Policy Research Working Paper 11170 The Asymmetric Bank Distress Amplifier of Recessions Dohan Kim Development Economics Prospects Group July 2025 Policy Research Working Paper 11170 Abstract One defining feature of financial crises, evident in U.S. uncertainty about the solvency of banks, thereby pushing and international data, is asymmetric bank distress—con- banks to deleverage. Quantitative analysis shows that the centrated losses on a subset of banks. This paper proposes bank distress amplifier doubles investment decline and a model in which shocks to borrowers’ productivity dis- increases the spread by 2.5 times during the Great Reces- persion lead to asymmetric bank losses. The framework sion compared to a standard financial accelerator model. exhibits a “bank distress amplifier,” exacerbating economic The mechanism helps explain how a seemingly small shock downturns by causing costly bank failures and raising can sometimes trigger a large crisis. This paper is a product of the Prospects Group, Development Economics. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The author may be contacted at dkim23@worldbank.org. The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent. Produced by the Research Support Team The Asymmetric Bank Distress Amplifier of Recessions Dohan Kim* Keywords: Bank Distress, Financial Crises, Financial Accelerator, Uncertainty. JEL Classification: D82, D86, E32, E44, G01, G20, G33. * World Bank Group (email: dkim23@worldbank.org). I am grateful to Enrique G. Mendoza, Guillermo L. Or- doñez, and Frank Schorfheide for their invaluable guidance and support. I also thank seminar participants at the University of Pennsylvania and the Federal Reserve Board of Governors for their helpful comments and suggestions. This paper was completed as part of my Ph.D. dissertation prior to my joining the World Bank. The views expressed herein are those of the author and do not necessarily reflect those of the World Bank, its Executive Directors, or the countries they represent. Part of this work was written while I was a dissertation intern at the Federal Reserve Board, whose hospitality I gratefully acknowledge. All errors are my own. 1. Introduction Recessions with a financial crisis (financial recessions) are more severe than other recessions (e.g., Reinhart and Rogoff, 2009; Jordà, Schularick, and Taylor, 2013). Many economists attribute this severity to bank distress, which amplifies the extent of the economic decline by contracting the supply of credit (e.g., Bernanke, 1983; Calomiris and Mason, 2003; Baron, Verner, and Xiong, 2021). Although a large empirical literature supports this view (e.g., Peek and Rosengren, 2000; Ivashina and Scharfstein, 2010; Becker and Ivashina, 2014; Chodorow-Reich, 2014), a general equilibrium model is required to evaluate the quantitative importance of this amplification effect. It is, however, challenging to examine the macroeconomic effects of bank distress using existing general equilibrium frameworks because they abstract from frictions between banks and depositors or combine the balance sheets of borrowers and banks. The former case results in the absence of banks; in the latter case, bank distress cannot be isolated from non-financial borrowers’ distress. This paper makes empirical, theoretical, and quantitative contributions to the growing litera- ture on financial amplification. Empirically, it documents that one defining feature of financial crises is asymmetric bank distress—the concentration of losses on a subset of banks—which has been overlooked in the literature.1 Theoretically, it proposes a model built on the Bernanke-Gertler costly-state-verification framework of financial frictions by incorporating a similar friction affect- ing deposit contracts between banks and depositors, and by introducing two types of borrowers exposed to different magnitudes of idiosyncratic risk. Banks are characterized by the fraction of riskier entrepreneurs in their loan portfolios, implying that they are heterogeneous and not perfectly diversified. Adverse shocks to riskier borrowers generate asymmetric bank distress because losses due to rising default rates are concentrated on banks heavily exposed to riskier borrowers. This asymmetry in bank distress amplifies economic downturns through two channels. First, the con- centration of losses on some banks causes costly bank failures, reducing the aggregate net worth of banks to a greater extent. Second, it impairs banks’ leverage capacity by raising uncertainty about their repayment capacity due to asymmetric information between banks and depositors. This re- sults in a contraction in the supply of loans, which generates a rise in spreads and a subsequent drop in investment. Quantitative analysis suggests that this amplification mechanism contributed sig- nificantly to the observed fluctuations in macro and financial variables during the Great Recession. 1 Some studies have documented heterogeneity in bank losses during the Great Depression and the Great Recession, although it was not the main focus of those papers. For example, Wicker (1980) documents that failures of banks in 1930 were local phenomena reflecting regional shocks or the specific risk exposure of a small subset of banks. Ivashina and Scharfstein (2010) and Chodorow-Reich (2014) exploit this heterogeneity to isolate supply shocks during the Great Recession. Liu, Moon, and Schorfheide (2022) also find heterogeneity in loan losses across U.S. banks even after controlling for regional economic conditions. However, none of these papers focus on asymmetry in bank distress or explore its macroeconomic consequences. 1 The mechanism accounts for 50% of the drop in investment and 60% of the increase in spreads. This paper starts by documenting that asymmetry in bank distress was observed both in the United States and across countries during financial crises. Profitability and loan quality deteriorated substantially more for the bottom 25% of U.S. bank holding companies during the Great Recession than for the others, and the cross-sectional distribution of stock returns for U.S. financial firms was significantly more dispersed and left-skewed during the recessions of the early 1990s and 2007– 2009. Similar patterns are also observed in the cross-sectional distribution of stock returns for commercial banks in other countries during the Asian financial crisis in the late 1990s, the global financial crisis, and the Eurozone debt crisis. To capture this empirical regularity, this paper examines a DSGE model in which shocks to the cross-sectional dispersion of borrowers’ productivity generate asymmetric losses in credit- supplying banks. The model separates two sectors—non-financial borrowers and banks—to study the joint role of their balance-sheet frictions, building on models of the financial accelerator pi- oneered by Bernanke and Gertler (1989) (hereafter BG), Bernanke, Gertler, and Gilchrist (1999) (hereafter BGG), and Christiano, Motto, and Rostagno (2014) (hereafter CMR), and widely used in the literature. The underlying financial contract used in those models is modified to add frictions on the banks’ side and then embedded into an otherwise standard RBC model.2 In the standard financial accelerator models, borrowers’ net worth plays a critical role, but banks are just a pure pass-through of funds because (i) they do not bear aggregate risk; (ii) they perfectly diversify id- iosyncratic risk; and (iii) they do not face financial frictions in raising external funds. As a result, the loan supply curve is fully elastic at the risk-free rate in these models. In contrast, the model in this paper gives a nontrivial role to the banking sector and generates an upward (downward) sloping loan supply (demand) curve with respect to the required return on loans. The first key model element is a non-state-contingent loan contract, in which aggregate risk must be shared between borrowers (entrepreneurs) and financial intermediaries (bankers). They sign a loan contract based on their expectations of future economic conditions. Bankers’ returns are not predetermined, but the contractual lending rate is fixed. In contrast, in the standard BG contract, the lending rate is state-contingent and bankers’ returns are predetermined because entrepreneurs guarantee a safe rate of return to the bankers. This standard contract generates a financial accelerator effect associated with entrepreneurs’ net worth, but the loan contract in this paper adds a new financial accelerator effect related to bankers’ net worth. Loan defaults induce losses for bankers, thereby reducing their ability to extend loans to entrepreneurs. 2 The main friction in these models is “costly state verification,” by which lenders must pay a fixed monitoring cost in order to observe an individual borrower’s realized return that is private information (see also Townsend, 1979 and Gale and Hellwig, 1985). 2 The second essential ingredient is the existence of different types of entrepreneurs and bankers, which enables the model to deviate from the perfect diversification assumption and motivates the financial friction between bankers and depositors. There are two types of entrepreneurs: low- and high-volatility types. While both types’ idiosyncratic productivity has the same mean, the high- volatility type is exposed to higher cross-sectional dispersion. Bankers also differ in type, and their type is characterized by the fraction of high-volatility entrepreneurs in their loan portfolios, which implies that each banker’s loan portfolio deviates from being perfectly diversified. The riskiness of bankers is inherently different, but their types are unknown ex-ante and inferred privately ex- post.3 This adds another layer of asymmetric information between bankers and households, thereby constraining bankers’ capacity to leverage. The fact that banks are not perfectly diversified contrasts with Diamond (1984), which shows that depositors are not incentivized to monitor the bank due to perfect diversification. In this model, however, depositors do have the incentive to monitor bankers. Despite imperfect diversification, bankers can ensure a risk-free return to depositors by diversifying idiosyncratic risk up to a certain degree and absorbing unexpected losses with their net worth. Diversification, although not perfect, allows bankers to take on much higher leverage than non-financial borrowers at a lower borrow- ing rate. Asymmetric information between bankers and households, which is well supported in the literature, is critical for the mechanism to operate. Theoretically, bank assets are opaque as a by-product of diversification (Diamond, 1984), and this opacity is essential for banks to produce money-like safe liquidity (Dang, Gorton, Holmström, and Ordoñez, 2017). This opacity, however, bites them back during financial crises. When some banks are in distress due to the concentration of losses, borrowing costs rise for all banks because depositors do not know which banks will default, and therefore all banks deleverage. The main shock in the model is an increase in the cross-sectional dispersion of idiosyncratic productivity of high-volatility entrepreneurs, because it generates asymmetric bank-level loan qual- ity. This shock contracts loan supply through two separate mechanisms: the deleveraging effect and the net worth effect. Via the deleveraging effect, the shock impairs bankers’ leverage capacity by increasing uncertainty about the repayment possibilities of banks, as it lowers the return on the loan portfolio of the worst-performing banker the most. The bankers’ funding rate remains at the risk-free rate as long as their leverage does not exceed the level that the worst-performing banker is expected to be able to repay. Once a banker’s leverage exceeds that threshold, the funding rate rises sharply due to the possibility of default. Thus, bankers’ optimal leverage is closely tied to the return of the worst-performing banker. However, the dispersion shock deteriorates the loan quality 3 The heterogeneity here can be interpreted as publicly unobservable screening efficiency (Boissay, Collard, and Smets, 2016) or risk-taking behavior (Coimbra and Rey, 2021). 