation and capital adequacy regulation are tightly interlinked because both influence banks’ ability to create inside money by stretching balance sheets. - Credit-risk channel of monetary policy: when monetary policy is "too loose", banks increase leverage in search of yields, raising fragility (solvency risk). - Fragility and nonlinearities can produce crisis dynamics (credit crunches, deleveraging) without large exogenous shocks if banks’ initial risk perceptions are overly optimistic. - Fiscal expansions via public debt issuance can raise the price level through endogenous money creation: - When the government issues a bond and spends the proceeds, it can contribute to creating broad money, increasing nominal demand beyond the impact implied by the risk-free (policy) rate. - This excess nominal demand persists until either the price level increases sufficiently to absorb the extra public debt, or private credit is crowded out and broad money returns to its initial level. - Monetary policy can defend price stability by tightening sufficiently and raising private lending rates, at the cost of private-sector deleveraging. - The model uncovers macro-banking determinants of the neutral policy rate. ### Policy messages and actionable insights - Monetary and macroprudential tools interact; regulators and central banks should consider joint effects on banks’ inside-money creation incentives. - Central bank collateral eligibility policy materially affects system-wide liquidity risk and hence should be a core part of macroprudential and monetary analysis. - Overly loose monetary policy can induce excessive bank leverage and systemic fragility; policy design should account for credit-risk amplification. - Fiscal expansions financed by public debt can be inflationary through banking-sector money creation; monetary policy can counteract this but may induce private deleveraging. - Narrow banking reforms that shut down private money endogeneity can reduce some risks but may forgo benefits from liquidity transformation; tradeoffs must be explicitly evaluated. ### Modeling setup: financial sector — structure and risks - Banks issue one type of financial liability (liquid, short-term, low risk, medium of exchange) to fund purchase of assets that are illiquid, long-term, and riskier — modeling liquidity, maturity and credit transformation. - Bank assets: 푄푡, 푆푡, 퐿푡푆, 퐿푡𝑅 (liquidity/reserve balances, risk-free government securities, private safe/secured loans, private risky/unsecured loans). - Bank liabilities and capital: 퐷푡, 푅푡, 퐸푡 (retail deposits, wholesale funding / central bank refinancing loans, equity/capital). - Central bank items: 푅𝑡𝑋 and 퐶푀𝑡 (FX reserves/net foreign assets and currency: cash and potentially CBDC). - Key uncertainties driving bank decisions: - [1] collateral-induced wholesale liquidity stress/runs, - [2] panic-induced retail bank runs, - [3] credit risk-induced default (nonperforming risky loans). - Banks maximize risk-adjusted after-tax return on equity (ROE) over loan maturity by choosing balance sheet quantities and interest rates, subject to demand and regulatory constraints. - Without demand/regulatory constraints, bank optimization implies infinite balance-sheet expansion and equity ratio at 0% (infeasible), so the model introduces downward-sloping demand functions and regulatory constraints to obtain finite equilibria. ### Financial-sector mechanisms: runs, collateral, and defaults - Demand constraints: 푆푡 ≤ 푓푆(푖푡𝑆), 퐿푡𝑆 ≤ 푓푆퐿(푖𝑡𝑆퐿), 퐿푡𝑅 ≤ 푓푅퐿(푖푡𝑅퐿), and 푖푡𝐷 ≥ 푓𝐷(푖푡𝑅); CES property: ∂푓𝑥/∂푖𝑡𝑥 = −휀𝑥 푓𝑥/푖𝑡𝑥. - Retail run model: depositors acquire information if cost 훾 < expected benefit; probability that a fear-induced retail bank run does NOT happen: 퐹휎 ≡ 퐹휎̂( γ / (1+푖𝑡𝐶𝐵 − φ) ). CBDC features affect 푖𝑡𝐶𝐵 and φ and hence 퐹휎. - Collateral constraint: central bank collateral base 퐶퐵푡 ≡ (η𝑆 푆푡 + η𝑆퐿 퐿𝑡𝑆 + η𝑅퐿 퐿𝑡𝑅)/(1+푖𝑡𝑅) must satisfy 퐶퐵푡 ≥ 푅𝑡. η’s define collateral-eligibility; government securities often η𝑆 = 1, loans often η𝑆퐿 = η𝑅퐿 = 0. - Probability of NOT having collateral sufficiency issues: 퐹𝑐 defined as 퐹𝑐̂( ... ) (expression in source). If collateral constraint binds → wholesale bank run → spike in wholesale funding costs; penalty parameter 휒 captures lender-of-last-resort or market penalty. - Credit risk: realized default fraction 훿̂ ~ 퐹훿̂. No bank default if 훿̂ 퐿𝑡𝑅 ≤ 퐸𝑡 − e𝑚 RWA𝑡 where RWA𝑡 = ω𝑆 푆𝑡 + ω𝑆퐿 퐿𝑡𝑆 + ω𝑅퐿 퐿𝑡𝑅. Probability of NO bank-default: 퐹훿 ≡ 퐹훿̂( 퐸𝑡 − e𝑚 RWA𝑡 / 퐿𝑡𝑅 ). - Lending-rate decomposition: lending rates compensate for (1) monopolistic markup 휀/(휀−1), (2) expected borrower credit risk δ, (3) retail funding costs, (4) wholesale funding costs under outflows, (5) penalty rates 휒, (6) liquidity/collateral premium, (7) credit-risk premium (marginal impact on default probability). Interlinkages are nonlinear; liquidity and credit premia prominently affect pricing even with risk-neutral banks. ### Banks’ equity dynamics and buffers - Law of motion for equity: - 퐸𝑡+1 = 퐹훿 ( 퐸𝑡 + (ROE𝑡 − Dividend𝑡) 퐸𝑡 + 휖𝑡𝐾 ) + (1− 퐹훿) 퐸𝑡 - 휖𝑡𝐾 is exogenous equity injection/destruction term. - Banks internalize effect of equity ratio on default probability → choose positive voluntary buffers above regulatory minimum e𝑚. Example calibration: voluntary equity ratio ~10% while e𝑚 = 8%. - Surviving banks’ shareholders reallocate equity after period-end so solvent banks start next period with same equity positions — preserves tractability. - Managers assumed not to internalize long-run impact of current interest-rate choices on future capital law-of-motion (short-term managerial focus). ### Modeling setup: nonfinancial sector — money demand linkage - Nonfinancial sector standard DSGE structure, except loans and deposits/money are jump choice variables creating an endogenous feedback with banks. - Households: - Utility: ∑ β^t [ (1−h) (C_t − h C̅_{t−1})^{1−σ}/(1−σ) − N_t^{η+1}/(η+1) ]. - Money balances and payment technology embed quadratic adjustment costs when spending is not backed by beginning-of-period money balances, generating demand for loans. - M_t = L_t^S − E_{t−1} + R_{t−1}^X + S_t; i_{t−1}^M = ĉ_t i_t^{CB} + (1−ĉ_t) i_t^D. - First-order conditions link consumption, money balances, and loan demand (e.g., L_t^S = (φ_{SL} − i_t^{SL} ξ_{SL}) P_t C_t + E_{t−1} − R_{t−1}^X − S_t). - Log-linearized IS contains money-balance term: ĉ_t = h/(1+h) ĉ_{t−1} + 1/(1+h) E_t ĉ_{t+1} − (1−h)/(1+h) σ ( i_t^M − E_t π_{t+1} − ρ − ξ_{SL}/φ_{SL} ( m̂_t − m̂_{t+1} ) ). - Producers, distributors, entrepreneurs: - Producers: Y_t = A_t N_t^{α} K_t^{1−α}; factor FOCs as usual. - Retailers: monopolistic competition and price adjustment costs yield a New-Keynesian Phillips Curve. - Entrepreneurs: investment and unsecured risky loans L_t^R; default variable δ̂_t enters entrepreneurs’ first-order conditions; loan demand L_t^R = ( φ_{RL} − i_t^{RL} − δ̂_t ξ_{RL} ) P_t I_t. - Macro policies: - Monetary policy: Taylor-type rule i_t^P = ρ_i i_{t−1}^P + (1−ρ_i)( r^P + π + η_π ( E_t π_{t+1} − π ) + η_y m ĉ_t ). - Central bank liquidity operations specify collateral and haircuts. - Macroprudential: minimum capital adequacy ratio and risk-weights. - Fiscal policy: budget deficit stabilizes public debt toward an implicit target. ### Calibration — selected parameters (as reported) - Financial-sector distributions: - σ̅ = 5%/4 - c̅ = 25%/4 - δ̅ = 5%/4 - Liquidity risk premium parameters: - γ = 20%/4 - φ = 10%/4 - η_S = 100% - η_SL = 0% - η_RL = 0% - rr = 0% - χ = 10%/4 - Competition elasticities: - ε_D = 5 - ε_S = 10 - ε_SL = 5 - ε_RL = 5 - Capital adequacy parameters: - e_m = 8% - ω_S = 0% - ω_SL = 100% - ω_RL = 100% - d = 15%/4 - ψ = 2% - Steady state macro policy variables: - π = 2%/4 - R_X = 0 - τ = 20% - BD = 0% - S = 5 - Persistence parameters: - ρ_i = 0.9 - ρ_{RX} = 0.9 - ρ_{BD} = 0.7 - ρ_c = 0.7 - ρ_A = 0.9 - ρ_{φ_SL} = 0.7 - ρ_{φ_RL} = 0.7 - Nonfinancial sector parameters: - h = 0.7 - σ = 2 - β = 0.995 - η = 0 - μ = 2 - ξ_W = 20 - ξ_P = 400 - ξ_I = 10 - ξ_SL = 0.5 - ξ_RL = 0.5 - φ_SL = 1 - φ_RL = 1 - A = 1 - α = 0.5 - ε_P = 1.5 - δ_K = 3% - η_π = 1.5 ### Key policy-relevant simulations and channels - Impulse responses to a temporary quarterly policy rate shock of 1 percentage point (annualized 4 pp) show transmission broadly in line with standard models, but with additional credit-risk and liquidity-risk channels. - Credit risk channel and bank risk-taking (monetary easing): - Monetary easing initially compresses interest margins and ROE; banks reduce capital buffers toward e_m. - Bank default probability rises from < 0.5% to about 1.5% temporarily (on impact). - Risky loans increase by about 5 pp; investment increases almost 3% on impact. - As policy rate returns, banks tighten lending more than the policy rate increase → risky loans decline and investment weakens persistently. - Net effect: initial easing leads to a permanent price level increase and permanent inflation of nominal money and credit, with adverse medium-term real effects (real asset prices jump then crash below initial steady state). - Liquidity risk channel (currency demand shock pushing collateral constraint): - Cash-demand shock pushes liquidity demand close to collateral constraint; central bank provides liquidity only against collateral. - Loan rates jump ~8 pp on impact (sudden deleveraging); lending portfolio and broad money fall almost 5% on impact. - Output contracts by about 1% persistently; investment declines >2%. - Banks double capital adequacy buffer; default risk goes to essentially zero; secured loans recover faster than risky loans. - Government securities yields decline by about 5 pp on impact (government securities provide liquidity insurance). - Banks reappraising previously-underpriced credit risk: - Secured lending drops ~8%; risky credit declines 12%; broad money declines 6–7%. - Private-sector lending interest rates jump 10–15 pp (annualized). - Capital adequacy ratio rises from near 8% to over 13% in a couple of years. - Output declines about 0.7%; investment declines about 1.2%. - If money demand and velocity were fixed, output contraction could be 5.8% with ~5 years to recover. - Reserve requirement ratio (RRR) and narrow banking: - Increasing RRR from 0% to 10% → broad money creation declines by 2.5%. - Increasing RRR to 20% (additional 