## Appendix I. The Lending Problem ## Source details **Canonical URL:** [Appendix I. The Lending Problem](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2014/_wp1430.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2014/_wp1430.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2014/_wp1430.pdf.json) --- ### Context and role in the paper - Appendix I provides the entrepreneurs’ optimization problem and the derivation of the loan contract conditions that underpin the financial accelerator and lending dynamics used throughout the model economy. - It links entrepreneurs’ first-order conditions, the definition of the risk premium, recovery rates, and the evolution of entrepreneurs’ net worth and consumption to the main text’s force of default and liquidation mechanisms. ### Core equations and definitions (as presented) - Entrepreneurs’ maximization problem: equation label (A.1). - First-order conditions with respect to control variables: (A.2) and (A.3). - Combined condition (from combining (A.2) and (A.3)): (A.4). - Risk premium definition: (A.5). - Real-terms transformation of relevant conditions: (A.6). - Recovery rate expression: (A.7). - Net worth evolution of entrepreneurs: (A.8). - Entrepreneurs’ consumption rule: (A.9). ### Key mechanisms summarized - Default and recovery: default decisions are characterized by the cutoff that determines repayment versus default; recovery rates enter entrepreneurs’ net worth and lenders’ break-even conditions through the expressions in (A.7) and (A.8). - Risk premium and real terms: the risk premium faced by entrepreneurs is explicitly defined (A.5) and then transformed into real terms (A.6) to feed into real-economy pricing and borrowing dynamics. - Link to liquidity and fire-sales: the outcomes in Appendix I feed into the liquidity intermediaries’ problem (Appendix II) through the amount of defaulted capital and the payments realizable after liquidation, which determine effective recovery and lender returns. *Source: _wp1430 - Appendix I. The Lending Problem (PDF).* ### Appendix I. The Lending Problem ........................................................................................ ### Appendix I. The Lending Problem ### Appendix listings and page references - Appendix I. The Lending Problem ..........................................................................................37 - Appendix II. The Problem of Liquidity Intermediaries ...........................................................37 - Appendix III. Price Rigidities, the Phillips Curve, and Aggregation of Final Goods Demand ...................................................................................................................................38 - Appendix IV. Complete Set of Equilibrium Conditions ..........................................................40 - Appendix V. Extension with Foreign Funding for Lending Intermediaries and Enterpreneurs ..........................................................................................................................43 - Appendix VI. Financial Dollarization of the Entrepreneurs’ Loans ........................................44 *Source: _wp1430 - Appendix I. The Lending Problem (PDF).* ### References................................................................................................33 ### _wp1430 - References................................................................................................33 ### I. INTRODUCTION — objectives, context, and contributions - Context: Shift toward macro-prudential, system-wide supervision following the recent financial crisis (Bernanke, 2008; Blanchard et. al. 2010); macro-prudential policies interact with other macro policies, especially monetary policy (Caruana, 2011; IMF, 2013). - Policy problem studied: Interactions between monetary and macroprudential policies, focusing on distortions they attempt to mitigate and management of swings in capital flows (Ostry et al, 2011). - Empirical motivation: - Emerging market economies have used macroprudential instruments countercyclically in response to capital flow swings (Lim et al, 2011; Federico et al, 2013 for reserve requirements). - Illustration: Figure 1 shows countercyclical use of reserve requirements for four EMs around Lehman’s bankruptcy. All four slashed policy rates immediately after Lehman; Brazil and Peru reduced reserve requirements dramatically even before cutting rates; following unconventional monetary policies abroad, all four raised reserve requirements and increased policy rates—with the exception of Turkey. - Footnote: Changes in average reserve requirements in Colombia underestimate actual impact because they don’t capture changes in marginal rates and remuneration (Vargas et al, 2010). - Literature: Most papers assume government minimizes a loss function combining credit growth volatility, output, and inflation; findings generally: macroprudential instruments help