3 of the most vulnerable banker due to its exposure to high-volatility entrepreneurs. The dispersion shock reduces bankers’ net worth by lowering realized returns on loans. This net worth effect has a nonlinear and asymmetric impact on the economy. In particular, since losses due to the higher default rate are concentrated on fragile bankers, it causes higher banker failures. The bankers’ defaults then raise the funding rate, which in turn induces more defaults. Thus, a kind of multiplier effect arises: initial shocks are amplified because they cause bank failures. The magnitude of this amplification effect depends largely on the size of the bankruptcies triggered by the initial shock, implying that the net worth effect is nonlinear. On the other hand, the effect of a decrease in dispersion does not amplify because it does not lead to changes in the banker default rate. Hence, the net worth effect is asymmetric. For this reason, the shock contracts loan supply by undermining both the bankers’ ability to leverage and the aggregate net worth of bankers. One of the novel features of the model is that asymmetric bank distress is explicitly linked to fluctuations in credit spreads for non-financial borrowers. This is consistent with the empirical findings of Gilchrist and Zakrajšek (2012), who show that a substantial portion of the increase in U.S. corporate credit spreads during the Great Recession can be attributed to a deterioration in the balance sheet conditions of financial intermediaries—referred to as the “excess bond pre- mium (EBP).” Saunders, Spina, Steffen, and Streitz (2021) confirm a similar pattern in the U.S. secondary corporate loan market, which they refer to as the “excess loan premium (ELP).” In the model, the credit spread can be decomposed into two parts: the default risk premium and the excess bond/loan premium. In particular, while the default risk premium rises in response to an increase in entrepreneurs’ default risk, the excess bond/loan premium is strongly associated with bank distress. This latter link, which can be more powerful than the former mechanism, has not been considered in existing quantitative macro-finance models. To evaluate the quantitative importance of the model’s amplification mechanism, the model is calibrated to the U.S. economy for the period 1985:Q1–2014:Q4. First, the behavior of economic aggregates in the model is compared against the data. Given the observed data and the model so- lution with calibrated parameters, innovations are first extracted using an inversion filter. Then, the path of endogenous variables is recovered by feeding the extracted innovations into the model. The model accounts reasonably well for the observed fluctuations in macro and financial variables. Next, the extracted innovations are fed into a version of the model with unconstrained banks, which can be seen as a standard financial accelerator model, and the responses are compared with those produced by the full model to highlight the quantitative effects of the asymmetric bank distress amplification mechanism. The results show that the amplification mechanism contributed signif- icantly to the observed fluctuations in macro and financial variables during the Great Recession. A comparison of the two models shows that the model with asymmetric bank distress doubles the 4 decline in investment and produces 2.5 times the rise in spreads, highlighting the quantitative im- portance of the asymmetric bank distress mechanism. In addition, the model with asymmetric bank distress generates reasonable dynamics of financial variables such as leverage and default in the banking sector, which is infeasible with standard financial accelerator models. Related Literature. This paper is related to several strands of the macro-finance literature. First, it contributes to the literature on quantitative macro models with financial frictions (see, e.g., Bernanke and Gertler, 1989; Kiyotaki and Moore, 1997; Carlstrom and Fuerst, 1997; Bernanke, Gertler, and Gilchrist, 1999; Mendoza, 2010; Gertler and Kiyotaki, 2011; Gertler and Karadi, 2011; He and Krishnamurthy, 2013; Brunnermeier and Sannikov, 2014; Gertler, Kiyotaki, and Prestipino, 2016; Nuño and Thomas, 2017; Di Tella, 2017; Gertler, Kiyotaki, and Prestipino, 2020). Since these models assume no friction between banks and depositors, the financial accelerator effect is associated only with the net worth of the non-financial sector.4 This paper adds an additional fi- nancial accelerator effect arising from banks’ net worth to such a model. This allows the model to address the joint dynamics of balance sheet frictions of non-financial borrowers and banks. A weakening of borrowers’ balance sheets deteriorates banks’ balance sheets, which impairs banks’ ability to intermediate. This leads to a higher cost of external financing for borrowers, which weak- ens borrowers further. Second, this paper builds on the literature that studies the macroeconomic effects of uncer- tainty shocks in models with financial frictions (see, e.g., Ordoñez, 2013; Christiano, Motto, and Rostagno, 2014; Gilchrist, Sim, and Zakrajšek, 2014; Arellano, Bai, and Kehoe, 2019). Along with the real option channel (Bloom, 2009; Schaal, 2017; Bloom, Floetotto, Jaimovich, Saporta-Eksten, and Terry, 2018) and the precautionary savings channel (Fernández-Villaverde, Guerrón-Quintana, Rubio-Ramírez, and Uribe, 2011), the financial frictions channel is identified in these models as a significant mechanism through which uncertainty shocks are transmitted to the real economy. This paper highlights the importance of uncertainty in the financial sector during financial recessions. Third, this paper contributes to the existing literature showing the importance of “financial shocks”—exogenous shocks that directly shift a parameter of the credit constraint—in aggregate fluctuations (see, e.g., Jermann and Quadrini, 2012; Khan and Thomas, 2013; Boz and Mendoza, 2014). In contrast to these models, the borrowers’ borrowing condition in this paper is endoge- nously linked to cross-sectional dispersion shocks through fluctuations in both the default risk premium and the excess loan premium. When banks’ capacity to leverage and their net worth deteriorates, banks require a higher return on loans (a rise in the excess bond premium), which can 4 Some of these studies focus on frictions between banks and depositors but combine the balance sheets of borrowers and banks. Hence, it is hard to evaluate whether an aggregate decline in bank credit is due to a contraction in credit supply or simply low demand for credit. 5 be interpreted as a “financial shock.” Fourth, the mechanism of the model in this paper is consistent with other empirical studies that emphasize the link between credit spreads and economic activity (see, e.g., Gilchrist and Zakrajšek, 2012; Krishnamurthy and Muir, 2020; Akinci and Queralto, 2022). In the model, a sharp rise in credit spreads operates as a financial shock that amplifies economic downturns. In particular, the model incorporates the findings of Gilchrist and Zakrajšek (2012) and Saunders, Spina, Steffen, and Streitz (2021) in showing that a large part of the increase in credit spreads—independent of borrowers’ default risk—is attributed to bank distress during financial crises. Fifth, this paper is closely related to several recent studies that separate two sectors and show that a powerful financial accelerator mechanism arises from intermediaries’ net worth dynamics (e.g., Elenev, Landvoigt, and Nieuwerburgh, 2021; Mendicino, Nikolov, Rubio-Ramirez, Suarez, and Supera, 2020). However, their models differ from the one examined here in that banks issue risk-free debt guaranteed by the government and are subject to a regulatory leverage constraint. In contrast, in this paper’s model, banks face an endogenous leverage constraint that is linked to uncertainty in the banking sector, and fluctuations in this leverage constraint are essential to the mechanism. Sixth, this paper also belongs to the literature focusing on heterogeneity in financial intermedi- aries. Ferrante (2018), Ma (2019), and Begenau and Landvoigt (2022) explore the role of institu- tional heterogeneity in determining financial stability, asset prices, and the optimal capital require- ment, respectively. Coimbra and Rey (2021) study the risk-taking channel of monetary policy with heterogeneous banks due to different risk attitudes. Corbae and D’Erasmo (2021) analyze how bank concentration interacts with capital requirements using a model with imperfect competition in loan markets. In Bianchi and Bigio (2022), who study how monetary policy operates through the banking system, banks are heterogeneous due to random liquidity shocks. This paper highlights ex-post asymmetry in realized losses during financial recessions. Finally, this paper supports the empirical findings of Ferreira (2016, 2019), who shows that financial dispersion and skewness are powerful predictors of business cycle fluctuations using U.S. stock returns data. While Ferreira (2016, 2019) focuses on U.S. stock market data, this paper confirms similar patterns using U.S. banks’ balance sheet data and international data. The rest of the paper is organized as follows. Section 2 documents the main stylized fact related to asymmetric bank distress during crises. Section 3 develops the financial contract framework in a partial equilibrium setting and describes the model’s mechanism. Section 4 lays out the general equilibrium model. Section 5 presents the calibration and quantitative results. Section 6 concludes. 6 2. Asymmetric Bank Distress during Financial Crises This section documents the considerable asymmetry in bank distress observed in the United States and across countries during financial crises. Three data sets are used: (1) U.S. bank holding company (BHC) balance sheet data from the Consolidated Financial Statements for Bank Holding Companies (FR Y-9C); (2) U.S. stock market returns from the Center for Research in Security Prices (CRSP) database; and (3) stock market returns for 21 countries, including the United States, from the Worldscope database. Figure 1 shows the cross-sectional distribution of two key performance indicators of BHCs: (a) Net Income to Assets Ratio (ROA) (b) Net Charge-off to Loans Ratio (c) Net Income to Assets Ratio (ROA) (d) Net Charge-off to Loans Ratio Figure 1. Cross-sectional distribution of U.S. BHCs Performance-Indicators Notes: Each year includes only BHCs that survived the entire year, and values below the 1 percent and above the 99 percent level are not plotted. The number of observations varies from 3,988 to 4,118 and from 6,701 to 8,553 for the 2007-2009 and 2001 recessions, respectively. Source: FR Y9-C 7 return on assets (i.e., the net income-to-assets ratio) and the net charge-off-to-loans ratio during the recessions of 2001 and 2007–2009. The 2001 recession is considered a non-financial recession, while the 2007–2009 recession is viewed as a financial recession. Plots (a) and (b) correspond to the latter, and (c) and (d) to the former. The goal is to compare the cross-sectional distribution of BHCs’ performance during the financial recession with that during the non-financial recession. Panels (a) and (b) show that BHCs’ performance is asymmetric during the financial recession, with the bottom 25% of BHCs performing extremely poorly. Deterioration in their profitability and loan quality is substantial.5 This pattern, on the other hand, is not observed during the 2001 recession (see panels (c) and (d)). Hence, these plots suggest that asymmetric bank distress is a striking feature of the 2007–2009 financial recession, distinguishing it from the non-financial recession of 2001. Consider next the cross-sectional distribution of stock returns taken from the CRSP. The sample