10 pp) → decline in broad money creation exceeds 10%. - More than 10% drop in broad money (essentially permanent) translates into permanent and significant declines in output, consumption and investments; money velocity increases to mute nominal effects. - Narrow banking (RRR = 100%) yields strong deleveraging and permanent decline in output >3%; investments decline >6% permanently; bank default probability drops to 0%. - Fiscal mitigation of narrow-banking costs: - Increasing government debt portfolio offsets narrow-banking deleveraging by creating broad money and relaxing collateral constraints; joint calibration eliminates permanent declines in output, consumption and investments in simulation. - Deposit market competition and neutral policy rate: - Permanent reduction in deposit-market competition raises bank profits/ROE initially; banks lower lending rates and expand balance sheets; central bank tightens policy. - Neutral policy rate shifts permanently (example: neutral rate becomes 3 pp higher to restore banking-sector interest rate levels). - Government debt, crowding out, and price-level links: - Permanent 20% increase in nominal government debt without higher promised primary surpluses: - Broad money remains stable as increased government money is offset by reduced private credit creation (credit-demand channel). - Real economy returns to initial steady state with virtually no price-level impact. - Government securities yields increase permanently by almost 2 pp; private-sector lending rates decline permanently by about 0.5 pp. ### Additional simulation insights (selected) - QE need not change broad money if central bank buys collateral-eligible government securities and banks are risk-neutral (no term premium). QE that buys assets outside collateral base or from nonfinancial sector can affect liquidity risk premia and inside money creation. - FX interventions: a one-off FXI worth 2% of annual GDP can increase output ~0.2% if the central bank buys FX, or reduce output ~1% if it sells FX (channel via liquidity risk premium). - Profit tax, investor required returns, bank overhead costs, competition, and risk-weight changes all have nuanced effects on ROE, capital buffers, risk-taking, lending rates, and the neutral policy rate — summarized in 36 numbered model implications in Appendix 1 (selected highlights included above). ### Conclusions, limitations, and research avenues - Model contributions: - Treats banks as major creators of inside money with solvency, liquidity and maturity risks, and regulatory and demand constraints tightly and nonlinearly linked. - Provides causal answers to policy-relevant questions at the intersection of macroeconomics and financial stability. - Two key financial frictions: central banks’ collateral base constraint (liquidity risk premium) and capital adequacy constraint (credit risk premium). - Limitations: - Rudimentary nonfinancial sector; closed-economy focus; exclusion of FX lending and FX deposits; no sovereign default modeling or term-premium generation in current version. - Potential extensions: - Incorporate open-economy features, FX deposit creation, more detailed lender-of-last-resort and collateral policy interactions, sovereign default risk, and estimation/filtering for data-based policy use. *Source: Appendix 1 – summary of all model implications, and Sections 3–6 of "Inside (the) Money Machine: Modeling Liquidity, Maturity and Credit Transformations."* ### Appendix 1 – summary of all model implications ......................................................................... ### Appendix 1 – summary of all model implications ### Purpose and overview - Objective: develop a tractable macro-banking theory that simultaneously and endogenously incorporates banks’ core real-life function — endogenous (inside) money creation by liquidity, maturity and credit transformations — subject to regulatory and demand constraints. - Motivation: fill a gap in the literature where no existing tractable equilibrium macro-banking model addresses all five tightly and nonlinearly interlinked aspects (money creation, solvency, liquidity, maturity risks, regulatory and demand constraints). - Use cases: monetary and macroprudential policy analysis, counterfactual simulations (e.g., introduction of a central bank digital currency (CBDC)), forecasting, and tail-risk identification for central banks. ### Key model features - Endogenous inside-money creation via banks’ balance-sheet expansions (liquidity, maturity, credit transformations). - Simultaneous treatment of: - Solvency risk (capital adequacy). - Liquidity risk (including dependence on central bank collateral eligibility policy). - Maturity transformation. - Regulatory constraints (capital adequacy requirements, liquidity requirements, central bank collateral eligibility policy, reserve requirements). - Demand constraints (macroeconomic conditions and nonfinancial sector reactions to interest rate adjustments). - Nonlinear dynamics capable of generating: - Excessive leverage during loose monetary policy. - Credit crunches or sharp deleveraging even without large shocks if initial conditions are fragile. - Asymmetric speed of impacts from monetary tightening versus easing. - Incorporates interactions between monetary policy, macroprudential policy, and fiscal policy in driving broad money and price-level dynamics. ### Main implications and mechanisms - Liquidity regulation and capital adequacy regulation are tightly interlinked because both influence banks’ ability to create inside money by stretching balance sheets. - Credit-risk channel of monetary policy: when monetary policy is "too loose", banks increase leverage in search of yields, raising fragility (solvency risk). - Fragility and nonlinearities can produce crisis dynamics (credit crunches, deleveraging) without large exogenous shocks if banks’ initial risk perceptions are overly optimistic. - Fiscal expansions via public debt issuance can raise the price level through endogenous money creation: - When the government issues a bond and spends the proceeds, it can contribute to creating broad money, increasing nominal demand beyond the impact implied by the risk-free (policy) rate. - This excess nominal demand persists until either the price level increases sufficiently to absorb the extra public debt, or private credit is crowded out and broad money returns to its initial level. - Monetary policy retains the ability to defend price stability (in contrast to some readings of the fiscal theory of the price level (FTPL)): - By tightening policy sufficiently and raising private lending rates, monetary policy can moderate extra money creation induced by public debt expansion, albeit potentially causing costly private-sector deleveraging. - The model uncovers macro-banking determinants of the neutral policy rate. ### Policy messages and actionable insights - Monetary and macroprudential tools interact; regulators and central banks should consider joint effects on banks’ inside-money creation incentives. - Central bank collateral eligibility policy materially affects system-wide liquidity risk and hence should be a core part of macroprudential and monetary analysis. - Overly loose monetary policy can induce excessive bank leverage and systemic fragility; policy design should account for credit-risk amplification. - Fiscal expansions financed by public debt can be inflationary through banking-sector money creation; monetary policy can counteract this but may induce private deleveraging. - Narrow banking reforms that shut down private money endogeneity can reduce some risks but may forgo benefits from liquidity transformation; tradeoffs must be explicitly evaluated. ### Literature context and novelty - Existing models (Bernanke et al (1999); Carlstrom and Fuerst (1997); Kiyotaki and Moore (1997); Gertler and Kiyotaki (2010); Gertler and Karadi (2011); Gertler and Kiyotaki (2015); Cúrdia and Woodford (2010); Benes et al (2014); Jakab and Kumhof (2015); Brunnermeier and Sannikov (2016)) capture subsets of banks’ functions but typically: - Treat banks as veils or intermediators of real resources, not as endogenous creators of broad money. - Abstract from one or more of solvency, liquidity, maturity transformations, or central bank operational links. - Omit central bank lender-of-last-resort operations or central bank collateral eligibility’s role for liquidity risk. - This paper departs from many DSGE approaches by starting the banking block from micro-founded money-creation, liquidity and maturity transformation behavior, and embedding regulatory and demand constraints to generate policy-relevant nonlinear dynamics. ### Uses, limitations, and extensions - Usability: model is tractable for policy simulations in normal times and crisis episodes and can highlight overlooked channels and tail risks. - Limitations: current version abstracts from some details (e.g., open-economy considerations); model requires time and scrutiny to prove real-time policy usefulness. - Potential extensions: incorporate open-economy features and further refine lender-of-last-resort and collateral policy interactions. *Source: Appendix 1 – summary of all model implications, from "Inside (the) Money Machine: Modeling Liquidity, Maturity and Credit Transformations."