price and financial stability especially for financial shocks, but trade-offs exist between monetary and macroprudential instruments for demand/productivity shocks (Glocker and Towbin, 2012; Mimic et al, 2013; Kannan et al, 2012; Unsal, 2013). - Three innovations of this paper: - Explicit modeling of nominal and financial frictions with microfounded financial sector providing credit and liquidity services. - Calculation of model-based welfare measures to rank policy arrangements. - Focus on a world interest rate shock that keeps rates low for an extended period and is later undone, inducing inflows followed by reversal—relevant to unwinding of unconventional monetary policies. - High-level result summary: - A large and protracted reduction in world interest rates produces large capital inflows, increases in aggregate demand, activity, the real exchange rate and asset prices in the “natural” economy (no frictions). - Introducing frictions magnifies cyclical fluctuations, particularly asset prices and credit. - Welfare ranking results: - Pure inflation targeting (IT) often dominates a standard Taylor-rule but generates excessive asset price volatility and may or may not be dominated by an adjusted Taylor-rule reacting to credit growth (Christiano et al, 2010). - All examined monetary regimes are dominated in welfare terms by a regime that combines a counter-cyclical reserve requirement (targeting the financial friction) with a pure IT rule for monetary policy (targeting nominal friction). Interpretation: Tinbergen principle—one instrument per objective—and Mundell’s principle of effective market classification (Glocker and Towbin, 2012; Beau et al, 2012). - Coordination implication: Once a macroprudential instrument is used, the policy rate evolution deviates substantially from a Taylor rule, suggesting close coordination of instruments; specifically, while the “natural” interest rate declines with the world rate, the policy rate may need to be increased to accommodate reserve requirements (contrast with Turkey). - Paper organization (sections described): - Next: model economy with focus on financial sector and four monetary/macroprudential frameworks. - Section III: baseline calibration and welfare metric. - Section IV: policy responses to world interest rate shocks; impulse responses and welfare rankings. - Section V: robustness. - Section VI: conclusion. ### II. MODEL ECONOMY — structure and key mechanisms - Modeling approach: Extension of Bernanke, Gertler and Gilchrist (1999) financial accelerator to a small open economy New Keynesian framework with Calvo-pricing nominal rigidities (Gali and Monacelli, 2005); integrates Choi and Cook (2012) mechanism by making liquidation require time and excess reserves as well as real resources. - Key frictional mechanisms: - Price rigidities induce role for monetary policy to affect real interest rate and correct nominal distortion. - Financial friction arises from cost of monitoring defaulted borrowers and fire-sale/liquidation dynamics; reserve requirements can manage intensity of financial cycle because liquidation uses excess reserves and real resources. - Agents and components detailed: - Households: intertemporal preferences include consumption basket, labor supply, and real money balances; budget constraint describes consumption and savings; savings invested in deposits (return in domestic), foreign bonds (foreign-currency return), and money balances; foreign bonds denominated in foreign currency; risk premium for foreign bonds is a function of total indebtedness of the economy; define real exchange rate. - Optimality conditions provided: - Deposits Euler equation (equation (1)). - Foreign bonds Euler equation (equation (2)). - Money demand by households (equation (3)). - Labor supply condition (equation (4)). - Production and capital accumulation: - Competitive firms produce domestic and capital goods; aggregate production given by equation (5); labor and capital demand equations (6) and (7); capital depreciates at rate delta; defaulted capital and conversion probability: probability pi of turning one unit of defaulted capital into productive one; evolution of productive capital in equation (8); investment demand from capital goods producers in equation (9). - Financial sector architecture: - Two representative intermediaries operating competitively: lending intermediaries and liquidity intermediaries interacting through an interbank market. - Lending intermediaries: - Provide credit to entrepreneurs solving agency-cost problem as in BGG (1999). - Entrepreneurs use net worth and loans to purchase installed capital; idiosyncratic shock (unit-mean, log-normal with standard deviation sigma_a) converts installed capital into realized output; default occurs if realization below cutoff. - Successful entrepreneurs sell capital at end of period and repay loans; unsuccessful entrepreneurs default, lending intermediary seizes capital