covers the period from 1960 to 2019, but the analysis focuses on the period between 1984:Q1 and 2019:Q4, following several data adjustments, because it has been widely documented that there was a structural break in the volatility of U.S. business cycle fluctuations and major regulatory changes in the U.S. financial system around 1984 (see, e.g., Jermann and Quadrini, 2006; Fuentes-Albero, 2019). Figure 2 shows how the cross-sectional dispersion in the non-financial and financial sectors Figure 2. Dispersion in U.S. non-financial and financial firm equity returns Notes: Each series represents the 4-quarter moving average of the interquartile range of quarterly stock returns for non-financial and financial firms, respectively. The gray areas indicate periods classified as recessions by the NBER. Source: The Center for Research in Security Prices (CRSP) 5 Note that this is not simply a case of a large number of small banks performing badly. Figure D.1 in Appendix D confirms that a similar pattern is observed when the sample is restricted to BHCs with assets above $5 billion, and there is indeed a negative correlation between a BHC’s return on assets and its size. 8 evolves over the business cycle, measured by the interquartile range of quarterly log stock returns.6 The figure is in line with the view that recessions are associated with positive second-moment shocks in the non-financial sector, as documented in previous studies (e.g., Bloom, 2009; Bloom et al., 2018). The cross-sectional dispersion in the non-financial sector indeed rises for all three recessions during the period. In contrast, the cross-sectional dispersion of returns in the financial sector increases only during the two financial recessions: the early 1990s and the Great Recession.7 As the comparison of BHC performance between the 2001 and 2008 recessions in Figure 1 suggests, this pattern also supports the view that financial recessions differ from non-financial recessions in that they involve substantial uncertainty regarding the state of the financial sector. Focusing only on the cross-sectional dispersion of returns in the non-financial sector makes it difficult to explain why the 2008 recession was much more severe than the 2001 recession. Figure 3. Distribution of cross-country equity returns: Non-crisis vs Crises Notes: Crisis periods are defined by Baron, Verner, and Xiong (2021) and 21 countries are covered: AUS, AUT, DNK, FRA, DEU, GRC, HKG, IND, IDN, ITA, JPN, KOR, MYS, NOR, PHL, ESP, CHE, TWN, THA, GBR, USA. The period spans from 1985 to 2015 but many early period observations are missing for many countries due to the lack of data. See Table D.1 in Appendix D for details. Source: Worldscope 6 Firms are classified first into two categories based on NAICS codes, following Ferreira (2016). Firms with NAICS codes 522 (credit intermediation and related activities) or 523 (securities, commodity contracts, and other finan- cial investments and related activities) are classified as financial firms, and firms with other codes—excluding 524 (insurance carriers and related activities) and 525 (funds, trusts, and other financial vehicles)—are classified as non- financial firms. SIC codes from the U.S. Census website associated with the 3-digit NAICS codes 522, 523, 524, and 525 are used for firms without NAICS codes to classify them accordingly. Firms that switched between sectors during the sample period are eliminated. Firms with fewer than five consecutive years of data and observations with returns higher than 50% or lower than –50% are also dropped. In each quarter, firms are dropped if they have fewer than three monthly return observations. Finally, monthly returns for each quarter are compounded to construct quarterly returns for each firm. 7 Note that this dispersion is associated with asymmetry, as it is left-skewed. This can be seen in Figure D.2 in Appendix D, which displays the cross-sectional distribution for two specific quarters—before the recession and during the peak of the recession—for each episode. 9 Similar patterns are observed in other countries during financial crises. This is documented us- ing annual stock returns across 21 countries, taken from the Worldscope database.8 Figure 3 shows how the cross-sectional distribution of annual equity returns for non-financial firms and commer- cial banks changes during financial crises across 21 countries.9 As in the U.S. case, the distribution is more dispersed and left-skewed during crises, and this pattern is particularly pronounced for the banking sector. In addition, Figure 4 presents similar patterns for selected individual countries’ cri- sis episodes. The first column of Figure 4 confirms the pattern for the Republic of Korea, Malaysia, and Thailand during the Asian Financial Crisis in the late 1990s. The second column corresponds to advanced countries (the United States, the United Kingdom, and Germany) during the Global Financial Crisis. The third column displays the pattern for Spain, Italy, and Greece during the European Debt Crisis. Finally, the last column shows Japan during the early 1990s crisis and the Asian Financial Crisis, and Denmark during the 1990s financial crisis in the Nordic countries. Finally, Table 1 examines more systematically the relationship between the cross-sectional dis- tribution of returns in the financial sector and financial crises. Columns (1)–(3) focus on U.S. stock Figure 4. Cross-sectional distribution of commercial bank equity returns for selected events Source: Worldscope 8 These countries are selected based on data availability. Starting with the list of countries included in the analysis of banking crisis episodes by Baron, Verner, and Xiong (2021), countries with too few commercial banks in the sample—roughly 10—for most of the sample period are excluded. Some periods for countries included in the sample are also dropped when there are not enough banks to compute the cross-sectional dispersion. See Table D.1 in Appendix D for details. 9 The median value of each distribution is subtracted for comparison. 10 Table 1. Dispersed and skewed bank distress during financial crises US CRSP Worldscope (1) (2) (3) (4) (5) (6) IQR KSK Skewness IQR KSK Skewness Crisis 0.137∗∗∗ -0.154∗∗∗ -0.126 0.161∗∗∗ -0.185∗∗ -0.660∗∗∗ [11.1] [-3.70] [-0.50] [2.91] [-2.38] [-3.32] Quarter fixed effects ✓ ✓ ✓ Country fixed effects ✓ ✓ ✓ Adj. R2 (within) 0.46 0.12 0.08 0.03 0.02 0.01 Frequency Quarterly Quarterly Quarterly Annual Annual Annual Years 1984-2019 1984-2019 1984-2019 1985-2015 1985-2015 1985-2015 Num of Countries 1 1 1 21 21 21 Observations 144 144 144 679 675 659 Underlying Obs. 77,854 77,854 77,854 34,214 34,214 34,214 Note: Crisis periods are defined by Baron, Verner, and Xiong (2021). IQR = p75 − p25 is the interquartile range of quarterly log stock return and KSK = (p90−pp 50)−(p50−p10) 90−p10 is the Kelley skewness. Numbers in brackets are t-statistics and *,**,*** indicate significance at the 0.1, 0.05, and 0.01 levels, respectively. returns. In column (1), the interquartile range (IQR) of quarterly stock returns for financial firms, which measures cross-sectional dispersion, is regressed on a financial crisis indicator. The regres- sion coefficient is statistically significant, with a point estimate of 0.137, indicating that a quarter classified as a financial crisis is associated with a 13.7% increase in the dispersion of financial firms’ equity returns. In columns (2) and (3), the Kelley skewness and the standard skewness of stock returns are regressed on the same financial crisis indicator, respectively.10 The statistically significant negative coefficient in column (2) suggests that the distribution is negatively skewed and that dispersion is mainly driven by the left tail, confirming asymmetry in the financial sector. The negative coefficient in column (3) supports the result in column (2), although it is statistically insignificant at the 10% level. Columns (4)–(6) report the results of the same regressions using annual stock returns for commercial banks across 21 countries and find similar patterns. Overall, the results indicate that there is considerable asymmetry in bank distress during financial crises. 3. A Model of Asymmetric Bank Distress This section describes the core mechanism of the model by laying out the loan and deposit contracts under a costly state verification problem in a partial equilibrium setting. The price of capital, entrepreneurs’ net worth, bankers’ net worth, the expected aggregate return to capital, and 10 The Kelley skewness is defined as (p90−pp 50)−(p50−p10) 90−p10 and varies between –1 and 1. It provides a simple measure to evaluate whether dispersion is mainly accounted for by the left or right tails of the distribution. For example, a negative (positive) value indicates that the distribution is negatively (positively) skewed, with more than half of the dispersion accounted for by the left (right) tail. 11 the risk-free interest rate are exogenously given. These variables, however, will be endogenously determined in general equilibrium later in the paper. The analysis first shows that these financial contracts generate a downward-sloping loan demand curve and an upward-sloping loan supply curve, which together characterize the equilibrium of the loan market. The effects of cross-sectional dispersion shocks on this equilibrium are then examined. In the next section, this framework is embedded into an otherwise standard Real Business Cycle (RBC) model. 3.1. Environment Bankers and Entrepreneurs: There is a continuum of mass one of risk-neutral entrepreneurs, in- dexed by i ∈ (0, 1), each with net worth ni t in units of consumption goods, and a continuum of mass one of risk-neutral bankers, indexed by j ∈ (0, 1), each with net worth nj b,t in units of consumption goods. Each entrepreneur has access to a stochastic technology that transforms one unit of capital goods i i into ωt +1 Rk,t+1 units of consumption goods in the next period, where ωt+1 is the idiosyncratic component of entrepreneur i’s return and Rk,t+1 is the aggregate return to capital. The two are +1 ∈ [0, ∞) follows a unit-mean log-normal distribution that is independently i uncorrelated, and ωt drawn across time and across entrepreneurs. There are two types of entrepreneurs: low-(L) and high-(H ) volatility types. These different types are introduced to depart from the standard BG assumption that banks perfectly diversify idiosyncratic risk. Let the entrepreneur’s type be denoted by θ ∈ {L, H }. While both types have idiosyncratic return components with the same mean of unity, the H types are exposed to higher cross-sectional dispersion: 1 2 2 i log(ωθ,t ) ∼ N − σθ,t , σθ,t where θ ∈ {L, H } and σH,t > σL,t . 2 Here, σθ,t is the time-t cross-sectional standard deviation of idiosyncratic productivity for each type, commonly referred to as “uncertainty,” “volatility,” or “risk” in the literature. For simplicity, σH,t and σL,t are assumed to be independent. Denote the probability density function and cumulative distribution function of the log-normal distribution of idiosyncratic productivity for each type at time t by ftθ and Ftθ , respectively. i At time t, each entrepreneur i purchases physical capital kt +1 for use at t + 1 at the unit market price Qt and this purchase is financed with her net worth ni i t and with external financing bt from a banker through the loan contract specified below. Thus, each entrepreneur’s balance sheet at time t is given by i i i Q t kt +1 = nt + bt . 