* ### 3. Modeling setup: financial sector ### 3. Modeling setup: financial sector ### Balance sheets and timeline - Banks issue one type of financial liability (liquid, short-term, low risk, medium of exchange) to fund purchase of assets that are illiquid, long-term, and riskier — modeling liquidity, maturity and credit transformation. - Each commercial bank and the central bank have balance sheets; aggregate stock-flow consistent representation (Figure 1 in source). - Bank assets: 푄푡, 푆푡, 퐿푡푆, 퐿푡푅 (liquidity/reserve balances, risk-free government securities, private safe/secured loans, private risky/unsecured loans). - Bank liabilities and capital: 퐷푡, 푅푡, 퐸푡 (retail deposits, wholesale funding / central bank refinancing loans, equity/capital). - Central bank items: 푅푡𝑋 and 퐶푀푡 (FX reserves/net foreign assets and currency: cash and potentially CBDC). - Key uncertainties driving bank decisions: - [1] collateral-induced wholesale liquidity stress/runs (natural/structural demand for central bank money relative to collateral base), - [2] panic-induced retail bank runs (retail depositors’ trust), - [3] credit risk-induced default (nonperforming risky loans). ### Objective function of banks - Banks maximize risk-adjusted after-tax return on equity (ROE) over loan maturity by choosing balance sheet quantities and interest rates, subject to demand and regulatory constraints. - Base optimization (no uncertainty or constraints): - maximize expected after-tax ROE: - objective (symbolic): maximize 피푡 [ ... ] − 휓 퐸푡 (1−휏) subject to balance sheet identity 푄푡+푆푡+퐿푡푆+퐿푡푅 = 퐷푡+푅푡+퐸푡. - Interest rate notation: 푖푡푆, 푖푡푆퐿, 푖푡푅퐿, 푖푡퐷, 푖푡푅 (securities, safe loans, risky loans, deposits, central bank discount/refinancing). - Without uncertainty or demand/regulatory constraints, solution implies infinite balance sheet expansion and equity ratio at 0% (infeasible/implausible). ### Introducing demand constraints to banks - Downward-sloping demand for loans and deposit competition impose limits: - 푆푡 ≤ 푓푆(푖푡푆) - 퐿푡푆 ≤ 푓푆퐿(푖푡푆퐿) - 퐿푡푅 ≤ 푓푅퐿(푖푡푅퐿) - 푖푡퐷 ≥ 푓퐷(푖푡푅) - CES property for credit demand: ∂푓𝑥(푖𝑡𝑥)/∂푖𝑡𝑥 = −휀𝑥 푓𝑥(푖𝑡𝑥) / 푖𝑡𝑥, for 푥 = 푆, 퐿𝑆, 퐿𝑅. - Deposit rate lower bound example: 푖푡퐷 ≥ 휀퐷−1/휀퐷 푖푡푅 (markdown on central bank policy rate). - With a constant fraction 푐̂ of deposits withdrawn post-lending, balance sheet relationships: - 푅푡 − 푄푡 = 푐̂ (푇퐴푡 − 퐸푡) − (1−푐̂) 푅푡𝑋 - 퐷푡 = (1−푐̂) (푇퐴푡 − 퐸푡 + 푅푡𝑋), where 푇퐴푡 = 푆푡 + 퐿푡푆 + 퐿푡푅 - First-order pricing equation (with demand constraints): - 푖푡𝑥 = 휀𝑥/(휀𝑥−1) [ (1−푐̂) 푖푡퐷 + 푐̂ 푖푡푅 ] for 푥 = 푆, 퐿푆, 퐿푅. - Demand constraints yield finite balance sheet sizes; liquidity and credit risk will compress volumes further. ### Introducing exogenous bank runs - Panic-induced retail bank run modeled: depositors acquire information if cost 훾 is less than expected loss from not acquiring information. - Run condition (symbolic): - (1−휎̂𝑡) 퐷𝑡 (1+푖𝑡𝐷) ≤ (1−휎̂𝑡) 퐷𝑡 (1+푖𝑡𝐷) + 휎̂𝑡 퐷𝑡 (1+푖𝑡𝐶𝐵 − φ) − γ 퐷𝑡 - where 푖𝑡𝐶𝐵 interest rate on central bank money (cash/CBDC), φ cost of holding cash/CBDC, γ cost of information acquisition. - Probability that a fear-induced retail bank run does NOT happen: - 퐹휎 ≡ 퐹휎̂( γ / (1+푖𝑡𝐶𝐵 − φ) ) - If central bank passively provides liquidity at policy rate without collateral constraint, a bank run forces banks to switch to more costly wholesale funding; updated pricing: - 푖𝑡𝑥 = 휀𝑥/(휀𝑥−1) [ 퐹휎 (1−푐̂) 푖𝑡𝐷 + (1− 퐹휎 (1−푐̂)) 푖𝑡𝑅 ]. - Effect on loan rates is marginal if 퐹휎 close to one; CBDC features can materially affect 퐹휎 by changing 푖𝑡𝐶𝐵 and φ. ### Introducing endogenous bank runs and collateral constraint - Collateral base for central bank operations depends on haircut- and policy-adjusted sum of collateral-eligible assets: - 퐶퐵푡 ≡ η𝑆 푆푡 + η𝑆퐿 퐿푡𝑆 + η𝑅퐿 퐿푡𝑅 / (1+푖𝑡𝑅) ≥ 푅𝑡 - η’s define collateral-eligibility (one minus haircuts); government securities often have η𝑠 = 1, loans often η𝑆퐿 = η𝑅퐿 = 0. - Using central bank and bank balance sheet identities and assuming currency demand fraction 푐̂ ~ 퐹𝑐̂, sufficient collateral condition leads to: - 푅𝑡 = 푐̂ (푇퐴푡 − 퐸푡 + 푅𝑡𝑋) + 푄푡 − 푅𝑡𝑋 ≤ 퐶퐵푡 → 푐̂ ≤ [ expression in source ]. - Probability of NOT having collateral sufficiency issues (if no panic-induced retail run): - 퐹𝑐 ≡ 퐹𝑐̂( 퐶퐵푡 − 푟𝑟 (푇퐴푡 − 퐸푡) + (1−푟𝑟) 푅𝑡𝑋 / (푇퐴푡 − 퐸푡) − ... ) - Conditional probabilities: - 퐹𝑅|휎 = 1 if 퐶퐵푡 ≥ 푇퐴푡 − 퐸푡, else 퐹𝑐. - 퐹𝑅|1−휎 = 1 if 퐶퐵푡 ≥ 푇퐴푡 − 퐸푡, else 0. - If collateral constraint binds → wholesale bank run → spike in wholesale funding costs (lender of last resort at penalty rate or market penalization). Penalty parameter 휒 captures penalty fraction of 푅𝑡. - Collateral constraint introduces a nonlinear liquidity risk premium into interest rates. ### Introducing bank defaults (credit risk) - Nonfinancial sector default modeled exogenously: realized default fraction 훿̂ ~ 퐹훿̂. - No bank default if: - 훿̂ 퐿푡𝑅 ≤ 퐸𝑡 − e𝑚 RWA𝑡 - where e𝑚 is minimum capital adequacy ratio; RWA𝑡 = ω𝑆 푆𝑡 + ω𝑆퐿 퐿𝑡𝑆 + ω𝑅퐿 퐿𝑡𝑅 (ω’s are risk weights). - Probability of NO bank-default: - 퐹훿 ≡ 퐹훿̂( 퐸𝑡 − e𝑚 RWA𝑡 / 퐿𝑡𝑅 ). - Inclusion of default risk induces banks to hold voluntary equity buffers and incorporate a nonlinear credit risk premium into pricing. ### Putting everything together: profit states and equilibrium interest rates - Profit outcomes depend on combinations of panic-induced retail run (휎), collateral sufficiency (푐), and borrower defaults (훿); profits: - 훱𝑐|휎 = 푖𝑡𝑆 푆𝑡 + 푖𝑡𝑆퐿 퐿𝑡𝑆 + (푖𝑡𝑅퐿 − 훿̂) 퐿𝑡𝑅 − 푖𝑡𝐷 퐷𝑡 − 푖𝑡𝑅 (푅𝑡 − 푄𝑡) − 휓 - 훱1−𝑐|1−휎 = same revenues − (푖𝑡𝑅 + 휒) 푅𝑡 − 휓 - Expected (default-probability adjusted) ROE (no default weighting): - 퐹훿 ROE𝑡 = 퐹훿 [ 퐹휎 퐹𝑐 훱𝑐|휎 + (1− 퐹휎 퐹𝑐) 훱1−𝑐|1−휎 ] / [퐸𝑡 (1−τ)] - Rewriting profits using balance sheet identities yields optimization problem (Equation (22) in source) subject to demand constraints and definitions of CDFs. - Equilibrium interest rate equations (summary form, full expressions in source): - Securities yields: - 푖𝑡𝑆 = 휀𝑆/(휀𝑆−1) ( [퐹휎 퐹𝑐 (1−푐) 푖𝑡𝐷 + (1−퐹휎 퐹𝑐 (1−푐)) 푖𝑡𝑅 + (1−퐹휎 퐹𝑐) 휒 + liquidity/collateral premium + credit-risk term × 푅𝑂𝐸𝑡/(1−τ) ] ) - Safe loans: - 푖𝑡𝑆퐿 analogous (Equation (28) in source). - Risky loans: - 푖𝑡𝑅퐿 = 휀𝑅퐿/(휀𝑅퐿−1) ( δ + [same weighted funding costs and premiums as above] + credit-risk premium term × 푅𝑂𝐸𝑡/(1−τ) ) - Components that lending rates compensate for: 1. monopolistic competition markup (휀/(휀−1)), 2. expected borrower credit risk (δ), 3. retail funding costs when no run (퐹휎 퐹𝑐 (1−푐) 푖𝐷), 4. wholesale funding costs when retail outflow occurs, 5. penalty rates for liquidity crises (휒), 6. liquidity risk (collateral service) premium, 7. credit risk premium (marginal impact on default probability). - Interlinkages are nonlinear; even with risk-neutral banks, liquidity and credit premia prominently affect pricing. ### Law of motion for banks’ equity - Bank equity follows retention-driven law of motion (banks cannot easily issue new equity short-run): - 퐸𝑡+1 = 퐹훿 ( 퐸𝑡 + (ROE𝑡 − Dividend𝑡) 퐸𝑡 + 휖𝑡𝐾 ) + (1− 퐹훿) 퐸𝑡 - where 휖𝑡𝐾 is exogenous equity injection/destruction term (not a bank choice). - Banks internalize effect of equity ratio on default probability → choose positive buffer above regulatory minimum. - Example illustration: bank voluntarily chooses equity ratio ~10% while regulatory minimum e𝑚 = 8% in calibration (figure in source). - Assumption: surviving banks’ shareholders reallocate equity after period-end so solvent banks start next period with same equity positions — restores ex-ante symmetry and tractability. - Banks do not internalize long-run impact of current interest-rate decisions on future capital law-of-motion (short-term managerial focus assumption). *Source: IMF Working Paper — “Inside (the) Money Machine: Modeling Liquidity, Maturity and Credit Transformations” — Section 3: Modeling setup: financial sector (content unit).* ### 4. Modeling setup: nonfinancial sector ### 4. Modeling setup: nonfinancial sector ### Overview - Nonfinancial sector is standard relative to other DSGE models, with standard notation for variables, except for one single part that generates a simple link with the banking sector: demand for loans (money balances for payment purposes) that depends on consumption/investment expenditures and loan interest rates. - This link (highlighted in the source as red in the optimization problems) is minimally necessary to create an endogenous (and nonlinear) feedback loop between the macroeconomy and banks. - The specification makes loans and deposits/money jump variables (choice variables) rather than predetermined ones, enabling endogenous money creation through household and entrepreneur borrowing rather than accumulation of “loanable funds.” ### 4.1. Households - Households maximize lifetime utility with external habit formation and labor disutility: - Utility: sum over t of β^t [ (1−h) (C_t − h C̅_{t−1})^{1−σ}/(1−σ) − N_t^{η+1}/(η+1) ]. - Budget and payment technology: - Period budget constraint includes a quadratic adjustment cost when spending needs are not adequately backed by beginning-of-period money balances, generating demand for loans. - Money balances definitions: - M_t = L_t^S − E_{t−1} + R_{t−1}^X + S_t is the beginning-of-the-period (domestic currency) money balances held by the household. - M_t′ are end-of-the-period money balances. - i_{t−1}^M = ĉ_t i_t^{CB} + (1−ĉ_t) i_t^D is the interest rate on money balances (weighted average of central bank money and bank deposits). - Households’ borrowing from banks is assumed to have no credit risk (superscript S for “safe” or “secured”). - Red (new) part: quadratic adjustment costs represent money as medium of exchange; marginal cost of obtaining money equals a loan interest rate. This makes consumption depend on loan extension and deposit creation. - First-order conditions (selected): - M_t′: 1/(1+i_t^M) = β Λ_{t+1}/Λ_t - C_t: (1−h)(C_t − h C̅_{t−1})^{−σ} ≈ Λ_t P_t (1−ξ_{SL} φ_{SL} (M_t P_t C_t^{−φ_{SL}} ) ) - L_t^S: L_t^S = (φ_{SL} − i_t^{SL} ξ_{SL}) P_t C_t + E_{t−1} − R_{t−1}^X − S_t - W_t: μ N_t^{η} Λ_t W_{t−1}^{−1} ≈ (μ−1) ξ_W (Ω_{W,t} − β Ω_{W,t+1}) - Log-linearized equilibrium conditions (illustrative): - ĉ_t = h/(1+h) ĉ_{t−1} + 1/(1+h) E_t ĉ_{t+1} − (1−h)/(1+h) σ ( i_t^M − E_t π_{t+1} − ρ − ξ_{SL}/φ_{SL} ( m̂_t − m̂_{t+1} ) ) - π_{t}^W = 1/(1+β) π_{t−1}^W + β/(1+β) E_t π_{t+1}^W + 1/((1+β)(μ−1) ξ_W) ( η n̂_t + σ/(1−h) ( ĉ_t − h ĉ_{t−1} ) − r ŵ_t ) - where m̂_t = (M_t/(P_t C_t) − φ_{SL}) is deviation of real money balances from long-run equilibrium (φ_{SL} fraction of C_t). - Interpretation: - Real money balances enter a dynamic IS curve in addition to permanent income and long-term rates. - φ_{SL} can be time-varying, equivalent to time-varying money velocity. ### 4.2. Producers - Competitive producers maximize profit choosing output and inputs subject to production function: - Y_t = A_t N_t^{α} K_t^{1−α} - First-order conditions: - N_t: α P_t^Y Y_t = W_t N_t - K_t: (1−α) P_t^Y Y_t = Q_t K_t - Price marginal cost under competition: - P_t^Y = 1/α^{α} (1−α)^{1−α} A_t W_t^{α} Q_t^{1−α} ### 4.3. Distributors (Retailers) - Retail distributors operate under monopolistic competition, repackaging producer output and generating price stickiness. - Retailer problem includes price inflation adjustment cost Ω_{P,t} = log(P_t/P_{t−1}) − log(P̅_{t−1}/P̅_{t−2}). - Demand and FOC