and sells to liquidity intermediary. - Average rate of return of capital defined in equation (11). - Endogenous cutoff q determines default vs repayment; lending rate r_L moves with cutoff (equation (12)). - When default occurs, the actual payment to lending intermediary from a defaulted loan is determined by fire-sale price p^f (notation as in text). - Lending intermediary zero-profit condition given in equation (13). - Cost of default endogenous and depends on difference between market prices of installed and defaulted capital (equation (14)). - Share of returns to lending intermediaries and entrepreneurs given by equations (15) and (16). - Optimal loan contract conditions: arbitrage/break-even conditions given by equations (17) and (18); risk premium interpretation in equation (19). - Entrepreneur net worth dynamics: fraction 1-psi die and are replaced; surviving entrepreneurs receive net wealth transfer n; net worth evolves per equation (20); dying entrepreneurs consumption in equation (21). - Liquidity intermediaries: - Receive deposits from households, pay gross rate r_D, lend in interbank market at r_IB (and to monetary authority at r_M). - Use “excess reserves” and final goods to provide liquidity services to restructure defaulted capital. - Demand for liquidation services related to stock of defaulted capital: D_t = kappa * K^D_t (equation (22)). - Evolution of defaulted capital given by equation (23) with new defaulted capital defined in (24). - Production technology for liquidity services combines excess reserves and final goods in a complementary way (equation (25)). - Liquidity intermediary profit maximization problem described; optimality conditions determine fire-sales price of defaulted capital (equation (26)). - Marginal costs of liquidation services tied to final goods and excess reserves drive spread between interbank and deposit rates (equations (27) and (28)). - Spread can be expressed in terms of time-varying reserve requirement rr_t (assuming φ_rr > 0). - Interbank market equilibrium equality: fraction of entrepreneurs’ debt financed in interbank equals fraction of real deposits lent in interbank (equation (29)). - Aggregation and price rigidities: - Total demand for final goods given by equation (30); final goods composite of domestic and imported goods per equation (31). - Real marginal cost of final goods given in equation (32) with share of foreign goods omega. - Relative demand for domestic vs foreign goods per equation (33); real exchange rate denoted q_t, elasticity of substitution between domestic and foreign goods is η. - Calvo (1983) price-setting for wholesale firms with partial indexation: log-linear Phillips curve of final good inflation in equation (34). - Balance of payments identity in equation (35): stock of foreign debt B_t, gross foreign interest rate R*_t, foreign inflation pi*_t. - Foreign demand for exports specified in equation (36). - Exogenous evolution of foreign interest rate follows stochastic process in equation (37). ### E. Alternative monetary and macro-prudential frameworks — instruments and rules - Instruments: - Monetary policy instrument: interbank market rate r_IB,t. - Macro-prudential tool: time-varying reserve requirement rr_t. - Policy rules: - Model evaluates alternative arrangements by setting different rules for r_IB,t and rr_t to complete the economy. - “Natural” allocation: specification removing price rigidities and financial frictions used as benchmark (natural economy). - Policy scenarios analyzed (as described in paper structure and results): - Pure inflation targeting (IT). - Standard Taylor-rule. - Taylor-rule augmented with a credit growth argument (as in Christiano et al, 2010). - Combination of pure IT monetary policy with a counter-cyclical reserve requirement (macroprudential rule). - Principal policy conclusions (from model-based welfare rankings and dynamics): - Regimes that combine a counter-cyclical reserve requirement with pure IT for monetary policy yield higher welfare than alternatives considered. - Use of a macroprudential instrument changes optimal policy-rate dynamics and requires coordination—policy rate may need to increase even when the natural rate declines with world rate due to reserve requirement effects. _italics: Source: _wp1430 - References................................................................................................33_ ### 1. Standard Taylor-type rule and constant reserve requirement. In this case monetary ### 1. Standard Taylor-type rule and constant reserve requirement. In this case monetary policy is characterized by the following reaction rule for the interbank rate (in annual terms): ### Policy regimes analyzed - Standard Taylor-type rule with constant reserve requirement equal to steady state value. - Inflation Targeting (IT) regime with constant reserve requirement equal to steady state level; monetary policy modeled as an implicit contingent rule