12 In contrast to entrepreneurs, bankers do not have access to the production technology but can borrow resources from households, who are willing to lend at the risk-free interest rate Rf,t . Thus, at time t, each banker j borrows dj t from households through the deposit contract to be specified later and lends bj t to a continuum of entrepreneurs. Their balance sheet is: bj j j t = nb,t + dt . The existence of bankers in this economy can be justified for two reasons. First, bankers may be more efficient than households at monitoring entrepreneurs. For example, Diamond and Rajan (2001) show that banks monitor borrowers more effectively than others because their fragile deposit structure gives them the right incentives to do so. Second, banks can ensure a risk-free return to depositors by diversifying idiosyncratic risk up to a certain degree and absorbing unexpected losses with their net worth.11 Note that bankers in the model do not have perfectly diversified loan portfolios. This contrasts with Diamond (1984), who shows that depositors are not incentivized to monitor the bank due to perfect diversification. Information Structure and Financial Frictions: Denote the fraction of high-type entrepreneurs in the economy by πH , which is assumed to be common knowledge. Each entrepreneur i draws i i her type θt +1 before her idiosyncratic productivity ωt+1 is realized. The entrepreneurs’ types are unknown and unverifiable to all agents, including the entrepreneurs themselves. However, en- i trepreneurs can observe their own ωt +1 ex-post privately, while bankers must pay a fixed monitoring i i i cost µωt +1 Rk,t+1 Qt kt+1 to observe ωt+1 . This assumption generates ex-post asymmetric informa- tion between entrepreneurs and bankers regarding the repayment possibilities of entrepreneurs. To motivate a nontrivial financial friction for the deposit contract, assume that there is a con- tinuum of types of bankers, where type represents the quality of the loan portfolio and is ex-post t ∼ U [0, 1], which represents the private information. Each banker’s type is a random draw pj percentile ranking of the bank’s loan portfolio when loan quality is sorted from lowest to highest (i.e., p = 1 is the highest quality, and p = 0 is the lowest). Specifically, type p is characterized by a fraction h(p) of high-volatility-type entrepreneurs in the loan portfolio, where h(p) satisfies h(0) = 1, h(1) = 0, h′ (p) < 0, and h′′ (p) > 0.12 This assumption parsimoniously captures the idea that financial intermediaries often hold assets with correlated risks, leading to a deviation from perfectly diversified loan portfolios.13 Because high-volatility-type entrepreneurs are exposed to 11 To keep the problem tractable, aggregate bankers’ net worth is assumed to be sufficiently large to absorb any losses on loans so that the banking system as a whole does not collapse in equilibrium. 12 Assuming the convexity of the function h yields a concave shape of cross-sectional returns on bank loans, as observed in the data (see Figure 1). The assumptions h(0) = 1 and h(1) = 0 are not essential and are made for simplicity. 13 There are a number of explanations for why intermediaries hold correlated assets. For example, Phelan (2017) 13 Figure 5. Banker’s type Notes: p ∈ [0, 1] denotes bankers’ type and h(p) indicates a fraction of high-type entrepreneurs in the loan portfolio of banker type-p. greater cross-sectional dispersion, their default rate is higher than that of low-volatility-type en- trepreneurs. Thus, the risk of the loan portfolio decreases with p. This relationship is illustrated t ∼ U [0, 1] is assumed to be independently drawn across time and in Figure 5. For tractability, pj across bankers. To be consistent with the assumption that the fraction of H -type entrepreneurs in 1 the economy is πH , it is required that 0 h(p) dp = πH . Suppose the expected return on the perfectly diversified loan portfolio in this economy is given by Rt+1 . Then, the expected return on the loan portfolio for a banker of type p can be defined as q (p)Rt+1 , where q (p) captures the idiosyncratic return component and satisfies q ′ (p) > 0. The equation for q (p) and the derivation of q ′ (p) > 0 and q ′′ (p) < 0 are introduced below in the context of the deposit contract. Bankers can infer their own type ex-post privately based on default rates in their loan portfo- lio but households must pay a fixed “monitoring cost” µb q (pj j t )Rt+1 bt to learn it. Similar to the monitoring cost between entrepreneurs and bankers, this assumption yields ex-post asymmetric information between bankers and households about the repayment possibilities of bankers. This assumption is consistent with the idea that financial intermediaries’ assets are opaque and hard to evaluate (see e.g., Dang, Gorton, Holmström, and Ordoñez, 2017). Finally, all lenders (bankers and households) are assumed to be committed to monitoring if borrowers (entrepreneurs and bankers) default, and stochastic monitoring is ruled out. As a result, the standard debt contract is optimal, as in Townsend (1979) and Gale and Hellwig (1985). In what develop a theory showing that financial intermediaries hold correlated assets because knowing how loan payoffs are correlated is valuable for managing investments in a costly enforcement setting. Other explanations are based on fixed costs of developing expertise in a specific industry or geographic region. 14 follows, the optimal loan and deposit contracts under no uncertainty are first described, and then the contracting framework is modified to incorporate the possibility of uncertainty. 3.2. Loan Contract without Aggregate Risk There are two parties to the loan contract: an entrepreneur i with net worth ni t and a banker j . Since both parties are risk-neutral, they care only about expected returns. After the entrepreneur borrows bi i t from the banker, her ability to repay the loan depends on the ex-post return to capital, ωt+1 Rk,t+1 . That is, the entrepreneur decides whether to repay the loan or default after observing the realization i i of ωt +1 , given Rk,t+1 . Let Rb,t denote the promised gross loan rate. The entrepreneur defaults for i i i sufficiently low values of the idiosyncratic shock, ωt ¯t +1 < ω ¯t +1 , where ω +1 is a default threshold defined as: t − 1) i Rb,t (ℓi +1 Rk,t+1 Qt kt+1 = Rb,t bt ⇐⇒ ω i i i i i ¯t ω ¯t +1 = , (1) Rk,t+1 ℓit t ≡ Qt kt+1 /nt is the leverage ratio of entrepreneur i. Using this expression for the default where ℓi i i threshold, the banker’s expected gross return on the loan can be written as follows: ∞ ω ¯ti +1 πθ i i Rb,t bt dFtθ +1 (ω ) + (1 − µ)ωRk,t+1 Qt kt i θ +1 dFt+1 (ω ) ω ¯ti 0 θ ∈{H,L} +1 ¯t ω i ω ¯ti +1 +1 = i Rk,t+1 Qt kt +1 πθ ¯t ω i +1 1− Ftθ +1 (¯ ωti +1 ) + ω +1 (ω ) −µ dFtθ ω dFtθ +1 (ω ) θ ∈{H,L} 0 0 ≡ Γ(¯ ωti +1 , σθ,t+1 ) ≡ G(¯ ωti +1 , σθ,t+1 ) +1 , σθ,t+1 ) − µG(¯ i i i i = Rk,t+1 ωt πθ Γ(¯ ωt +1 , σθ,t+1 ) ℓt nt , θ ∈{H,L} i i ωt where Γ(¯ ωt +1 , σθ,t+1 ) and µG(¯ +1 , σθ,t+1 ) denote, respectively, the expected gross share of total revenues going to the banker and the expected monitoring costs when the entrepreneur’s type is θ ∈ {L, H }. Thus, θ ∈{H,L} πθ Γ(¯ ωti +1 , σθ,t+1 ) − µG(¯ ωti +1 , σθ,t+1 ) represents the net share of total revenues going to the banker, and θ ∈{H,L} πθ 1 − Γ(¯ ωti +1 , σθ,t+1 ) represents the net share going to the entrepreneur. The expected gross return to the entrepreneur can be similarly defined by ∞ +1 − Rb,t bt dFt+1 (ω ) πθ 1 − Γ(¯ i i i θ i i i πθ ωRk,t+1 Qt kt = Rk,t+1 ωt +1 , σθ,t+1 ) ℓt nt . ω ¯ti θ ∈{H,L} +1 θ ∈{H,L} 15 Notice that both parties’ expected returns are linear in the entrepreneur’s net worth ni t . Given that the only known heterogeneity among entrepreneurs at the time the contract is signed is their level of net worth, the loan contract will be independent of entrepreneur-specific variables. That is, the loan contract is constant across entrepreneurs, implying that all entrepreneurs choose the same leverage ratio. Because of this linearity, only the aggregate net worth of entrepreneurs needs to be tracked.14 Hence, the entrepreneur index i and the individual net worth variable ni t can be dropped. Time subscripts are also omitted for notational convenience, unless necessary. Each party’s surplus from the loan contract can be defined as follows: Banker’s surplus : V B ≡ Rk ω , σθ ) ℓ − Rf (ℓ − 1), ω , σθ ) − µG(¯ πθ Γ(¯ (2) θ Entrepreneur’s surplus : V E ≡ Rk ω , σ θ ) ℓ − Rk . πθ 1 − Γ(¯ (3) θ Note that Rf in Equation (2) refers to both the risk-free rate and the banker’s funding rate, under the assumption that the banker can currently borrow at the risk-free rate. The second term in Equa- tion (3) reflects the fact that the entrepreneur can be self-financed. The loan contract environment described so far is equivalent to that in BGG, except for the presence of two types of entrepreneurs. Following BGG, the optimal loan contract maximizes the entrepreneur’s surplus subject to the banker’s participation constraint. Therefore, the loan contract is given by the solution to: max Rk ω , σ θ ) ℓ − Rk πθ 1 − Γ(¯ ℓ, ω ¯ θ s.t. Rk ω , σθ ) ℓ ≥ R(ℓ − 1), ω , σθ ) − µG(¯ πθ Γ(¯ θ where R ∈ [Rf , Rk ] is the required gross return on the loan by the banker. This is one of the key deviations of this model from other financial accelerator models in the literature, including BGG, in which R = Rf . Note that for any R ∈ (Rf , Rk ), the banker and entrepreneur share the surplus (i.e., VB > 0 and VE > 0). Similar to BGG, assume that Rk (1 − µ) < R to prevent the entrepreneur from obtaining unbounded profits by defaulting with probability one, and only equilibria without rationing are considered.15 While R is taken as given in the loan contract, it will be endogenously determined in equilibrium by aggregate loan supply and demand. Fluctuations in R are, in fact, the main channel through which disruptions in the financial sector are transmitted to the real sector of the economy. The first-order conditions of the above problem reduce to: 14 In general equilibrium, a proportional relationship between net worth and capital demand at the firm level is obtained by assuming a constant-returns-to-scale production technology, which yields a horizontal marginal benefit curve for investment. 15 The proof of the existence of an interior solution, which can be shown similarly to BGG, is omitted. 16 1 ℓ= s , (4) 1− b ω , σθ ) − µG(¯ θ πθ Γ(¯ ω , σθ ) s s λb (¯ ω) = , (5) θ πθ 1 − Γ(¯ ω , σθ ) − µG(¯ sb ω , σθ ) + λb (¯ ω ) θ πθ Γ(¯ ω , σθ ) where λb (¯ ω) = θ πθ Γω (¯ ω , σθ ) θ ω , σθ ) − µGω (¯ πθ {Γω (¯ ω , σθ )} is the Lagrange multiplier on the constraint, Γω (¯ ω , σθ ) = ∂ Γ(¯ ω , σθ )/∂ ω ¯ , Gω (¯ ω , σθ ) = ∂G(¯ ¯ , and s ≡ Rk /R and ω , σθ )/∂ ω sb ≡ R/Rf denote the “external finance premia” or the discounted return to capital and to lending for the entrepreneur and the banker, respectively. Note that s > sb and sb ≥ 1 in the competitive equilibrium in which entrepreneurs borrow from bankers to purchase capital (Rk > R), and bankers are willing to intermediate funds from depositors to entrepreneurs (R ≥ Rf ). Given that s > sb and sb ≥ 1, the first-order conditions above yield the following expression for the entrepreneur’s optimal leverage ratio: s ∂ψ ∂ψ s 1 ℓ≡ψ , with > 0 and <0 for ∈ 1, . (6) sb ∂s ∂sb sb 1−µ Equation (6) establishes two critical relationships in the model: a positive relationship between the entrepreneur’s external finance premium and the leverage ratio, and a negative relationship between the banker’s external finance premium and the entrepreneur’s leverage ratio. The former is the central insight of BG and BGG, while the latter is a key result of this paper. All else equal, a rise in the banker’s external finance premium sb leads the entrepreneur to take on less debt and reduce capital expenditures. From the entrepreneur’s perspective, an increase in sb can be interpreted as a financial shock, as the banker suddenly requires a higher return on loans regardless of the entrepreneur’s balance sheet condition. In the model’s equilibrium, sb increases when the banking sector is in distress. This mechanism will be explained in detail below. Notice that s can also be interpreted as the total surplus generated from the loan contract. Simi- larly, s/sb may denote the share of the surplus that the entrepreneur can keep for herself, and sb may represent the share of the surplus that the banker requires. Figure 6 displays how the total surplus generated from the loan contract is divided between two parties depending