yield a New-Keynesian Phillips Curve: - P_t: ε P_t^Y P_{t−1}^{−1} ≈ (ε−1) ξ_P (Ω_{P,t} − β Ω_{P,t+1}) - Overall demand: Z_t = C_t + I_t + BD_t, and equilibrium requires Z_t = Y_t (BD_t is government budget deficit). ### 4.4. Entrepreneurs - Entrepreneurs invest to accumulate capital rented to producers; profit and budget constraints include quadratic adjustment costs if investment expenditures are not adequately backed by beginning-of-period money balances, generating loan demand sensitive to nominal investment and loan interest rates. - Optimization problem includes investment adjustment costs Ω_{I,t} = log(I_t/I_{t−1}) and a random variable δ̂_t representing share of failed projects (when failure occurs entrepreneurs default). - Loans to entrepreneurs are unsecured and risky (superscript R). - First-order conditions (selected): - K_{t+1}: Θ_t = 1/(1+i_t^M) ( Q_{t+1} + (1−δ_K) Θ_{t+1} ) - I_t: (1−δ̂) Θ_t P_t − 1 + ξ_{RL} φ_{RL} ( L_t^R/(P_t I_t) − φ_{RL} ) ≈ ξ_I (Ω_{I,t} − β Ω_{I,t+1}) - L_t^R: L_t^R = ( φ_{RL} − i_t^{RL} − δ̂_t ξ_{RL} ) P_t I_t ### 4.5. Macro policies - Central bank monetary policy: Taylor-type reaction function reacting to one-period ahead inflation forecast: - i_t^P = ρ_i i_{t−1}^P + (1−ρ_i)( r^P + π + η_π ( E_t π_{t+1} − π ) + η_y m ĉ_t ) - m ĉ_t is marginal cost gap (real economy cyclical position). - Central bank liquidity operations specify types of collateral and haircuts. - Macroprudential policy sets minimum capital adequacy ratio and risk-weights for bank assets. - Fiscal policy determines budget deficit to stabilize public debt toward an implicit target. ### 5. Key policy-relevant simulations (overview) - The model is calibrated for quarterly dynamics and steady states. - Simulations include impulse response analysis to temporary shocks, comparative statics, and transition dynamics to permanent shocks/recalibrations. - Calibration of financial sector partly based on regulatory/policy setups; other parameters based on judgment and empirical literature with robustness checks. ### 5.1. Calibration — selected parameters (as reported) - Fin. sector distributions: - σ̅ = 5%/4 (Mean of the retail bank run probability (exponential) distribution F_{σ̂}) - c̅ = 25%/4 (Mean of the central bank liquidity demand (exponential) distribution F_{ĉ}) - δ̅ = 5%/4 (Mean of nonperforming loans/borrower defaults (exponential) distribution F_{δ̂}) - Liquidity risk premium–related parameters: - γ = 20%/4 (Cost of information acquisition about a bank’s balance sheet health) - φ = 10%/4 (Full cost of handling cash (can be different for CBDC)) - η_S = 100% (Portion of government securities included in the central bank collateral) - η_SL = 0% (Portion of secured retail loans included in the central bank collateral) - η_RL = 0% (Portion of risky retail loans included in the central bank collateral) - rr = 0% (Reserve requirement ratio) - χ = 10%/4 (Penalty rate due to lending of last resort or market cost of liquidity stress) - Competition in financial sector: - ε_D = 5 (Elasticity of substitution in the deposit market) - ε_S = 10 (Elasticity of substitution in the government securities market) - ε_SL = 5 (Elasticity of substitution in the secured retail loans market) - ε_RL = 5 (Elasticity of substitution in the risky retail loans market) - Parameters for optimal capital adequacy: - e_m = 8% (Minimum capital adequacy ratio requirement (in % of RWA)) - ω_S = 0% (Risk weight applied to government securities) - ω_SL = 100% (Risk weight applied to secured retail loans) - ω_RL = 100% (Risk weight applied to risky retail loans) - d = 15%/4 (Steady state dividends per equity (investors’ required ROE in SS)) - ψ = 2% (Overhead costs for banks per equity) - Steady state macro policy variables: - π = 2%/4 (Quarterly inflation target) - R_X = 0 (Steady state level of central banks net FX reserves) - τ = 20% (Corporate profit tax rate applied to banks) - BD = 0% (Steady state budget deficit) - S = 5 (Steady state government securities overall portfolio) - Persistence parameters: - ρ_i = 0.9 (Persistence of policy rate) - ρ_{RX} = 0.9 (Persistence of FX reserves process) - ρ_{BD} = 0.7 (Persistence of budget deficit process) - ρ_c = 0.7 (Persistence of currency demand process) - ρ_A = 0.9 (Persistence of technology process) - ρ_{φ_SL} = 0.7 (Persistence of consumption expenditures’ share to be covered by money) - ρ_{φ_RL} = 0.7 (Persistence of investment expenditures’ share to be covered by money) - Nonfinancial sector parameters: - h = 0.7 (Habit persistence) - σ = 2 (Relative risk aversion of households) - β = 0.995 (Discount factor) - η = 0 (Elasticity of labor supply) - μ = 2 (Elasticity of substitution in the labor market) - ξ_W = 20 (Adjustment cost applied to wage changes) - ξ_P = 400 (Adjustment cost applied to prices changes) - ξ_I = 10 (Adjustment cost applied to investment changes) - ξ_SL = 0.5 (Adjustment cost applied to mismatch between money and consumption) - ξ_RL = 0.5 (Adjustment cost applied to mismatch between money and investment) - φ_SL = 1 (Share of consumption expenditures to be covered by money balances) - φ_RL = 1 (Share of investment expenditures to be covered by money balances) - A = 1 (Steady state of technology process) - α = 0.5 (Share of labor in the production function) - ε_P = 1.5 (Elasticity of substitution in the product market) - δ_K = 3% (Depreciation of physical capital) - η_π = 1.5 (Policy rate’s reaction to inflation) *IMF Working Paper — Inside (the) Money Machine: Modeling Liquidity, Maturity and Credit Transformations — Section 4 (Modeling setup: nonfinancial sector).* ### 0.5 Policy rate’s reaction to real marginal cost gap ### 0.5 Policy rate’s reaction to real marginal cost gap ### 5.2. Impulse response analysis - Impulse responses are computed to a temporary quarterly policy rate shock of the size of 1 percentage point (annualized 4 pp). - Transmission to output and prices is broadly in line with standard models, but the model includes additional channels absent in most standard macro models (credit risk and liquidity risk channels). - Central bank has a variety of instruments in the model with different transmission to the economy. ### Credit risk channel and bank risk-taking - Monetary easing initially compresses interest margins and ROE because loan interest rates decline more than deposit rates. - Banks reduce capital buffers and capital adequacy ratio moves closer to the regulatory minimum. - Bank default probability increases from less than 0.5% to about 1.5% temporarily (on impact). - All loan types accelerate, but risky loans pick up the most (by about 5 pp). - Risky loans finance entrepreneurs, so investment increases the most by almost 3% on impact. - As the policy rate returns to steady state, compressed margins and profitability lead banks to tighten lending more than the policy rate increase, causing risky loans to decline and investment to weaken persistently. - Net effect: initial monetary easing leads to a permanent price level increase (inflating nominal money and credit permanently) without benefit for the real economy; real asset prices jump initially then crash later below their initial steady state and remain persistently lower. - Conclusion: excessive monetary easing can be costly in the medium term due to negative implications for financial stability and credit risk-taking. ### Liquidity risk channel (cash demand shock pushing collateral constraint) - Simulation: nonfinancial sector massively switches to outside money (central-bank-created) from inside money (bank-created), pushing overall liquidity demand close to the central bank’s collateral constraint. - Central bank provides liquidity only against collateral; collateral base is finite and usually much less than deposit portfolio. - When liquidity demand approaches the collateral constraint, banks cannot borrow extra from the central bank for future deposit outflows, precipitating fear of runs and higher liquidity risk premium. - Impact on interest rates and quantities: - Interest rates on both types of loans jump up on impact, by about 8 pp – a sudden deleveraging. - Lending portfolio and broad money creation decline by almost 5% on impact. - Output contracts by about 1%, quite persistently. - Investments decline by more than 2%. - Banks respond by deleveraging and doubling the capital adequacy buffer in this simulation, reducing bank default risk essentially to zero. - Recovery: secured loans recover much faster than risky loans; broad money and lending gradually recover once liquidity buffers are rebuilt. - Government securities yields decline significantly on impact, by about 5 pp, because government securities provide liquidity risk insurance and banks accept low yields to obtain this insurance—resembles GFC dynamics. ### Banks reappraising previously-underpriced credit risk - Simulation: banks initially underestimate standard deviation of borrowers’ default distribution; after period 0 they realize true (larger) standard deviation and take a couple of quarters to adjust lending interest rates. - Results: - Secured lending drops by about 8%. - Risky credit declines by 12%. - Broad money declines by only 6-7% (bank lending to government/securities portfolio is unchanged). - Private sector lending interest rates jump up by 10-15 pp (annualized), with risky loan rates hit particularly hard. - Capital adequacy ratio rises from close to regulatory minimum of 8% initially to over 13% in a couple of years (via temporarily higher retained earnings). - Output declines by about 0.7%. - Investments (financed by risky loans) decline by 1.2%. - If money demand and velocity are held stable, the simulated output contraction would be 5.8% with recovery taking about 5 years (illustrating sensitivity to money demand/velocity assumptions). - During crises money demand may increase, worsening output decline; in baseline the private sector economizes on money balances (increases velocity), muting output effects and leaving policy rate response marginal. ### 5.3. Transition dynamics: reserve requirement ratio (RRR) and narrow banking - Permanent increase in RRR from current calibration of 0% to higher values has real effects in this model because central bank liquidity provision is collateral-constrained. - Simulation