that achieves full stabilization of inflation in every period and every state. - Augmented Taylor-type rule: standard Taylor rule extended with a countercyclical reaction to fluctuations in entrepreneurs’ loan; reserve requirement constant at its steady-state level. - IT regime combined with a countercyclical reserve requirement: inflation fully stabilized each period/state; reserve requirement adjusted counter-cyclically according to the model specification. ### Key parameters and calibration (quarterly frequency) - Steady state inflation rate: zero. - (Gross) deposit rate at steady state: approximately 4 percent on an annual basis. - Calvo parameter: average duration of not having optimally reset prices is four quarters (full indexation with parameter set to full indexation). - Ratio of net exports to GDP: 0.5 percent → implies foreign debt to annual GDP around 12.4 percent. - Share of foreign goods in final goods composite: 30 percent. - Elasticity of substitution between home and foreign goods: less than one. - Investment adjustment costs calibrated so they do not affect steady-state allocations; follow Smets and Wouters (2007) style parameterization. - Quarterly depreciation rate of capital: 2.5 percent (quarterly). - Probability of selling defaulted capital: parameter implies on average defaulted capital takes one year to be restructured and productive again. - Production technology: capital share around one third. - Reserve requirement at steady state: value denoted in text, implying a steady state interbank rate of 4.5 percent on an annual basis. - Financial contract parameters: - Annual default rate: 3 percent. - Leverage ratio: 40 percent. - Average cost of liquidation: calibrated (value stated in text). - These financial parameters imply: - Risk premium at steady state: stated in text. - Recovery rate: around 36 percent. - Return to capital: around 15 percent in annualized terms. - Lending rate: around 7 percent in annualized terms. - Entrepreneurs’ death rate: 1 percent quarterly. - With this parameter, entrepreneurs’ debt and deposits (as percent of GDP, annual terms): about 55 percent and 61 percent, respectively. - Liquidation services calibration consistent with Choi and Cook (2012). - Excess reserves normalized such that excess reserves correspond to 0.25 percent of deposits → excess reserves around 0.15 percent as percentage of annual GDP. - Numerical solution method: - Dynamics solved with first-order approximation around deterministic steady state. - Welfare computed with second-order approximation to obtain welfare ranking among policy frameworks. - Only household utility matters for welfare (entrepreneurs are risk neutral). - Weight of real money balances in household utility set very small. ### Welfare calculation - Welfare under policy framework P computed as the discounted sum of household utility over T quarters under policy P (formula specified in source). - Losses/gains relative to the natural equilibrium measured as steady-state consumption equivalent that solves the specified equation using deterministic steady-state consumption and employment. - Natural equilibrium defined as the equilibrium when nominal and financial frictions are removed while keeping the same deterministic steady state. ### Shock scenario studied - Transitory reduction in the foreign interest rate perceived to last according to a persistence coefficient (set for welfare analysis in the text); after twelve quarters foreign interest unexpectedly rises to original level. - Scenario associated with large capital inflows and a reversal after twelve quarters. - Qualitative response: - Reduction in world interest rates triggers sharp increase in aggregate demand, GDP, and asset prices. - Natural (interbank) rate follows the world rate, inducing current account deficit and real exchange rate appreciation. - When world rate increases back, model intrinsic dynamics deliver a slowdown in the increase in foreign debt rather than a sudden stop. ### Simulation findings and welfare comparisons - Under Standard Taylor-type rule: - Policy (interbank) rate does not fall in the first quarter → sharp increase in real interest rate → deflationary cycle and stronger real exchange rate appreciation. - Policy rate falls after first quarter, even beyond natural rate, inducing sharp increase in credit (entrepreneur debt). - Under pure Inflation Targeting (IT) regime: - Inflation fully stabilized. - Aggregate demand and asset price fluctuations exacerbated relative to Taylor rule. - Entrepreneur debt increases less than under Taylor rule due to sharp increase in price of capital (“Tobin Q”) which increases net worth and reduces need for external funds. - Higher asset price volatility leads to sharper swings in default and recovery rates and a highly procyclical cost of liquidation (contrasting with constant one in BGG, 1999). - Excess reserves fall in the first three years and are restored when