on the banker’s external finance premium sb . As the banker requires a higher share of the total surplus (an increase in sb ), the entrepreneur’s share falls. Thus, the entrepreneur borrows less and invests less. This results in a decline in the total surplus. On the other hand, the banker faces a trade-off. A rise in sb increases the banker’s share of the total surplus but reduces the total surplus. The former effect dominates the 17 Figure 6. Bank external finance premium and surplus sharing latter effect for sb < sb b M but the latter effect begins to dominate the former effect once s exceeds M . s ∈ [1, sM ] in the competitive equilibrium because the banker has no incentive to require sb b b sb > s b M. The fact that the banker’s surplus is maximized at sb b M implies that sM corresponds to the share required by a monopolistic banker. On the other hand, the case in which sb = 1 can be interpreted as a perfectly competitive banking sector. This is indeed the case in standard financial accelerator models such as BG and BGG, which assume that all bargaining power is given to the entrepreneur due to perfect competition in the banking sector. In contrast, Afanasyeva and Güntner (2020) con- sider a monopolistic loan market and assign market power to the banker by formulating the loan contract problem from the lender’s perspective. In this regard, the loan market in this model is sit- uated between these two extremes. In particular, the relative market power between the two parties is endogenously determined by aggregate loan supply and demand, and varies over the business cycle depending on bankers’ intermediation capacity. Greater competition in the banking sector— resulting from improved intermediation capacity during economic upturns—reduces their relative market power, and the opposite occurs during downturns. Thus, loan margins are countercyclical.16 ¯) Equation (1) shows that there is a one-to-one relationship between the default threshold (ω and the lending rate (Rb ). Thus, the loan contract can be fully characterized by the pair (ℓ, Rb ) that satisfies the following conditions: 16 This countercyclicality of loan margins has been documented across a large number of OECD countries by Olivero (2010), who shows that it plays an important role in the international transmission of aggregate TFP. 18 θ πθ Γ(¯ω , σθ ) − µG(¯ ω , σθ ) ℓ = 1 + λb (¯ ω) , (7) θ πθ 1 − Γ(¯ ω , σθ ) ℓ ¯ Rk Rb = ω , (8) ℓ−1 Equation (7) is obtained by combining Equation (4) and (5). Panel (a) of Figure 7 illustrates the loan contract in (ℓ, Rb )-space. It shows that the loan spread (Rb − Rf ) can be decomposed into two components: a component capturing the default risk premium (DRL, Rb − R), and a component representing the excess loan premium (ELP, R − Rf ). A positive relationship between the en- trepreneur’s leverage and DRL implies that higher leverage leads to higher lending rates, reflecting the increase in expected default costs associated with a greater debt-to-net-worth ratio. In contrast, ELP is constant with respect to leverage, indicating that it is orthogonal to the quality of the en- trepreneur’s balance sheet. Fluctuations in ELP reflect shifts in the supply of loans. For example, bank distress transmits to the real sector by raising ELP, which shifts the entire curve upward. Panel (a) of Figure 7 also shows that DRL increases when dispersion is higher for the same leverage ratio, because greater dispersion raises the default probability. The effect of dispersion is nonlinear in the entrepreneur’s leverage ratio, as default becomes more likely when leverage is higher for a given negative shock. The positive relationship between lending rates and dispersion can be characterized analytically using Propositions 1 and 2 below. Consider the banker’s partic- ipation constraint. If dispersion increases, the banker’s net share, θ ω , σθ ) − µG(¯ πθ {Γ(¯ ω , σθ )}, ¯ . Then, given ℓ, Rk , and R, the default threshold ω declines for a given ω ¯ must rise to satisfy the constraint.17 This leads to an increase in the lending rate Rb , as shown in Equation (8). ω , σθ ) − µG(¯ Proposition 1. Γ(¯ ω , σθ ) increases in ω ¯ for ω ¯ ∗ is a global maximizer. ¯ ∗ , where ω ¯<ω Proof. See Appendix A.1. ω , σθ ) − µG(¯ Proposition 2. Γ(¯ ¯ < 1. ω , σθ ) decreases in σθ for ω Proof. See Appendix A.2. 17 Note that the conditions for Propositions 1 and 2 always hold under the parameter values used later, because the ¯ is sufficiently low. As explained analysis is restricted to non-rationing equilibria in which the equilibrium value of ω by BGG, the borrower is rationed from the market if the equilibrium value of ω ¯ exceeds a global maximizer ω ¯ ∗ , as no value of ω¯ can satisfy the lender’s participation constraint in that case. 19 (a) Loan Contract (b) Deposit Contract Figure 7. Illustration of the loan and deposit contracts Notes: The following parameter values are used: Rk = 1.03, Rf = 1.005, µ = 0.2, µb = 0.4, πH = 0.1, σL = 0.2 and σH = 0.45. 3.3. Deposit Contract without Aggregate Risk The parties to the deposit contract are the bankers and the households. This partial equilibrium setting abstracts from the households’ problem by assuming that they are willing to lend fully elastically to bankers at the exogenous risk-free interest rate Rf , and that bankers ensure this risk- free return to households using their own net worth. From the banker’s participation constraint in the loan contract, the banker’s required return on the loan, R, can be written as ℓ R = Rk ω , σθ ) − µG(¯ πθ Γ(¯ ω , σθ ) . (9) ℓ−1 θ Notice that R is also the expected return on the perfectly diversified loan portfolio, or the aver- age return on loans, because the share of high-type entrepreneurs in this portfolio is πH —which is exactly the share of high-type entrepreneurs in the economy. Since the share of high-type en- trepreneurs in the loan portfolio of a banker of type p is h(p), the return on the loan portfolio for banker type p can be written as: ¯ , σ H , σ L )R q (p, ω ℓ = Rk ω , σH ) + (1 − h(p)) Γ(¯ ω , σH ) − µG(¯ h(p) Γ(¯ ω , σL ) − µG(¯ ω , σL ) , (10) ℓ−1 20 ¯ , σH , σL ) is the idiosyncratic return on the loan portfolio of a banker of type p. Then, where q (p, ω ¯ , σH , σL ) can be expressed as follows: using Equation (9) and (10), q (p, ω ω , σH ) + (1 − h(p)) Γ(¯ ω , σH ) − µG(¯ h(p) Γ(¯ ω , σL ) − µG(¯ ω , σL ) ¯ , σH , σL ) = q (p, ω . (11) πH Γ(¯ ω , σH ) + (1 − πH ) Γ(¯ ω , σH ) − µG(¯ ω , σL ) − µG(¯ω , σL ) Equation (11) shows that the return on loans for a banker of type p deviates from R (i.e., q (p) ̸= 1) ¯ , σH , σL ) measures the return on the loan portfolio of banker type unless h(p) = πH . Thus, q (p, ω p relative to the average return on loans. ¯ , σH , σL ) is increasing in p and concave: It can be easily shown that q (p, ω Rk ℓ q ′ ( p) = h ′ ( p) ω , σH ) − Γ(¯ ω , σH ) − µG(¯ Γ(¯ ω , σL ) − µG(¯ ω , σL ) > 0, R ℓ−1 (− ) (−) (+) Rk k q ′′ (p) = h′′ (p) ω , σH ) − Γ(¯ ω , σH ) − µG(¯ Γ(¯ ω , σL ) − µG(¯ ω , σL ) < 0. R k−1 (+) (− ) (+) More importantly, asymmetric bank losses arise when σH increases, because the decrease in q (p) is larger for lower values of p. An increase in σH causes high-type entrepreneurs to default more, thereby lowering the return on loans—especially for bankers who are heavily exposed to high-type entrepreneurs (i.e., those with lower p). This is illustrated in Figure 8. Similar to the loan contract, banker j ’s ability to repay the deposit depends on the return on loans, q (pj , ω ¯j can be defined as: ¯ , σH , σL )R. Given R, a default threshold p (ℓb − 1) j j Rd j q (¯ p ,ω ¯ , σH , σL )Rb = j j j Rd d ⇐⇒ q (¯ p ,ω j ¯ , σH , σL ) = , (12) Rℓjb b ≡ b /nb is the leverage ratio of banker j , and Rd is banker j ’s funding rate, which where ℓj j j j depends on the expected default rate. Note that the banker’s expected default rate is zero if ¯ , σH , σL )Rbj ≥ Rd q (0, ω j j d. The household’s expected return on the deposit is given by 1 ¯j p j j Rd d dp + (1 − µb )q (p, ω ¯ , σH , σL )Rb j dp ¯j p 0 21 Figure 8. Idiosyncratic return on loans Notes: p ∈ [0, 1] denotes bankers’ type and q (p) indicates idiosyncratic return on the loan portfolio of banker type-p. The dashed-line displays idiosyncratic returns when σH increases. ¯j p ¯j p = Rb j q (¯ j ¯ , σH , σL ) 1 − p p ,ω ¯ j + ¯ , σH , σL ) dp −µb q (p, ω ¯ , σH , σL ) dp q (p, ω 0 0 ≡ Y (¯ p j, ω ¯ , σH , σL ) ≡ L(¯ p j, ω ¯ , σH , σL ) = R Y (¯ ¯ , σH , σL ) − µb L(¯ p j, ω ¯ , σH , σL ) ℓbj nbj , p j, ω and the banker j ’s return on the deposit is given by 1 ¯ , σH , σL )Rb j − Rd q (p, ω d dp = R 1 − Y (¯ j j ¯ , σH , σL ) ℓbj nbj , p j, ω ¯j p where Y (¯ ¯ , σH , σL ) and µb L(¯ pj , ω pj , ω ¯ , σH , σL ) denote the expected gross share of total revenues going to the household and the expected monitoring costs, respectively. For the same reason dis- cussed in the loan contract, the banker index j and the net worth variable nj b can be dropped. All the economic rents generated by the deposit contract are assumed to accrue to the banker. Thus, the household’s required return is equal to the risk-free rate Rf , and the optimal deposit 22 contract solves: max R 1 − Y (¯ ¯ , σ H , σ L ) ℓb − R p, ω ¯ ℓb , p s.t. R Y (¯ ¯ , σH , σL ) − µb L(¯ p, ω ¯ , σH , σL ) ℓb ≥ Rf (ℓb − 1). p, ω The first-order conditions of this problem reduce to 1 ℓb = , (13) 1 − sb Y (¯ ¯ , σH , σL ) − µb L(¯ p, ω ¯ , σH , σL ) p, ω λd sb = , (14) 1 − Y (¯ ¯ , σH , σL ) + λd Y (¯ p, ω ¯ , σH , σL ) − µb L(¯ p, ω ¯ , σH , σL ) p, ω where λd = Yp (¯ ¯ , σH , σL ) {Yp (¯ p, ω ¯ , σH , σL ) − µb Lp (¯ p, ω ¯ , σH , σL )} is the Lagrange multiplier p, ω on the household’s participation constraint. The term sb {Y (¯ ¯ , σH , σL ) − µb L(¯ p, ω ¯ , σH , σL )} is p, ω assumed to be sufficiently less than 1 to prevent the banker from achieving an infinite level of leverage. The implied banker’s funding rate can be driven from Equation (12):   Rd = q (¯ ℓb ¯ , σ H , σ L )R p, ω ¯> 0 if p ℓb − 1 (15)   Rd = Rf ¯ ≤ 0. if p Panel (b) of Figure 7 illustrates the deposit contract in (ℓb , Rd )-space. One noticeable difference from the loan contract is that bankers are able to borrow at the risk-free rate up to a certain level of leverage, because their expected default rate is zero for leverage ratios below the kink point in the figure. This is due to the fact that bankers’ returns on loans are bounded as a result of diversification: q (p)R ∈ [q (0)R, q (1)R]. Bankers with sufficiently low leverage ratios are always able to deliver the promised return to households. The kink point represents the maximum leverage ratio at which bankers can fund themselves at the risk-free interest rate. This point is closely tied to the value of ¯ , σH , σL ), which corresponds to the idiosyncratic return on the loan portfolio of the worst- q (0, ω performing banker. Once bankers’ leverage exceeds this threshold, their funding rate begins to rise sharply, reflecting the increase in expected default costs. Panel (b) of Figure 7 also shows how the deposit contract responds to increases in σH and σL . As σH increases (or σL increases), the maximum leverage that bankers can achieve at the risk- free funding rate falls (or rises). To understand the mechanism behind this result, it is essential 23 ¯ , σH , σL )—the idiosyncratic return on the loan portfolio of the lowest-type to examine how q (0, ω banker—responds to fluctuations in σH and σL . This variable plays a central role in transmitting dispersion shocks to the banker’s leverage capacity. ¯ , σH , σL ) is given by Recall that q (0, ω Γ(¯ω , σH ) − µG(¯ ω , σL ) ¯ , σH , σL ) = q (0, ω . πH ω , σH ) + (1 − πH ) Γ(¯ ω , σH ) − µG(¯ Γ(¯ ω , σL ) − µG(¯ ω , σL ) ω , σθ ) − µG(¯ As shown in Proposition 2, Γ(¯ ω , σθ ) decreases in σθ . Accordingly, an increase in ¯ , σH , σL ), while an increase in σL raises it. Intuitively, whether bankers can fund σH lowers q (0, ω themselves at the risk-free rate depends on the returns of the type-0 banker. As long as the type-0 banker is able to ensure the risk-free return to households, the expected default rate is zero. For this reason, households are concerned only with the loan quality of the lowest-type banker. While σH is strongly associated with the loan quality of bankers in the bottom percentiles, σL is more closely related to the loan quality of those in the top percentiles. Thus, only an increase in σH significantly undermines bankers’ capacity to leverage. Finally, observe that the solution above yields the following expression for the banker’s optimal leverage ratio: ∂ψb 1 ℓ b ≡ ψ b ( sb ) , with >0 for sb ∈ 1, . (16) ∂sb Y (¯ ¯ , σH , σL ) − µb L(¯ p, ω ¯ , σH , σL ) p, ω Along with Equation (6), Equation (16) describes a key relationship in the model: a positive rela- tionship between the banker’s external finance premium and the banker’s leverage. The banker’s optimal leverage ratio is positively associated with the profitability of its loan portfolio. 