outcomes: - Increasing RRR from 0% to 10%: broad money creation declines by 2.5%. - Increasing RRR by another 10 pp (total 20%): decline in broad money creation exceeds 10% (more than quadruples). - More than 10% drop in broad money (essentially permanent) translates into permanent and significant declines in output, consumption and investments. - Money velocity increases as agents economize on money balances; without the velocity increase nominal output decline would be close to 10%. - Switching to narrow banking (RRR = 100%) for given size of government securities portfolio leads to strong deleveraging and a permanent decline in output by more than 3%; investments decline permanently by more than 6%. - Mechanism: banks constrained on liquidity transformation switch to equity financing, lowering solvency risk (bank default probability drops to 0%) but reducing risky lending and investment. ### Fiscal mitigation of narrow banking costs - Fiscal policy can offset narrow-banking costs by increasing the government debt portfolio (deficit spending). - Larger government securities holdings: - Directly create broad money and relax collateral constraints on central bank operations. - In the simulation, coupling higher RRR with a permanent increase in government debt eliminates permanent declines in output, consumption and investments and may boost short-term output. - Political economy and distributional questions remain: who receives newly created money matters for allocation efficiency; perceived sovereign risk from large fiscal expansion is not modeled. ### Deposit market competition and neutral policy rate - Permanent reduction in degree of competition in the deposit market increases bank profits/ROE for several years due to decline in deposit interest rates. - Countervailing forces: - Banks reduce lending interest rates to pass part of profit to borrowers, expanding balance sheets. - Central bank tightens policy rate in response to moderate boom, raising interest rates over time and generating some bank losses from maturity/interest-rate risk. - Long-run outcome: most variables return close to initial steady states, but the neutral policy rate shifts permanently. - In the example, neutral policy rate becomes 3 pp higher (permanent shift) to restore initial banking-sector interest rate levels. ### Government debt, crowding out, and price level links - Model unifies money-centered and fiscal-theory views: price level depends on monetary policy-induced money creation (private credit) and fiscal policy-induced money creation (government debt). - Broad money tracks consolidated government “overall debt” (conventional government debt + outside money + privately created inside money less bank equity). - Simulation: permanent 20% increase in nominal government debt without higher promised primary surpluses: - Broad money remains stable as increased government-created money is counterbalanced by reduced private-sector credit creation (credit demand channel). - Real economy quickly returns to initial steady state with virtually no impact on price level. - Permanent adjustments occur in banking-sector interest rates: - Government securities yields increase permanently by almost 2 pp. - Private-sector lending interest rates decline permanently by about 0.5 pp (secured and risky loans decline similarly because changes originate from liquidity risk, not credit risk). - Interpretation: crowding out here is a credit demand story—private agents reduce borrowing because they receive more money from government debt expansion—leading to “immaculate deleveraging” when deficit spending goes directly to previous borrowers. If deficit spending goes to wealthy non-borrowers, deleveraging is uglier and would require stronger central bank actions. *Source: author’s construction* ### 6. Conclusions ### 6. Conclusions ### Model features and contributions - Developed a tractable macro-banking model that treats banks as the major creators of (inside) money. - Banks in the model face solvency, liquidity and maturity risks, and are subject to regulatory and demand constraints (i.e. incremental profit prospects). - The five aspects (inside money creation; solvency, liquidity and maturity risks; regulatory and demand constraints) are tightly and nonlinearly interlinked. - The model provides causal answers to many highly policy-relevant questions at the intersection of macroeconomics and financial stability. - Simulations demonstrate broad applicability of the model and its usefulness for understanding both normal as well as stress episodes. ### Key findings and policy-relevant frictions - Many of the policy messages coming out of the model have usually been overlooked. - Two financial frictions were key: - central banks’ collateral base constraint (generating a liquidity risk premium) - capital adequacy constraint (generating a credit risk premium) ### Limitations and avenues for future research - An important drawback relative to the literature is the model’s rudimentary treatment of the nonfinancial sector; a more realistic modeling of the nonfinancial sector can reveal even more intricacies relevant for policymaking. - The closed economy nature of the model limits the analysis of open economy–related questions. - The banking sector abstraction excludes FX lending and the creation of FX deposits, an issue particularly important for emerging and developing economies. - The creation of FX deposits by domestic banks, which are not covered by a central bank of that currency, means potentially more liquidity and (currency induced) credit risks, which are worth studying separately. - Introducing potential sovereign defaults, generating term premium, as well as stickiness in interest rate settings may also be interesting to study within this model. - Bringing the model to data for estimation/filtering purposes is an avenue for future research. *Source: wpiea2025166-source-pdf - 6. Conclusions* ### Appendix 1 – summary of all model implications ### Appendix 1 – summary of all model implications ### Model implications (numbered summaries) - [1] Credit risk channel of monetary policy: overly loose monetary policy can lead to banks taking on more credit and, hence, bank default risk (e.g. see Borio and Zhu, 2018 for empirical evidence). This works via excessively low rates compressing banks’ profitability and increasing search for yield. As a result, bank leverage increases and capital adequacy ratio declines. Therefore, it’s risky lending and, hence, risky borrowers’ (e.g. entrepreneurs) activity that experience the most pronounced boom-bust cycles. - [2] Strength of monetary tightening VS easing: in line with empirical evidence (e.g. Alessandri et al, 2025), a very strong monetary tightening is visibly more impactful on output relative to a similarly strong monetary easing. The reason is the asymmetric response of credit spreads to monetary tightening versus easing. When policy is tightened unexpectedly, this leads to banks seeing capital losses, to which they react more (in order to avoid breaching the capital requirement). On the other hand, capital gains due to an unexpected monetary easing do not lead to such abrupt reaction from banks. - [3] Demand for currency and liquidity risk premium: when demand for currency increases substantially, this can lead to banks tightening their lending conditions excessively. The reason is the collateral constraint on central bank’s liquidity provision. Namely, since only a portion of banks’ assets (usually government securities) are part of the central bank’s collateral base, if currency demand is big enough to lead to deposit outflows in value more than those particular assets, banks will find it difficult to borrow the needed liquidity from the standard central bank discount window (and, hence, on the market as well). To avoid the liquidity stress, banks increase lending interest rates to deleverage.58 This, in turn, reduces deposit (money) creation and, hence, dependence on runnable liabilities. - [4] Link between credit risk and liquidity risk: for banks, there is a certain degree of equivalence between accepting a credit risk and accepting a liquidity risk. For example, when banks want to reduce liquidity risks, they start deleveraging with the intent to build up capital position, reducing the bank default risk as well. The latter helps since the more capital they hold the less is dependence on run-prone information-insensitive liabilities like deposits. This, in turn, partially mitigates the liquidity risk. In other words, banks’ decisions about the size of the balance sheet usually affect credit and liquidity risks in the same direction, meaning that if an exogenous driver pushes only one of those risks up, banks will respond by reducing both. Hence, they will end up operating with still slightly more of one risk and slightly less of the other risk. Put differently, banks may tolerate higher credit risks if liquidity risks are kept low by something else (e.g. implicit guarantees from the central bank), or vice versa – a certain kind of within-sector regulatory arbitrage if credit and liquidity risks aren’t both properly and simultaneously regulated. - [5] Liquidity risk and collateral service: liquidity risk premium is usually a flip side of collateral service premium. Namely, while banks charge higher interest rates on those types of lending that create runnable deposits but are not part of the central bank’s collateral base, the same banks will charge lower interest rates on those types of lending that also create runnable deposits but are part of the central bank’s collateral base. The first type of lending, usually to the private sector, generates liquidity risk and, hence, incorporates liquidity risk premium (upward pressure on rates). The second type of lending, usually to governments, insures liquidity risk and, hence, incorporates collateral service premium (downward pressure on rates). Indeed, during the GFC risky rates jumped up, while risk-free ones dropped. - [6] Underpricing of risks and subsequent credit crunch: when banks underprice the credit risk of their borrowers, this leads to a gradual erosion of bank capital. With little capital buffers to begin with, a later reappraisal of the credit risk