world interest rates rise. - Welfare: IT regime yields higher welfare than standard Taylor rule. - Table 1 (Welfare Comparison: Standard Taylor vs IT) - 1 Standard Taylor type rule: Welfare = -17.1602 ; Losses relative to welfare of steady-state consumption in the Natural Equilibrium = 11.27% - 2 IT regime: Welfare = -16.9999 ; Losses relative to welfare of steady-state consumption in the Natural Equilibrium = 11.15% - Augmented Taylor Rule (adds credit growth term and increases weight on inflation): - Reduces volatility of asset prices and defaults relative to standard Taylor rule. - Still leads to a relatively fast increase in credit and is dominated in welfare terms by pure IT regime in this calibration. - Table 2 (Welfare Comparison: IT vs Augmented Taylor) - 2 IT regime: Welfare = -16.9999 ; Losses relative to welfare of steady-state consumption in the Natural Equilibrium = 11.15% - 3 Augmented Taylor type rule: Welfare = -17.0887 ; Losses relative to welfare of steady-state consumption in the Natural Equilibrium = 11.22% - Interpretation: adding credit growth with a positive coefficient moves the economy further away from natural path when initial shock lowers the natural interest rate. - IT regime combined with countercyclical reserve requirement (macroprudential instrument): - Reserve requirements rise substantially in the first two years: from 10 percent of deposits to just above 30 percent at the end of the first year. - After reversal in world interest rates, reserve requirements are reduced to less than the original 10 percent rate. - Combined monetary-macroprudential regime brings macro variables closer to natural levels and smoothes volatility of financial variables. - Much more effective in containing credit growth than augmented Taylor rule and delivers clear welfare gains relative to other regimes. - Table 3 (Welfare Comparison: Augmented Taylor vs IT + Countercyclical RR) - 3 Augmented Taylor type rule: Welfare = -17.0887 ; Losses relative to welfare of steady-state consumption in the Natural Equilibrium = 11.22% - 4 IT regime and Countercyclical RR: Welfare = -15.4649 ; Losses relative to welfare of steady-state consumption in the Natural Equilibrium = 10.00% - Policy interactions: - Use of macroprudential instrument affects monetary policy: while natural interest rate falls early, policy rate falls only marginally and is driven above its original level after one year to accommodate increased reserve requirement. - Emphasizes need to coordinate monetary and macroprudential policies when dealing with swings in capital flows. - Interpretation: macroprudential instrument mitigates financial friction while monetary policy addresses nominal friction, consistent with Tinbergen principle and Mundell’s principle of effective instrument-objective pairing. ### Robustness explored - Robustness section examines ways to enhance financial and nominal distortions and how changes affect policy rankings. - Also explores calibration where liquidity services represent a larger share of financial services. - Foreign borrowing and dollarization extension: - Previously only households held foreign liabilities; robustness check assumes entrepreneurs and lending intermediaries have direct access to external funding in world markets (levels of borrowing assumed constant to capture valuation/balance sheet effects). *Source: IMF working paper content unit provided.* ### Appendix V for details). We also allow, in a separate exercise, for dollarization of credit, i.e. ### Appendix V: Dollarization, Wage Rigidities, and Robustness Checks ### Dollarization of Credit (direct foreign funding) - Setup: - Half of the entrepreneurs’ borrowing can be done in dollar-indexed instruments. - Entrepreneurs and lending intermediaries have direct access to external funding. - Key dynamics: - Initial real exchange rate appreciation magnifies the increase in entrepreneurs’ net worth and requires less borrowing; there is an initial reduction in credit. - Greater financial integration transmits the world interest rate shock more, and lending interest rates fall substantially in the early periods. - Welfare comparison (Table 4: Welfare Losses relative to welfare of steady-state consumption in the Natural Equilibrium): - 1 Standard Taylor type rule: -16.9037 ; 10.85% - 2 IT Regime: -16.7712 ; 10.75% - 3 Augmented Taylor type rule: -16.8059 ; 10.78% - 4 IT regime and Countercyclical RR: -14.1297 ; 8.79% ### Partial dollarization in entrepreneurs’ loans - Setup: - Partial dollarization of entrepreneurs’ borrowing (half-dollar-indexed borrowing as above) exacerbates financial distortions. - Key dynamics: - Larger fluctuations in aggregate demand, GDP, real exchange rate, and financial variables. - Default rate spikes to 12 percent in the simulation; net worth increases by more than 30 percent of GDP after the initial shock. - Credit declines initially when partial dollarization enhances the financial