3.4. Loan Market Equilibrium without Aggregate Risk This section characterizes the loan market equilibrium. Recall that all entrepreneurs choose the same leverage ratio ℓ, and all bankers take the same leverage ratio ℓb . Thus, aggregate loan demand and supply can be expressed as functions of the banker’s external finance premium sb : s Loan demand curve : [ℓ − 1] N = ψ − 1 N, (17) sb Loan supply curve : ℓb (sb )Nb,t = ψb (sb )Nb , (18) where N and Nb denote the aggregate net worth of entrepreneurs and bankers, respectively. Since 24 ℓ = ψ (s/sb ) is a decreasing function of sb and ℓb is an increasing function of sb , the loan demand (supply) curve is downward (upward) sloping with respect to sb . Thus, there exists a unique s∗ b such that aggregate loan demand equals aggregate loan supply. The equilibrium is illustrated in Figure 9. Observe that there are two possible cases. The first case, in which the supply and demand curves intersect on the upward-sloping portion of the supply curve, is shown in Figure 9. In this case, s∗ b > 1. The other possible case is that they intersect on the horizontal portion of the supply curve. In this case, the equilibrium is equivalent to one in which banks are not financially constrained, and s∗ ∗ b = 1. Note that sb ≥ 1 because bankers do not want to intermediate funds from households to entrepreneurs if s∗ b < 1. This yields an occasionally binding constraint in the model:   s∗ = 1 if ℓb ≤ ψb (1) b (19)  s∗ > 1 if ℓb > ψb (1). b ψb (1) indicates the maximum leverage ratio that bankers can attain while still borrowing at the risk- free interest rate (i.e., when sb = 1). Bank leverage must also be consistent with the loan market equilibrium condition: ℓb = (ℓ − 1) N N b . Hence, s∗ b = 1 implies that loan demand is lower than the banking sector’s loan supply capacity. As a result, the loan market equilibrium becomes equivalent to that of a model with financially unconstrained banks—analogous to the standard BG setup, in which the loan supply curve is fully elastic at sb = 1. This suggests that the model’s dynamics should resemble those of standard financial accelerator models when sb = 1. Figure 9. Illustration of the Loan Market 25 3.5. Financial Contracts with Aggregate Risk The contracting framework is now modified to accommodate aggregate risk. This modification is necessary because unexpected aggregate shocks cause fluctuations in the net worth of entrepreneurs and bankers, which are critical to the dynamics of the model. Consider the loan contract first. BGG assume that the banker’s participation constraint is sat- isfied ex-post, implying that entrepreneurs bear all the risk in the economy. In other words, the lending rate is state-contingent to ensure predetermined returns to bankers. In contrast to BGG, this model allows aggregate risk to be shared between entrepreneurs and bankers, so that bankers’ returns are not predetermined and their participation constraint may not be satisfied ex-post. The loan contract reflecting this assumption is given by the solution to: max Et Rk,t+1 πθ 1 − Γ(¯ ωt+1 , σθ,t+1 ) ℓt ¯ t+1 ℓt , ω θ s.t. Et Rk,t+1 ωt+1 , σθ,t+1 ) ℓt ≥ Rt+1 (ℓt − 1), ωt+1 , σθ,t+1 ) − µG(¯ πθ Γ(¯ θ where expectations are taken with respect to the aggregate return to capital (Rk,t+1 ) and the dis- persion of idiosyncratic productivity (σL,t+1 and σH,t+1 ). Observe that the banker’s participation constraint must hold ex-ante but may not hold ex-post. As a consequence, given the predetermined ¯ t+1 , the ex-post default threshold will vary depending on the ex-post ex-ante default threshold ω realization of Rk,t+1 . This is one of the principal differences of this model with respect to the BGG model. Let ω ˜ k,t+1 denote the ex-post default threshold and the realized aggregate return on ˜ t+1 and R ¯ t+1 ) that capital, respectively. In period t, the lending rate Rb,t is predetermined by the pair (ℓt , ω ¯ t+1 Et [Rk,t+1 ] ℓt = Rb,t (ℓt − 1). In period t + 1, given the pre- solves the loan contract problem: ω determined lending rate Rb,t and leverage ratio ℓt , entrepreneurs default ex-post if the idiosyncratic productivity is below ω ˜ t+1 , where ω ˜ k,t+1 ℓt = Rb,t (ℓt − 1). This implies ˜ t+1 R Et [Rk,t+1 ] ˜ t+1 = ω ω ¯ t+1 . (20) R˜ k,t+1 Equation (20) shows that any deviation of the realized aggregate return on capital from the expected return results in a wedge between the ex-ante and ex-post default thresholds, thereby generating a gap between the ex-post realized default rate and the expected default rate. This also leads to a wedge between the expected (or required) return on loans, Rt , and the realized return for the banker, 26 ˜ t . The two are given by: R ℓ t −1 Rt = Et−1 Rk,t ωt , σθ,t ) − µG(¯ πθ Γ(¯ ωt , σθ,t ) , (21) θ ℓ t −1 − 1 ℓ t −1 ˜t = R R ˜ k,t ˜θ,t ) − µG(¯ πθ Γ(˜ ωt , σ ˜θ,t ) ωt , σ , (22) θ ℓ t −1 − 1 where σ ˜θ,t denotes the realized volatility in period t. In this partial equilibrium model, fluctuations in σ ˜θ,t are not associated with the realized return on capital R ˜ k,t . In a general equilibrium version of this model, however, any deviation of σ ˜ k,t by affecting ˜θ,t from its expected value would impact R the demand for capital. Regarding the deposit contract, bankers are assumed to absorb all risks. Thus, the household’s participation constraint must be satisfied ex-post. The deposit contract is given by the solution to: max Et ut+1 Rt+1 1 − Y (¯ ¯ t+1 , σH,t+1 , σL,t+1 )) ℓb,t − ut+1 Rt+1 pt+1 (ut+1 ), ω ¯t+1 (ut+1 ) ℓb,t , p s.t. ut+1 Rt+1 Y p ¯ t+1 , σH,t+1 , σL,t+1 − µb L p ¯t+1 (ut+1 ), ω ¯ t+1 , σH,t+1 , σL,t+1 ¯t+1 (ut+1 ), ω ℓb,t ≥ Rf,t (ℓb,t − 1) ∀ ut+1 , where ut+1 captures any deviations of the bankers’ realized return on loans from the expected return in period t + 1 because of aggregate shocks and Eu = 1. In contrast with the loan contract, the ¯t+1 is determined ex-post depending on the realization of aggregate banker’s default threshold p shocks so that the households’ participation constraint is satisfied in all possible states next period. Let R˜ t = ut Rt and p ˜t denote the ex-post realized aggregate return on loans and the ex-post ˜ t , the ex-post default threshold default threshold, respectively. Then, given the realized return R ˜t ≥ 0 is determined by the household’s participation constraint. p  p˜t = 0 if R ˜ t {Y (0, ω˜t, σ ˜L,t ) − µb L(0, ω ˜H,t , σ ˜t, σ ˜L,t )} ℓb,t−1 ≥ Rf,t−1 (ℓb,t−1 − 1) ˜H,t , σ (23) R˜ t {Y (˜ ˜t, σ pt , ω ˜L,t ) − µb L(˜ ˜H,t , σ pt , ω ˜t, σ ˜L,t )} ℓb,t−1 = Rf,t−1 (ℓb,t−1 − 1) o/w, ˜H,t , σ and the bankers’ funding rate follows: ¯ p ¯t )Rd,t Dt−1 + (1 − µb ) (1 − p q (p, ω ˜t, σ ˜H,t , σ ˜ t Bt−1 dp = Rf,t−1 Dt−1 , ˜L,t )R (24) 0 27 where Bt−1 and Dt−1 denote aggregate loans and deposits in period t − 1, respectively. The unique- ness of the point that satisfies Equation (23) is proved in Appendix A.3. Equation (23) introduces a ˜ ≥ 0. Note that p second occasionally binding constraint in the model, p ˜t also represents the ex-post default rate of bankers because pt ∼ U [0, 1]. Thus, Equation (23) implies that bankers may occa- sionally default in response to negative shocks that lower the realized aggregate return on loans. Furthermore, Equation (24) shows that the bankers’ funding rate rises above the risk-free rate when some banks default, indicating that costly bank failures lead to a greater reduction in the aggregate net worth of bankers. 3.6. Loan Market Equilibrium and Dispersion Shocks This part numerically illustrates how an unexpected increase in σH,t+1 affects the steady-state equilibrium in a partial equilibrium model. To obtain a well-defined steady-state equilibrium in a dynamic setting, an exogenous fraction (1 − γe ) of entrepreneurs’ net worth and an exogenous fraction (1 − γb ) of bankers’ net worth are assumed to be consumed each period. Figure 10 displays how an unexpected increase in σH,t+1 influences the equilibrium in the loan market. Starting from the steady state, Et , the value of σH,t+1 is unexpectedly and permanently increased to 0.55 from its steady-state value of 0.45. This is illustrated in panel (a) of Figure 10. Focus first on the loan supply curve. There are two distinct effects: the leverage effect and the net worth effect. First, the increase in σH,t+1 significantly impairs bankers’ capacity to leverage in period t + 1, as the loan quality of the lowest type (i.e., p = 0) deteriorates substantially. The (a) σH : 0.45 → 0.55 (b) σH : 0.45 → 0.50 Figure 10. Dispersion shocks and loan market Note: Rk = 1.03, Rf = 1.005, µ = 0.2, µb = 0.4, πH = 0.1, σL = 0.2, σH = 0.4, N = 1 and Nb = 0.1 28 maximum leverage that bankers can achieve at the risk-free rate shrinks considerably, contracting the loan supply. The net worth effect is highly nonlinear and asymmetric. The unexpected rise in σH,t+1 pushes the realized aggregate return on loans below the expected aggregate return due to the higher default rate of high-type entrepreneurs. As a result, the bankers’ aggregate net worth, Ntb+1 , falls below its steady-state level. In particular, since losses are concentrated among a small fraction of bankers, some of them default. These defaults raise the bankers’ funding rate, which in turn triggers further defaults. Thus, a kind of multiplier effect arises: the initial shock is amplified because it induces bank failures. The magnitude of this amplification depends largely on the size of the initial default triggered by the shock, implying that the net worth effect is nonlinear. As shown in panel (b) of Figure 10, the net worth effect is much weaker for a relatively small dispersion shock. Moreover, a decrease in σH,t+1 does not amplify the effect, as the bankers’ default rate remains at zero. Hence, the net worth effect is asymmetric. This asymmetric net worth effect may help explain the well- known asymmetric movements of economic variables over the business cycle reported by other studies (e.g., see Ordoñez, 2013). In contrast to the effect on the loan supply curve, the effect on the loan demand curve is fairly weak. Since the fraction of high-type entrepreneurs is small (πH = 0.1), the increase in σH,t+1 raises the overall standard deviation of idiosyncratic productivity by only 0.02. Thus, the loan contract tightens only marginally. The net worth effect is also negligible because losses from the unexpected rise in σH,t+1 fall primarily on the bankers. The entrepreneurs’ aggregate net worth barely changes, as some entrepreneurs benefit from higher dispersion even though defaults increase. Therefore, the bankers’ aggregate net worth plays a critical role in the dynamics of this partial equilibrium model—strikingly contrasting with the BG loan contract, where the entrepreneur’s net worth is central. Notice that the financial accelerator effect is muted in this partial equilibrium setting because Rk is fixed, implying that the price of capital remains constant. In general equilibrium, however, there will be a well-known financial accelerator effect associated with the entrepreneur’s net worth. While this effect is weaker compared to the BG model due to risk-sharing between entrepreneurs and bankers, another financial accelerator effect tied to the banker’s net worth emerges. 