becomes particularly impactful. Namely, to quickly rebuild the capital buffers and avoid the solvency crisis, the fastest way would be an immediate and sharp deleveraging (credit crunch). And unlike the liquidity stress, when secured as well as risky loan interest rates increase similarly (since they both similarly lead to more deposit creation and, hence, liquidity risk), during the solvency stress, risky loan interest rates increase particularly more. The aim is to cut back on the type of lending that was eating up capital buffers the most. - [7] Reserve requirements and money supply: it is already a widely held view that, under interest rate targeting framework, the level of reserve requirements doesn’t matter for money supply. Which is true in most cases. Yet our model shows that it does matter in some (potentially rare) cases. The issue is that central banks’ standard liquidity injection operations require eligible assets (for them to either buy or take as collateral). When liquidity demand, which is partly determined by the level of reserve requirements, becomes so high that these eligible assets start becoming very scarce, then these requirements will start having real effects, even under interest rate targeting frameworks. These real effects come from banks deleveraging in response to them realizing that liquidity risks are now too high. - [8] Narrow banking and credit crunch: whenever the size of a risk-free asset portfolio (usually, government securities) in the financial system is much less than the value of broad money, moving to a narrow banking approach may result in a sharp deleveraging. Narrow banking essentially means an end to banks’ liquidity transformation (creation) business. As a result, banks will gradually59 switch to equity financing, meaning no more inside money creation, which the economy depended on for transaction purposes. Since the central bank alone cannot usually create unlimited amounts of liquidity, unless it’s backed by risk-free assets (which are scarce), the overall broad money declines as well, permanently. Hence, the economy gets a hit permanently as well (output and/or prices). - [9] Fiscal expansion and costs of narrow banking: fiscal policy can mitigate the above-mentioned cost (credit crunch) of switching to a narrow banking approach, by significantly increasing the size of government debt portfolio. This would lead to pushing broad money back up again (counterbalancing the previously mentioned downward pressure on it). Banks holding more government securities on their balance sheets would also relax their collateral constraint on central bank operations. As a result, overall money creation doesn’t get hit and the real economic effects become minimal as well. On the positive side, as a byproduct, banks’ solvency risk goes to essentially zero. But what about political economy costs? The need to decide who receives the newly created money from such a huge deficit-spending (this is equivalent to centrally deciding on credit allocation)? Also, what if perceived probability of sovereign default increases in response to such a huge fiscal expansion? These questions are beyond this particular model. - [10] Deposit market competition and bank credit: increasing the share of unsophisticated depositors leads to less deposit market competition and, hence, lower deposit rates. This causes banks profitability to go up, but not very much due to two reasons: (i) in response, lending interest rates also decline (so borrowers also profit, not just banks, to increase loan demand); and (ii) the central bank hikes the policy rate (in response to accelerating credit and, eventually, inflation), resulting in negative revaluation effects due to interest rate (maturity) risk that banks have on their balance sheet. - [11] Macro-banking determinants of a neutral policy rate: the model clearly shows that any macro-banking adjustment that leads to permanent changes in deposit or lending interest rates (other than policy rate movements themselves), will lead to adjustments in a neutral level of the monetary policy rate. The reason is that its these deposit/lending interest rates that pin down the real economy variables, like consumption and investment. If their spread below/over the policy rate changes permanently, then the policy rate will have to be adjusted permanently as well, so that those banking sector interest rates go back to their steady states, equilibrating the real economy. - [12] Bank credit and the fiscal theory of the price level: our model provides a unifying framework where it incorporates a certain bridge between the money-centered view of the price level (emphasizing monetary policy) and the fiscal theory of the price level (emphasizing government’s expected future primary surpluses). Namely, price level in our model depends on both monetary policy-induced creation of money (via private bank credit) as well as fiscal policy-induced creation of money (via government debt). This means that, when the government increases its nominal debt, without promising to increase primary surpluses in the future, this may not necessarily lead to a permanent increase in the price level. Instead, more money creation via government debt may be counterbalanced by less money creation via private sector credit. In FTPL terms, the key is that deposit-money (created by private bank credit) is a kind of a contingent liability for governments: they (through a central bank) promise to ensure one-to-one convertibility from banks’ liability (deposits, or inside money) into consolidated government’s liability (cash/CBDC), as if it is outside money. - [13] Public debt and crowding out: crowding out literature usually emphasize that, by public debt expansion, governments are using up scarce loanable funds, which forces banks to supply less credit to the private sector. However, this is not what happens in our model. Instead, crowding out may happen because the government’s deficit spending leads to more money creation going in the private nonfinancial sector hands, meaning that the latter economic agents no longer need to borrow from banks as much (which is quite costly) in order to obtain money balances for their transactions. Hence, crowding out is not a credit supply effect, but a credit demand effect materializing. This is important to understand the impact on lending interest rates. - [14] Public debt and crowding in: on the other hand, public debt can even support crowding in, if the above-mentioned (crowding out) effect will be absent (which, of course, usually isn’t). The reason for this (potential) crowding in from government debt accumulation is that it leads to lower liquidity risk for banks. This means that government securities yields increase (since collateral service becomes marginally less valuable), while private sector lending interest rates decline (since the liquidity risk they generate is now marginally less scary). Lower lending rates may incentivize private credit. - [15] Immaculate deleveraging, or not: public debt-induced deleveraging (crowding out), in our model, happens with little cost to the real economy. But this is only due to our (strong) nonfinancial sector modeling assumptions (homogeneity of households and entrepreneurs). When the government spends the newly created money in our model, it goes directly to those households and entrepreneurs that were borrowing from the banks (at some interest cost). Hence, when these economic agents get (part of) the money they need for their transactions, they simply reduce borrowing from the banks, leading to an “immaculate deleveraging”. But if deficit spending goes to economic agents that are wealthy and not bank borrowers themselves, then the deleveraging would be much uglier – it would need to happen via strong deliberate central bank actions, increasing interest rates to engineer a sharp private sector deleveraging of other economic agents. - [16] QE and its impact on broad money: in the model, even if the central bank buys government securities and increases the liquidity supply via QE, this doesn’t necessarily impact broad money itself. The reason is that in our model, as banks are assumed to be risk-neutral, we do not have a term premium (even though we do have an interest rate risk). But without an adjustment to the term premium, there is no impact on bank credit, meaning that broad money doesn’t change either.60 In other words, this emphasizes the endogeneity of money creation via the banking system, instead of looking just at the supply of reserve balances by the central bank. On the other hand, if QE involves buying assets outside of central bank’s collateral base (especially if it’s bought from the nonfinancial sector), then this particular type of QE does affect liquidity risk premium, banking sector interest rates and inside money creation (broad money). - [17] Credit booms and their predictive power: after the GFC, it became already widely acknowledged that credit booms predict subsequent financial crises (e.g. Jordà et al, 2010). The model can endogenously understand this empirical regularity, as mentioned above. However, the model also incorporates an asymmetry in a credit boom-bust cycle. Namely, when banks start underpricing the credit risk, the credit boom that follows is relatively moderate, however once they start to (properly) re-appraise the credit risk then the credit crunch is disproportionately larger. The reason is the nonlinearity on the credit transformation side: when banks become overly optimistic they marginally reduce lending rates, not to hurt profitability too much, which may accelerate demand for credit only moderately; however, when they realize that capital buffers have declined too much (as ex-post credit risk continuously surprised on the upside) then they sharply increase interest rates to quickly rebuild their capital position via a sharp deleveraging.61 - [18] Credit market competition and bank risk-taking: when competition among banks (in credit extension) accelerates permanently, a trade-off emerges. The result of more intense competition is a permanent expansion of credit volumes (especially risky credit) and real economic activity. Borrowers enjoy lower lending interest rates. However, the downside