friction. - Policy ranking change: - IT regime with macroprudential policies continues to dominate monetary-policy-only regimes, but the Augmented Taylor rule now dominates the IT regime. - Augmented Taylor rule brings the economy closer to its “natural” state than pure IT under these conditions. - Welfare comparison (Table 5: Welfare Losses relative to welfare of steady-state consumption in the Natural Equilibrium): - 1 Standard Taylor type rule: -17.1063 ; 10.65% - 2 IT Regime: -16.8577 ; 10.58% - 3 Augmented Taylor type rule: -16.7650 ; 10.56% - 4 IT regime and Countercyclical RR: -10.8573 ; 6.27% ### Wage rigidities - Modeling approach: - Follow Blanchard and Galí (2007): only a fraction of households can reset nominal wages each period; the rest keep previous nominal wages. - Aggregate nominal wage inflation summarized by equation (G.2); aggregate real wage evolution by equation (G.3). - Calibration used: households set their nominal wages every 8 quarters (parameter value corresponds to this frequency). - Key dynamics: - Wage rigidities further exacerbate asset price and default/recovery fluctuations and exchange rate movements. - Welfare gains from countercyclical reserve requirement are larger than with only price rigidities—reaching 2 percent of steady state consumption in the model. - Welfare comparison (Table 6: Welfare Losses relative to welfare of steady-state consumption in the Natural Equilibrium): - 1 Standard Taylor type rule: -17.6437 ; 13.17% - 2 IT Regime: -16.9368 ; 12.63% - 3 Augmented Taylor type rule: -17.3656 ; 12.96% - 4 IT regime and Countercyclical RR: -14.3495 ; 10.65% ### Larger “excess reserves” - Context and calibration: - Baseline: 0.25 percent of deposits kept as excess reserves. - Alternative calibration: “excess reserves” set to 1.5 percent of deposits (six times baseline). - To support larger excess reserves, select a quarterly default rate of 4 percent and choose parameters so average recovery rate is around 31 percent. - Modify curvature of the liquidation technology accordingly; remaining liquidation-service parameters unchanged. - Key dynamics: - Financial frictions play a larger role: more volatile asset prices, default and recovery rates, and a loan rate that is more pro-cyclical. - Under this calibration, an inflation targeting regime may not require a fall in the monetary policy rate in the boom phase (contrary to baseline), and this response is not effective in stabilizing GDP, aggregate demand, or financial variables. - A more pro-cyclical financial sector constrains how loose monetary policy can be in response to a reduction in the world interest rate. - Using the reserve requirement requires a monetary policy rate that increases in the boom phase and is reduced in the downturn; monetary policy rate deviates more from the natural rate to accommodate countercyclical reserve requirements. - Welfare comparison (Table 7: Welfare Losses relative the welfare of steady-state consumption in the Natural Equilibrium): - 1 Standard Taylor type rule: -22.9513 ; 15.31% - 2 IT Regime: -22.6254 ; 15.04% - 3 Augmented Taylor type rule: -22.8156 ; 15.20% - 4 IT regime and Countercyclical RR: -22.1323 ; 14.63% ### Conclusions (policy implications and summary findings) - Main result: - The coordinated use of an inflation targeting (IT) regime with a countercyclical reserve requirement increases welfare, even in the absence of externalities, across a range of institutional features and alternative calibrations. - Policy coordination: - Important to coordinate monetary policy rate adjustments with the countercyclical macroprudential instrument. - Adding financial variables such as credit growth to a standard Taylor rule yields slight improvement over a pure Taylor rule but is still dominated by coordinated use of both instruments (monetary policy + countercyclical reserve requirement). - When both instruments are used, the monetary policy rate may deviate substantially from the pure IT regime rate and from the economy’s natural rate (which follows the world interest rate closely). - Cautions: - “Leaning against the wind” (monetary policy countering the credit cycle) can be perilous when macro-prudential policies are imperfect or in open-economy settings. - Responding with the policy rate to increases in credit growth may not be the right response when world interest rates are reduced dramatically. - Credit and asset prices may not move together after shocks; policy responses to asset prices and the exchange rate are left for further research. - Practical guideline: - Incorporate the shock’s impact on the “natural” interest rate and coordinate monetary policy responses with macro-prudential instruments when managing volatile capital flows. *Source: Excerpt from the IMF working paper (Appendix and robustness sections).