4. General Equilibrium The two contracting problems examined in the previous section are now embedded into an oth- erwise standard RBC model to quantify the general equilibrium effects of the mechanism. The model includes five types of agents: households, firms, capital producers, entrepreneurs, and 29 Figure 11. Model Overview bankers. A graphical overview of the model is provided in Figure 11. The representative house- f hold consumes the final good (Ct ), lends to bankers (Dt ) at the gross risk-free interest rate (Rt ), and supplies labor (Lt ) to firms at the real wage (wt ). Firms hire labor and rent capital in perfectly competitive markets to produce consumption goods using a Cobb-Douglas production function. Capital producers purchase consumption goods (It ) from firms and the non-depreciated capital stock (1 − δ )Kt from entrepreneurs, where δ is the depreciation rate. They then transform these into new capital Kt+1 , subject to adjustment costs, and sell the new capital to entrepreneurs at the unit price Qt . Since these parts are standard in the literature, a detailed specification is deferred to Appendix B. 4.1. Entrepreneurs and Bankers As they are described in detail in the previous section, the description below provides only the critical equations. Since only aggregate variables need to be tracked due to constant returns to scale and risk neutrality, the entrepreneur’s index i and the banker’s index j are omitted unless necessary. i k k The realized gross return to holding a unit of capital in period t is ωθ,t Rt , where Rt is the aggregate return to capital, given by 30 k rt + (1 − δ )Qt Rt = , (25) Q t −1 i and ωθ,t is an idiosyncratic random variable that is i.i.d. across time and entrepreneurs, where iid θ ∈ {L, H } denotes the entrepreneur type. Hence, log(ωθ,t i ) ∼ N −2 1 2 2 σθ,t , σθ,t , where σθ,t is the stochastic standard deviation of idiosyncratic productivity (i.e., “dispersion”) and follows a standard AR(1) process to be specified later. At time t, entrepreneurs use their accumulated net worth Nt and loans to purchase capital Kt+1 for use in the next period at the unit price Qt . Thus, the entrepreneurs’ leverage ratio ℓt is given by Qt Kt+1 ℓt = , (26) Nt ¯ t+1 satisfy the following optimality condition of the and ℓt and the ex-ante default threshold level ω loan contract:   −1  Rk  ℓ t = 1 − Et t+1 ωt+1 , σθ,t+1 ) − µG(¯ πθ Γ(¯ ωt+1 , σθ,t+1 ) , (27)  Rt+1  θ ∈{L,H } k Rt +1 λb t = , (28) Rt+1 θ πθ 1 − Γ(¯ ωt+1 , σθ,t+1 ) + λb t π ωt+1 , σθ,t+1 ) − µG(¯ θ θ Γ(¯ ωt+1 , σθ,t+1 ) t = { where λb θ ωt+1 , σθ,t+1 )} / Γω (¯ θ ωt+1 , σθ,t+1 ) − µGω (¯ Γω (¯ ωt+1 , σθ,t+1 ) is the multi- plier on the banker’s participation constraint, and Rt+1 is the required return on the loan by the b ep banker. Then the implied gross lending rate Rt ¯t and the ex-post default threshold ω satisfy ℓt b Rt ¯ t+1 Et Rt =ω k , (29) +1 ℓt − 1 ep E t −1 Rt k ¯t ω ¯t =ω k . (30) Rt As for the bankers, the realized aggregate gross return on lending a unit of consumption good in period t is given by ℓ t −1 ˜ t = Rk R ωt πθ Γ(¯ ep , σθ,t ) − µG(¯ ωtep , σθ,t ) . (31) t ℓ t −1 − 1 θ 31 t ∈ [0, 1] is determined ˜ t , the ex-post default threshold p Given the realized aggregate gross return R ¯ep by the household’s participation constraint of the deposit contract:  p ˜ t ℓ b ≥ Rt ¯ep −1 (ℓt−1 − 1) f b t = 0 if qt (0)R t −1 (32) R ˜ Y (¯pep ) − µ L (¯pep ) ℓb = Rf (ℓb − 1) otherwise, t t t b t t t −1 t −1 t −1 d and the bankers’ funding rate Rt satisfies the following condition: ¯ep p t (1 − ¯ep p d t )Rt Dt−1 + (1 − µb ) ˜ t Bt−1 dp = Rt qt ( p t ) R f −1 Dt−1 . (33) 0 f d Notice that Rt = Rt ¯ep −1 if p t = 0. The equation for qt (pt ) is given by ωt h(pt ) Γ(¯ ep , σH,t ) − µG(¯ ωtep , σH,t ) + (1 − h(pt )) Γ(¯ ωtep , σL,t ) − µG(¯ ωtep , σL,t ) qt ( p t ) = . πH Γ(¯ ωt , σH,t ) − µG(¯ ep ωt , σH,t ) + (1 − πH ) Γ(¯ ep ωt , σL,t ) − µG(¯ ep ep ωt , σL,t ) (34) At time t, bankers use their accumulated net worth Ntb and deposits Dt to lend Bt to en- trepreneurs. The banker’s leverage ratio ℓb t is defined by Bt ℓb t = , (35) Ntb and ℓb ¯t+1 satisfy the following optimality condition of the t and the ex-ante default threshold level p deposit contract: −1 Rt+1 ℓb t = 1 − Et f Y (¯ pt+1 ) − µb L(¯ pt+1 ) , (36) Rt Rt+1 λd t = , (37) f Rt pt+1 ) + λt Y (¯ 1 − Y (¯ d pt+1 ) − µb L(¯ pt+1 ) pt+1 )} / {Yp (¯ t = {Yp (¯ where λd pt+1 )} is the multiplier on the household’s partici- pt+1 ) − µb Lp (¯ pt+1 ) = Y (¯ pation constraint, and the notations are simplified: Y (¯ ¯ t+1 , σH,t+1 , σL,t+1 ) and pt+1 , ω pt+1 ) = L(¯ L(¯ ¯ t+1 , σH,t+1 , σL,t+1 ). pt+1 , ω Following BGG and CMR, entrepreneurs and bankers are assumed to consume an exogenous fraction of their net worth each period, respectively, to avoid self-financing in the long run. This 32 implies the following laws of motion for the aggregate net worth and consumption of entrepreneurs and bankers: Nt = γ Rk,t 1 − Γ(¯ ωtep e , σθ,t ) ℓt−1 Nt−1 + wt , (38) θ 1 Ntb = γ b ˜ t ℓb,t−1 − Rt qt ( p t ) R d (ℓb,t−1 − 1) dp Ntb−1 + wt b , (39) ¯ep p t Cte = (1 − γ ) Rk,t 1 − Γ(¯ ωtep , σθ,t ) ℓt−1 Nt−1 , (40) θ 1 Ctb = (1 − γ b ) ˜ t ℓb,t−1 − Rd (ℓb,t−1 − 1) dp N b . qt ( p t ) R t t −1 (41) ¯ep p t 4.2. Market Clearing and the Exogenous Processes The model is closed by the goods market-clearing condition and the specification of three exoge- nous shocks. The market-clearing condition for consumption goods is: Yt = Ct + Cte + Ctb + It + µRt k πθ G(¯ ωtep , σθ,t ) + µb L(¯ pep ˜ t ) Rt B t −1 , (42) θ ∈{H,L} k where µRt θ ∈{H,L} πθ G(¯ ωtep , σθ,t ) and µb L(¯ pep ˜ t )Rt Bt−1 represent aggregate costs of monitoring defaulting entrepreneurs and bankers, respectively. Three exogenous processes, At , σH,t , and σL,t , follow a standard AR(1) process: log(At ) = ρA log(At−1 ) + ϵA t ∼ N (0, σA ), ϵA 2 t , (43) σ log(σL,t−1 ) + (1 − ρσ ) log(σL,ss ) + ϵt , log(σL,t ) = ρL t ∼ N (0, (σσ ) ), L L ϵL L 2 (44) σ log(σH,t−1 ) + (1 − ρσ ) log(σH,ss ) + ϵt , log(σH,t ) = ρH t ∼ N (0, (σσ ) ), H H ϵH H 2 (45) where ϵA L H t , ϵt , and ϵt denote innovations to technology and idiosyncratic volatility for each type, and σL,ss and σH,ss are the steady-state values of the standard deviation of idiosyncratic productivity for each type. 33 5. Quantitative Analysis In this section, the general equilibrium model is calibrated to the U.S. economy for the period 1985:Q1–2014:Q4 to evaluate the quantitative importance of the model’s amplification mecha- nism.18 5.1. Calibration Most of the parameter values are standard and follow the existing literature. New parameters that cannot be determined this way are either calibrated using steady-state targets or through a method of simulated moments. Table 2 lists the calibrated parameter values of the baseline model. Overall, there are 23 parameters: 16 parameters are standard and taken from the literature, and 7 L H (πH , σL,ss , σH,ss , ρL , ρH , σσ , σσ ) are new and specific to this model. Standard Parameters: One period in the model corresponds to a quarter. The discount factor is β = 0.995, implying a steady-state real interest rate of 2% per annum. Utility is logarithmic, so σ = 1. The elasticity of labor is assumed to be 2 (η = 0.5), which is in the middle of the range of values used in other macroeconomic studies.19 The value of ψL is chosen so that hours worked equal 1 in the steady state. The capital share in the production function is set to αk = 0.35. The shares of entrepreneur and banker labor are set to be very small so that αh = 0.649, αe = 0.0009, and αb = 0.0001. The depreciation rate is set to δ = 0.025. The elasticity of the price of capital with respect to the investment-capital ratio is taken from BGG: ϕk = 0.25. Following CMR, the entrepreneur’s bankruptcy cost coefficient is set to µ = 0.21, which is within the 0.1–0.5 range found in the literature. For instance, BGG set this parameter at 0.12, and Carlstrom and Fuerst (1997) argue that 0.2–0.36 is the empirically relevant range. In addition, estimates by Leven, Natalucci, and Zakrajšek (2004) lie in the range 0.1–0.2 for the 1997–99 period and 0.3–0.5 for the 2000–03 period in the U.S. The banker’s bankruptcy cost coefficient is set based on Granja, Matvos, and Seru (2017), who report 0.28 for U.S. banks, so µb = 0.28. This is also close to the value of 0.3 used by Mendicino et al. (2020). Given other parameter values, the implied fractions of net worth consumed by entrepreneurs and bankers are 1 − γ = 1 − 0.986 and 1 − γ b = 1 − 0.938, respectively. These values are within the range used in the literature.20 Finally, the parameters (ρA , σA ) of aggregate productivity are chosen using the Federal Reserve Bank of San Francisco- 18 The model involves two occasionally binding constraints: the banker’s external finance premium cannot fall below 1, and the bank default rate cannot be negative. Due to the complexity of applying a global solution method to this model, as well as for the purpose of comparison with existing financial accelerator models, the model is solved using a piecewise linear perturbation method, which is a variant of the extended perfect-foresight path (EPFP) method. In particular, the model is solved using the OccBin toolbox developed by Guerrieri and Iacoviello (2015). 19 For example, 1, 2, and 3 are used by CMR, Arellano, Bai, and Kehoe (2019), and BGG, respectively. 20 For example, BGG and CMR set this parameter for entrepreneurs at 1−0.973 and 1−0.985, respectively. Afanasyeva and Güntner (2020) use 1 − 0.985 for entrepreneurs and 1 − 0.92 for a monopolistic bank. 34 Table 2. Parameter values Parameter Value Description Source or target Standard: β 0.995 quarterly discount factor 4 · (Rf − 1) = 2% σ 1 coefficient of relative risk aversion BGG / CMR η 0.5 inverse labor supply elasticity macro literature ψL 0.971 labor weight in utility function Lss = 1 αk 0.35 capital share in production macro literature αh 0.649 household labor share in production macro literature αe 0.0009 entr. labor share in production ad-hoc assumption αb 0.0001 banker labor share in production ad-hoc assumption δ 0.025 depreciation rate macro literature ϕk 0.25 elasticity of Q w.r.t. I /K BGG µ 0.21 entr. bankruptcy costs CMR µb 0.28 banker bankruptcy costs Granja-Matvos-Seru (2017) 1−γ 0.014 fraction of net worth consumed for entr. Implied 1 − γb 0.062 fraction of net worth consumed for banker Implied ρA 0.95 persistence of productivity shock FRBSF-CSIP, Fernald (2014) σA 0.006 std. dev. of productivity shock FRBSF-CSIP, Fernald (2014) New: πH 0.06 fraction of high type entr. target valuea σL,ss 0.22 std. dev. of idio. prod. for low type target valuea σH,ss 0.61 std. dev. of idio. prod. for high type target valuea ρL 0.76 persistence of risk shock for low type SMMb ρH 0.92 persistence of risk shock for high type SMMb L σσ 0.14 std. dev. of risk shock for low type SMMb H σσ 0.05 std. dev. of risk shock for high type SMMb a. These parameters are jointly determined to match the following three values in the steady state: an entrepreneur’s leverage ratio of 1.8; a banker’s leverage ratio of 16; and an annualized entrepreneurs’ default rate of 3.7%. b. These parameters are jointly calibrated using a method of simulated moments. Table 3. Targeted moments Data Model Standard deviations (%) Non-financial dispersion 15.4 15.3 Financial dispersion 28.6 28.6 Autocorrelation Non-financial dispersion 0.73 0.73 Financial dispersion 0.78 0.78 Notes: Non-financial dispersion is the standard deviation of quarterly return on equity for entrepreneurs, and financial dispersion is the interquartile range of quarterly return on equity for bankers. The data moments are calculated using the US stock market returns from the CRSP database for the period from 1985:Q1-2014:Q4. The model moments are computed in the same fashion as the data moments over 500 simulations of the same length as the data (120 quarters). In particular, only nonbankrupt entrepreneurs and bankers are included to be consistent with the data. 35 CSIP quarterly log TFP series (Fernald, 2014) over 1985:Q1–2014:Q4, after removing a linear trend: ρA = 0.95 and σA = 0.006. New Parameters: The three parameters (πH , σL,ss , σH,ss ) are jointly determined to match steady- state variables.21 Specifically, these parameters are set to match the following three targets: a steady-state entrepreneur’s leverage ratio of 1.8; a steady-state banker’s leverage ratio of 16; and an annualized entrepreneur default rate of 3.7% in the steady state. The entrepreneur’s leverage ratio in financial accelerator models typically falls within the range of 1.5–2, and 1.8 is a rough median of that range. The banker’s steady-state leverage ratio is based on the average ratio of assets to equity constructed using data from the U.S. Flow of Funds.22 As for the default rate, the target is based on the mean of the delinquency rate on all loans extended by commercial banks. This yields the following parameter values: πH = 0.06, σL,ss = 0.22, and σH,ss = 0.61. Assuming independence of σL and σH , the overall standard deviation of idiosyncratic productivity in the model is 0.26, which is close to values used in the literature (e.g., 0.28 and 0.26 in BGG and CMR, respectively). The remaining parameters are the persistence and standard deviation of dispersion shocks for L H each type of entrepreneur, (ρL , ρH , σσ , σσ ). These parameters govern the two dispersion processes Table 4. Selected Steady-State Values Variable Value Description K /(4 · Y ) 2.33 capital-output ratio B /(4 · Y ) 1.04 credit-output ratio C /Y 0.67 household consumption-output ratio C e /Y 0.08 entr. consumption-output ratio C b /Y 0.02 banker consumption-output ratio I /Y 0.23 investment-output ratio ℓ 1.80 leverage ratio of entr.