of this is that banks respond to this heightened competition by increasing their risk-taking behavior. As a result, they lower capital buffers and get closer to the regulatory minimum. This increases the chances of bank defaults down the road, potentially significantly. Hence the trade-off, which the model simulations directly show. - [19] Bank opacity and money creation: inside money trades at par with outside money. However, the former is created by banks and for it to be widely accepted its backing should never be questioned. Otherwise, there would be bank runs. That is why bank assets are optimally opaque (Gorton, 2013). Indeed, if we reduce the cost of banks’ information acquisition in our model, it predicts deleveraging, as banks start building up capital position to reduce dependence on runnable deposits. In other words, they will incorporate higher liquidity risk premium into lending interest rates. In addition, it will increase the banks’ weighted average cost of funds, since their expected dependence on more costly wholesale funding increases. The result is less money creation (deleveraging). - [20] Borrower default risk and the macroeconomy: borrower default risk is a powerful driver for the model economy. When it worsens permanently, along with banks’ perceptions about it, the risky bank lending sees a sharp permanent deleveraging, without a commensurate increase in other types of lending. The reason is the banks’ heightened desire to (voluntarily) accumulate more capital buffers, above the regulatory minimum. Hence, overall money creation declines, leading to a permanent drop in output and prices. Due to persistent disinflationary pressures, the policy rate also declines very persistently. Observationally, this is equivalent to saying that a worsening in the borrower default risk leads to a decline in neutral policy rate, making the probability of hitting the effective lower bound on interest rates, which this model does not cover, higher. - [21] Asymmetric effects of capital requirements: when a prudential authority changes its capital adequacy requirements, the effect on bank balance sheets within the model is very asymmetric. If the requirement is relaxed, the effect on credit and money creation is very gradual and initially small. On the other hand, if the requirement is tightened, the effect on credit and money creation is abrupt and quite large. That’s why tightening of these requirements are usually introduced very gradually and communicated early, so that banks have a chance to accumulate capital over time, without trying to generate an increase in a capital adequacy ratio via the denominator (i.e. deleveraging). - [22] Collateral base and its impact on credit allocation: it is frequently overlooked how the central bank’s collateral framework can have a significant impact on the private credit allocation. And it is even less well understood that this impact on credit allocation is quite nuanced, as captured by our model. Namely, when the central bank includes a new asset into its collateral base, it incentivizes banks to reallocate where they divert credit, even when nothing else changes. However, this reallocation depends on how much liquidity buffers change. If the central bank introducing a new asset into its collateral base increased liquidity buffers (i.e. the overall size of the collateral base) so much that liquidity risk premium declined essentially to zero, then the central bank has just incentivized banks to increase their exposure to those assets that generate liquidity risk, including to those assets that are still not part of the collateral base (which may initially have sounded counterintuitive). On the other hand, if the collateral base still remained small and liquidity risk premium still sizeable, then the central bank has just incentivized banks to increase their exposure to that particular asset that was recently included in the collateral base. - [23] Risk weights and its impact on credit allocation: within the capital adequacy regulation, and unlike collateral base changes, the impact that a risk weight change has on credit allocation has received more attention. However, similar to collateral base changes, its nuanced nature has been less well acknowledged. According to our model, if the change in the risk weight is such that it increases/frees up overall capital buffers (e.g. secured loans now receiving much lower risk weight, hence, instantaneously increasing equity/risk-weighted asset ratio), the impact on credit allocation would be to incentivize banks to reallocate credit more generally to those borrowers that generate credit risk. Sounds counterintuitive, but a closer look shows the underlying intuition. Indeed, if banks are now less worried about violating the capital adequacy regulation (after equity ratio suddenly jumping up), then they will exploit this by taking on more credit risk, instead of reducing it by switching to safer loans (i.e. defeating the purpose of the initial risk weight change). On the other hand, if the risk weight changes are such that the overall capital buffers do not change (e.g. secured loans now receiving lower risk weight, but risky loans now receiving higher risk weight), then the central bank has just incentivized banks to reallocate credit from risky borrowers to safer borrowers, as intended. - [24] Access to wholesale funding and credit growth: as would be expected, if the cost of accessing the wholesale funding market increases, then overall credit extension and money creation decline, even if there are no retail deposit outflows. The reason is that liquidity risk premium becomes higher. The latter happens not because of higher amount of liquidity risk that banks face, but because of higher price of liquidity risk. Banks would reduce credit in the hope of increasing the capital adequacy ratio and, hence, dependence on equity on the liability side. Naturally, because of less money creation, output and prices decline as well. - [25] FX interventions (FXI) and the real economy: FXI can have a significant impact on the real economy, depending on the size of a central bank’s collateral base, even without considering exchange rate-related effects. For example, within our model and its current calibration, a one-off FXI worth of 2% of annual GDP can either increase output by about 0.2% if the central bank buys FX or reduce it by about 1% if the central bank sells FX. The channel goes through our liquidity risk premium. That’s why the impact is asymmetric. Namely, when the central bank buys FX it supplies newly created liquidity to banks and reduces their dependence on central bank discount borrowing (which uses up their scarce collateral). But if liquidity buffers were adequate to begin with, then any extra buffers are not valued as much. Hence, liquidity risk premium declines only modestly and, therefore, credit and money creation accelerate only marginally as well. On the other hand, when the central bank sells FX it widens the structural liquidity deficit in the system, making the scarce collateral significantly more valuable, as banks are now forced to use the central bank discount window more aggressively. Hence, liquidity risk premium, because of its nonlinearity, jumps up and induces credit and money creation to decelerate significantly, hurting output. Weaker demand reduces prices, which would actually make this impact in line with the objectives of selling FX in a real world example (central banks usually sell FX when they want to defend the exchange rate due to inflation fears). However, this impact comes at a price of slowing down credit and output. - [26] Profit tax and bank credit: reducing the profit tax rate applied to banks leads to more risk-taking, on the side of banks, in terms of accelerating the risky credit extension in particular. However, this does not lead to a higher probability of bank default. If anything, this probability marginally declines. The reason for both (acceleration of risky loans but a reduction in bank default risk) happening at the same time is that lower tax rate implies higher after-tax ROE, other things equal. Higher retained earnings, in turn, lead to higher capital. More capital is partly used to take on more borrower risk, while another part is used to maintain higher capital adequacy ratio, implying more robustness of banks in facing the credit risk. Obviously, the downside of this is marginally higher price level and/or the need for monetary policy tightening and fiscal consolidation in other parts of the economy (as lower bank profit tax rate implies lower overall tax revenues, other things equal). Also, if banks suddenly increase dividend payouts, then the solvency risk would no longer decline. - [27] Investors’ required return and bank default risk: if investors suddenly and permanently increase their required returns on bank equity (which in the model is achieved through higher SS dividend payouts per equity share), the result will be banks increasing their lending interest rates to generate higher ROE. However, higher ROE doesn’t result in higher capital adequacy ratio, since this extra profit is not kept as retained earnings but rather paid out as dividends. As a result of higher dividend payouts, capital adequacy ratio actually declines, resulting in higher bank default probability on the margin. In other words, when investors increase their perception of banking sector risk (hence, requiring higher returns), the banking sector risk, in response, then actually does increase. - [28] Bank overhead costs and financial stability: a decline in banks’ overhead costs (or net non-interest costs) leads to higher ROE/retained earnings and, hence, an accumulation of capital buffers. These buffers are then partially used to extend more credit, particularly risky credit, without negatively impacting bank default risk (as capital is now higher than initially). The flip side of this up-leveraging is lower interest rates, hence borrowers benefiting from this as well. Unlike lowering profit tax, here there’s no fiscal cost. If anything, higher ROE means banks will be paying higher taxes as well. Hence, work on reducing bank overhead (net non-interest) costs may be a particularly efficient way of helping