* ### References ### _wp1430 - References ### References - Agénor, Pierre-Richard, Koray Alper, and Luiz Pereira da Silva, 2013, “Capital Regulation, Monetary Policy and Financial Stability,” International Journal of Central Banking, Vol. 9 No. 3 - Basel Committee on Banking Supervision, 2013, “Basel III: The Liquidity Coverage Ratio and liquidity risk monitoring tools,” BIS, January. - Beau, Deni, Laurent Clerc, and Benoit Mojon, 2012, “Macro-Prudential Policy and the Conduct of Monetary Policy,” Banque de France Working Paper No. 390. - Benigno, 2013, “Commentary on Macroprudential Policies,” International Journal of Central Banking, March, Vol. 9, No. 1. - Bernanke, Ben S., 2008, “Reducing Systemic Risk,” Speech at the Annual Economic Policy Symposium, August 22, Jackson Hole, Wyoming. - ______, Gertler, M., and Gilchrist, S., 1999, “The Financial Accelerator in a Quantitiative Business Cycle Framework,” in Handbook of Macroeconomics, Vol., 1C, edited by J.B. 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Betancourt, and Norberto Rodríguez, 2010, “Effects of Reserve Requirements in an Inflation Targeting Regime: The Case of Colombia,” Borradores de Economía, No. 582 (Colombia: Central Bank). ### Appendix I. The Lending Problem - The entrepreneurs’ maximization is stated with equation label (A.1). - First order conditions with respect to variables are given by (A.2) and (A.3). - Combining (A.2) and (A.3) yields (A.4). - Risk premium is defined in (A.5). - Real terms transformation leads to (A.6). - Recovery rate, net worth evolution, and entrepreneurs consumption are given by (A.7), (A.8), and (A.9), respectively. ### Appendix II. The Problem of Liquidity Intermediaries - Optimization problem and constraints are specified with labels (B.1) and (B.2). - Lagrangian and interpretation: lagrange multipliers are the marginal cost of providing liquidation services from final goods and excess reserves. - First order conditions are (B.4), (B.5), and (B.6). - Combining (B.5) and (B.6) gives (B.7). - Fire sales price expressed as marginal present discounted value of defaulted capital is given by (B.8). - Definition of the stochastic term in (B.9). - Expression for marginal cost of liquidation services and derivation for fire sales price net of liquidation cost is summarized yielding (B.10). ### Appendix III. Price Rigidities, the Phillips Curve, and Aggregation of Final Goods Demand - Final good production aggregation is given by (C.1). - Demand for each intermediate composite good is (C.2); aggregate price level of domestic demand is (C.3). - Intermediate composite producer technology is (C.4); cost minimization condition is (C.5); marginal cost in real terms is (C.6). - Aggregate relationships and total demand composition use (C.7). - Calvo pricing mechanism: fraction able to reset prices is denoted ; indexing weight is . - Optimal pricing FOC is (C.8); recursive definitions are (C.9) and (C.10); optimal condition restated as (C.11). - Price level aggregation under Calvo is (C.12). - Relationship between domestic demand and supply of final goods is (C.13); aggregate supply is (C.14). - Evolution of price dispersion term under Calvo is (C.15). ### Appendix IV. Complete Set of Equilibrium Conditions - The model equilibrium, given macroeconomic policy rules for policy variables ( and ), is characterized by a sequence satisfying conditions labeled (D.1) through (D.45). - The appendix lists the complete set of equilibrium conditions as equations (D.1)–(D.45). ### Appendix V. Extension with Foreign Funding for Lending Intermediaries and Entrepreneurs - Allows financial intermediaries to use external funds constant in foreign currency ( ) and entrepreneurs to use external funds constant in foreign currency ( ). - In equilibrium both lending intermediaries and entrepreneurs use the complete amount of external funds each period. - Presence of external funds implies a direct balance sheet effect of exchange rate movements and can magnify capital flows through the financial accelerator. - The loan contract maximization problem is stated as (E.1). - Equilibrium conditions and definitions include: - Arbitrage condition for loans to entrepreneurs: (E.2). - Definition of the recovery rate of financial intermediaries’ loans: (E.3). - Definition of the risk premium: (E.4). - Budget constraints of entrepreneurs: (E.5). - Break-even condition for financial intermediaries: (E.6). - Real net worth of entrepreneurs: (E.7). - Entrepreneurs’ consumption and lump-sum transfers including interest on foreign debt described in (E.8) and (E.9). ### Appendix VI. Financial Dollarization of the Entrepreneurs’ Loans - Introduces fraction of loan in domestic currency and fraction in foreign currency; denotes fraction with and and reports exercise with . - Loan contract problem with foreign-currency interest rate is stated as (F.1). - First order conditions are obtained analogously to Appendix I. *Source: _wp1430 - References (appendices and references list).* --- _Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2014/_wp1430.pdf_