credit while maintaining financial stability. - [29] Changes in potential output and bank leverage: potential output adjustments may have an impact on banks credit risk-taking, however the size of this impact depends on the nature of the driver behind such adjustments. For example, if the potential output changes due to one-off adjustments in the level of inputs into production (e.g. level shift in physical capital, labor force, or productivity), then output and prices move in opposite directions and bank credit doesn’t change much, since they depend on nominal output. However, if we instead have a change in the expectations about the future potential output without much changes instantaneously, then output and prices move in the same direction in this transition period and, hence, this may have an outsized impact on bank credit. If this is in the downward direction, then banks see a decline in profitability (as lower demand for credit leads to lower lending interest rates) and increase their risk-taking, in the form of reducing their (voluntary) capital buffers. - [30] Money velocity and liquidity/credit risk premia: in an interest rate targeting framework, volatility in money velocity should be absorbed by corresponding changes in the supply of money. As inside money (created by banks) represents the biggest part of the overall broad money, this means that changes in money velocity should be absorbed by corresponding changes in bank credit. If the change in money velocity is large, then bank credit will have to adjust in similarly large amount. However, an abrupt and substantial change in bank credit means liquidity and capital buffers also change in relation to the overall size of the bank balance sheet. E.g. if money velocity declines, bank credit should increase by the same amount, instantaneously. But as accumulating liquidity and capital buffers, on the other hand, takes time for the bank, it will then have to temporarily operate will higher risks on both liquidity as well as credit transformation sides. The result is higher risky lending rates (due to higher default risk premium) and lower risk-free yields (due to higher liquidity risk premium, which is the opposite of collateral service premium). - [31] FXI and financial stability: FX interventions by the central bank may affect not only the real economy, but the solvency risk of the banking system, at least during a transition period. Namely, if the central bank buys FX to hold it permanently (i.e. accumulates FX reserves), this means overall broad money demand is satisfied more with FX reserves stock and the need for private credit extension becomes less. Lower credit, for a given level of bank capital, means higher capital adequacy and less probability of bank default. It is true, the bank will then respond to this situation by relaxing credit conditions (reducing lending interest rates) and pushing bank default probability back to its equilibrium. However, at least during the transition period (i.e. when FXI is unanticipated, which in many cases they are), financial stability risks are somewhat mitigated. The reverse is true for FX sales. - [32] Interest rate risk and policy transmission: while the presence of interest rate risk may once in a while generate financial stability considerations, they can also strengthen the transmission of monetary policy. For instance, if policy rate unexpectedly tightens, this leads to banks experiencing losses due to negative realization of their interest rate risk. This leads to an erosion of capital and, hence, even more tightening of bank credit than it would otherwise lead. The reverse is also true when policy is eased, but maybe working at a slower pace. This is so since, in the model, bank capital being insufficient is much more risky, to which banks reacts immediately, while excess capital may be used for credit acceleration more gradually (unless investors ask for an immediate dividend payouts). In any case, whether policy rate changes are anticipated or unanticipated matters for the strength of the transmission.62 - [33] Inside money and its endogeneity: according to this model, there can be four aggregate variables that directly lead to inside money creation. These are private credit extension, government deficit spending (debt accumulation), central banks’ FX reserves accumulation and contractions in bank equity. In other words, while some models have rightly emphasized the key feature of money creation (that it is done via private bank credit), there are a couple of key factors that can generate leakages/divergence between money and private credit. This, as this model also underscores, shouldn’t make us think that money creation is a process exogenous to private bank credit decisions. - [34] Inflation target and financial stability: higher inflation is usually associated with higher seigniorage only for central banks. But the model underlines that commercial banks may also benefit from seigniorage income. This is the case since, as inflation increases, deposit interest rates remain partially compressed due to money demand, while lending interest rates increase more forcefully. In other words, other things equal, nominal credit spreads are positively related to inflation. This means that, if the central bank increases its inflation target or allows higher inflation, banks’ ROE increases pushing capital up and reducing the bank default probability. This is akin to “financial dominance”, when central banks tolerate higher inflation because of financial stability considerations. - [35] Money multiplier VS money divisor: does the model tell the mechanics of money creation process? It does and suggests that the so-called money multiplier understanding of money creation explains the process totally upside down. In the words of Goodhart (2017), while the money multiplier can be calculated based on an identity, “the causal direction works in exactly the reverse direction to that normally supposed”. More specifically, first banks lend and create broad money, and only then do central banks supply the necessary base money to support their interest rate target. Eventually, we may still have a picture where base money is a small fraction of broad money, but the process of getting there is completely in reverse. Put differently, “it is not so much a money multiplier, as a money divisor” (Goodhart, 2017). This means that, in principle, the process starts with banks and that’s why we need to understand their behavior first. - [36] Comovement of consumption and investment: a standard banking theory of loanable funds would suggest that for investment to pick up, first consumption should slow down. The reason is that those models confuse financial saving with real saving. Namely, in these models, households would need to save – in the form of less consumption – so that banks have sufficient funds to lend out later. And since investment depends on lending, we get a negative comovement between investment and consumption in a credit boom-bust cycle. Yet, our model shows that this is not necessarily the case. The reason is that in our model it is financial saving that funds investment.63 But financial saving is created by banks expanding the size of their balance sheets, which – while have limitations of their own – are not really constrained by the amount of real saving in the economy. This list, while very rich in policy implications, isn’t full, as we have only considered the impacts of single shocks or re-calibrations. Generating plausible scenarios (with multiple shocks) may generate some extra interesting policy implications, due to strong nonlinearities. All these emphasize the usefulness of the features that the model incorporates in terms of understanding the intricacies of the macro-banking realm. *Appendix 1 – summary of all model implications* ### Appendix 2 – deriving key interest rate equations ### Appendix 2 – deriving key interest rate equations ### Bank optimization problem - The bank’s optimization problem is stated as: max피푡 퐹훿 푅푂퐸푡 =피푡 퐹훿 퐼푅푡 − 퐹휎 퐹𝑐 [ (푖푡퐷 (1−푐̂) + 푖푡푅 ĉ)(푇퐴푡 − 퐸푡) − (푖푡푅 − 푖푡퐷)(1−푐̂)푅푡𝑋 ] − (1−퐹휎 퐹𝑐)(푖푡푅 + χ)(푇퐴푡 − 퐸푡) − ψ퐸푡(1−τ) - subject to: - 푆푡 ≤ 푓𝑆(푖푡𝑆) - 퐿푡𝑆 ≤ 푓𝑆𝐿(푖푡𝑆𝐿) - 퐿푡𝑅 ≤ 푓𝑅𝐿(푖푡𝑅𝐿) - 푖푡𝐷 ≥ 푓𝐷(푖푡𝑅) ### Key partial derivatives collected - For x = 푆푡, 퐿푡𝑆: - 휕퐼푅푡/휕푖푡𝑥 = (1−휀𝑥) 푥 - 휕푇퐴푡/휕푖푡𝑥 = −휀𝑥 푥 푖푡𝑥 - 휕퐶퐵푡/휕푖푡𝑥 = −휂𝑥 휀𝑥 푥 푖푡𝑥 - 휕𝑅𝑊𝐴𝑡/휕푖푡𝑥 = −휔𝑥 휀𝑥 푥 푖푡𝑥 - 휕𝐹𝑐/휕푖푡𝑥 = 푓𝑐̂( (퐶퐵푡 − 푟𝑟(푇퐴푡 − 퐸푡) + (1−푟𝑟)푅𝑡𝑋(1−푟𝑟)(푇퐴푡 − 퐸푡) + (1−푟𝑟)푅𝑡𝑋) 1 1−푟𝑟 ( 휕퐶퐵푡/휕푖푡𝑥 1 (푇퐴푡 − 퐸푡 + 푅𝑡𝑋) − 휕푇퐴푡/휕푖푡𝑥 퐶퐵푡 + 푅𝑡𝑋(푇퐴푡 − 퐸푡 + 푅𝑡𝑋)2 ) ) - where x = 푆푡, 퐿푡𝑆, 퐿푡𝑅 - 휕𝐹훿/휕푖푡𝑥 = 푓훿̂(퐸푡 − e^m 𝑅𝑊𝐴𝑡 퐿푡𝑅)(− e^m 퐿푡𝑅) - with 휕𝑅𝑊𝐴𝑡/휕푖푡𝑥 where x = 푆푡, 퐿푡𝑆 - For 푖푡𝑅𝐿: - 휕퐼푅푡/휕푖푡𝑅𝐿 = (1 − 휀𝑅𝐿 + 휀𝑅𝐿 훿̂ 푖푡𝑅𝐿) 퐿푡𝑅 - 휕𝐹훿/휕푖푡𝑅𝐿 = 푓훿̂(퐸푡 − e^m 𝑅𝑊𝐴𝑡 퐿푡𝑅)(− e^m 퐿푡𝑅 휕𝑅𝑊𝐴𝑡/휕푖푡𝑅𝐿 + 퐸푡 − e^m 𝑅𝑊𝐴𝑡 (퐿푡𝑅)2 휀𝑅𝐿 퐿푡𝑅 푖푡𝑅𝐿 ) ### First order condition (FOC) for the equilibrium interest rate on securities (푖푡𝑆) - General FOC prior to substitution: - 퐹훿 휕푅푂퐸푡/휕푖푡𝑆 + 휕퐹훿/휕푖푡𝑆 푅푂퐸푡 = 0 - Expanded: - 퐹훿(휕퐼푅푡/휕푖푡𝑆 − 퐹휎 퐹𝑐 (푖푡𝐷 (1−푐) + 푖푡𝑅 푐) 휕푇퐴푡/휕푖푡𝑆 − (1−퐹휎 퐹𝑐)(푖푡𝑅 + χ) 휕푇퐴푡/휕푖푡𝑆 − 퐹휎[(푖푡𝐷 (1−푐) + 푖푡𝑅 푐)(푇퐴푡 − 퐸푡) − (푖푡𝑅 − 푖푡𝐷)(1−푐)푅푡𝑋 − (푖푡𝑅 + χ)(푇퐴푡 − 퐸푡)] 휕퐹𝑐/휕푖푡𝑆) 퐸푡(1−τ) + 푓훿̂(퐸푡 − e^m 𝑅𝑊𝐴𝑡 퐿푡𝑅)(− e^m 퐿푡𝑅) 휕𝑅𝑊𝐴𝑡/휕푖푡𝑆 푅푂퐸푡 = 0 - Substituting the partial derivatives (intermediate expanded form shown in source) and multiplying both sides by 퐸푡 푖푡𝑆 (1−τ) / 푆푡, and denoting: - 푓𝑐 ≡ 푓𝑐̂(퐶퐵푡 − 푟𝑟(푇퐴푡 − 퐸푡) + (1−푟𝑟)푅푡𝑋(1−푟𝑟)(푇퐴푡 − 퐸푡) + (1−푟𝑟)푅푡𝑋) - 푓훿 ≡ 푓훿̂(퐸푡 − e^m 𝑅𝑊𝐴𝑡 퐿푡𝑅) - The rearranged closed-form for 푖푡𝑆 is: - 푖푡𝑆 = 휀𝑆/(휀𝑆 − 1) ( - 퐹휎 푓𝑐 (1−푐) 푖푡𝐷 - + (1 − 퐹휎 푓𝑐 (1−푐)) 푖푡𝑅 - + (1 − 퐹휎 푓𝑐) χ - + 1/(1−푟𝑟) 푓𝑐 퐹휎 (퐶퐵푡 + 푅푡𝑋 푇퐴푡 − 퐸푡 + 푅푡𝑋 − 휂𝑆) [ (푖푡𝑅 − 푖푡𝐷)(1−푐) + χ 푇퐴푡 − 퐸푡 푇퐴푡 − 퐸푡 + 푅푡𝑋 ] - + (푓훿/퐹훿) 휔𝑆 e^m 퐸푡 퐿푡𝑅 푅푂퐸푡/(1−τ) - ) - Note: the factor 휀𝑆/(휀𝑆 − 1) multiplies the full parenthesis. ### Additional notes - The appendix states: "We have similar equations for other interest rates set by banks." - The derivation uses the shorthand definitions: - 푓𝑐 ≡ 푓𝑐̂( ... ) as above - 푓훿 ≡ 푓훿̂( ... ) as above *Source: Appendix 2 of IMF Working Paper Inside (the) Money Machine: Modeling Liquidity, Maturity and Credit Transformations (Working Paper No. WP/2025/166).* --- _Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025166-source-pdf.pdf_