∗ ℓb 16.00 leverage ratio of bankers∗ Nb /N 5. 0% net worth ratio of bankers to entr. 4 · ( θ πθ F (¯ ω , σθ )) 3 . 7% annualized default rate of entr.∗ 4 · (Rk − 1) 5. 0% annualized rate of return on capital 4 · (R − 1) 3. 5% annualized return on bank loans 4 · (Rf − 1) 2.0% annualized risk-free interest rate * indicates values are targeted. 21 1− p The function h(p) is specified as h(p) = ap+1 , where a is the parameter that governs the shape of the function h(p). 1 Because πH = 0 h(p) dp, the value of a is determined by choosing the value of πH . 22 Data for assets and equity are taken from the U.S. Flow of Funds. The ratios of assets to equity computed for each financial sector—U.S.-chartered depository institutions, security brokers and dealers, and finance companies—are weighted using each sector’s share of total assets. 36 in the model. Since these stochastic processes are not directly observed in the data, they are jointly calibrated using a method of simulated moments. Specifically, this procedure minimizes an equal- weighted distance measure between moments from model simulations and selected targets from the actual data. The targets are the standard deviation and autocorrelation of two time series: the cross-sectional standard deviation of quarterly log stock returns in the non-financial sector (“non- financial dispersion”), and the cross-sectional interquartile range of quarterly log stock returns in L the financial sector (“financial dispersion”). This yields ρL = 0.76, ρH = 0.92, σσ = 0.14, H and σσ = 0.05. Table 3 compares the model’s fit to the targeted moments and shows that the model closely captures the overall time-series properties of measured dispersion in the data. The deterministic steady-state values of selected variables and ratios are summarized in Table 4. 5.2. Business Cycles Statistics To evaluate the performance of the model in explaining standard business cycle moments, the model is simulated 500 times, starting from the steady state, for 120 periods, and the mean of the moments is computed over the 500 simulations. This is done in particular to calculate the standard deviation and contemporaneous correlation with output of several variables. The same exercise is then performed with two alternative models to understand the contribution of the mechanism in the model.23 The first alternative model is a standard real business cycle model without any financial Table 5. Business Cycle Statistics Y I C L B CS Standard deviations (%) Data 1.08 5.56 0.89 1.79 1.51 1.00 Model 1.09 5.04 0.57 0.70 0.75 1.14 Model with unconstrained banks 1.08 3.43 0.41 0.52 0.51 0.49 Model with no financial frictions 1.12 3.15 0.33 0.55 - - Correlation with output Data 1.00 0.91 0.89 0.87 0.31 -0.21 Model 1.00 0.84 0.42 0.79 0.26 -0.24 Model with unconstrained banks 1.00 0.96 0.81 0.94 0.26 -0.05 Model with no financial frictions 1.00 0.99 0.92 0.99 - - Notes: All series except spread (CS), which is demeaned by their respective sample and expressed in percentage points, are in log-deviation from an HP trend with smoothing parameter 1600. Output (Y) is real gross domestic product (FRED GDPC1), investment (I) is real gross private domestic investment (GPDIC1), consumption (C) is real personal consumption expenditure (PCECC96), hours (L) is nonfarm business sector hours of all persons (HOANBS), credit (B) is the sum of debt securities and loans of nonfinancial corporate business (TCMILBSNNCB) and nonfinancial noncorporate business (TCMILBSNNB), both are deflated by the GDP deflator (GDPDEF), and spread (CS) is the average credit spread on senior unsecured bonds issued by nonfinancial firms constructed by Gilchrist and Zakrajšek (2012). 23 These models are calibrated separately but are almost identically parameterized. Some parameter values are in- evitably different. For example, the labor share parameters in a model with unconstrained banks are αh = 0.649, αe = 0.001, and αb = 0, and αh = 0.65 and αe = αb = 0 in a model with no financial frictions. 37 frictions. In this model, dispersion shocks have no impact because the mean of idiosyncratic pro- ductivity is always 1 and financial markets are complete. The second model includes the financial friction between entrepreneurs and bankers only. This is a standard financial accelerator model akin to the BG setup, where financial intermediaries are homogeneous, passive, and do not bear risk. This BG-like model is labeled the model with unconstrained banks. This alternative model and the baseline model have the same exogenous shocks: an aggregate productivity shock and two dispersion shocks. The only difference lies in the financial contracts used, and therefore the bank distress amplification mechanism is absent in the BG-like model. Table 5 presents the computed business cycle statistics. Overall, the three models are compa- rable, but the baseline model outperforms the others in explaining investment, loans, and credit spreads. It delivers higher investment volatility, which is closer to the data. As in the data, loans co-move with output, and credit spreads are countercyclical. On the other hand, the model under- performs in capturing the correlation between consumption and output, as the correlation is quite low compared to the data. This issue will be discussed in more detail below. 5.3. The Effects of Dispersion Shocks Consider next the dynamics of the model in response to unanticipated cross-sectional dispersion shocks (a one-standard-deviation increase (5%) in ϵH 0 and a one-standard-deviation increase (14%) in ϵL 0 , respectively) starting from the deterministic steady state. Figure 12 displays how dispersion shocks affect non-financial and financial dispersion (the lines with dots for ϵH 0 and the solid lines for ϵL L 0 ). The increase in ϵ0 significantly raises non-financial dispersion, but its impact on financial dispersion is negligible. This is because it increases default rates in the high-quality loan portfolio Figure 12. Dispersion shocks and financial and non-financial dispersion Notes: Non-financial dispersion is measured by the standard deviation of equity returns for entrepreneurs, and finan- cial dispersion is computed by the interquartile range of equity returns for bankers. In particular, only nonbankrupt entrepreneurs and bankers are included to be consistent with the data. 38 (high p) more than in the low-quality loan portfolio (low p). Therefore, financial dispersion actually decreases, although the magnitude is trivial. In contrast, the rise in ϵH 0 sharply increases financial dispersion but has a relatively small effect on non-financial dispersion. This occurs because higher ϵH 0 lowers the returns on the low-quality loan portfolio more than on the high-quality portfolio. In particular, it reduces the return on the lowest quality (i.e., p = 0) the most. Thus, the rise in financial dispersion in Panel (b) is mainly driven by lower returns on loans for banks below the bottom 25th percentile, indicating greater asymmetry in bank distress. The reason its impact on non-financial dispersion is much smaller than in the case of the increase in ϵL 0 is that the share of high-type entrepreneurs in the model is only about 6%. Figure 13 shows the dynamic responses of various variables to dispersion shocks in the two different models. The black lines with dots (black solid lines) represent the response of the baseline Figure 13. Impulse responses to dispersion shocks Notes: Responses to a one-standard-deviation increase in ϵH L 0 and ϵ0 , respectively. All series are expressed in terms of percent deviation from steady state, except for the spread, excess loan premium, default risk premium, entrepreneur default, and bank default, which are expressed in terms of percentage points deviation. 39 model to the increase in ϵH L 0 (ϵ0 ), and the blue lines with triangles (blue dashed lines) represent the response of the model with unconstrained banks. Focus first on the response of variables to the two shocks in the baseline model (the black lines with dots versus the black solid lines). The top panels of Figure 13 plot the responses of macro variables. They show that the increase in ϵH L 0 affects macro variables significantly more than ϵ0 , despite the fact that the share of high-type entrepreneurs is only 6%. Output, investment, and hours worked initially drop by 0.2, 2, and 0.3 percentage points more, respectively. The initial rise in consumption is an unattractive feature of the model with dispersion shocks only. Sharp drops in investment reflect the fall in the real interest rate, which makes immediate con- sumption more attractive. Bloom et al. (2018) and Khan and Thomas (2013) also find initial jumps in consumption in response to uncertainty shocks and credit shocks in their models, respectively, due to similar general equilibrium effects. CMR and Bloom et al. (2018) discuss this issue in detail. CMR argue that price and wage frictions are necessary to generate an initial consumption fall in response to a risk shock in their model, because the real interest rate falls by less than it does under flexible prices.24 Bloom et al. (2018) suggest that allowing households to save in other technolo- Figure 14. Impulse responses to all shocks Notes: Responses to a one-standard-deviation increase in both σH,0 and σL,0 and a one-standard-deviation decrease in productivity At,0 . All series are expressed in terms of percent deviation from steady state, except for the spread, which is expressed in terms of percentage points deviation. 24 CMR report that consumption indeed rises initially in response to a risk shock in the flexible wage and price version of their model. 40 gies, such as foreign assets, might help, but it is essential to consider both first- and second-moment shocks together to generate an empirically realistic consumption response in their model. Similarly, it is necessary to include TFP shocks to obtain a reasonable consumption response in this model. The addition of a negative one-standard-deviation shock (-0.6%) to productivity indeed dampens the initial consumption response close to zero in the model (see Figure 14). The bottom two panels of Figure 13 depict the response of financial variables to dispersion H shocks. They help explain why the increase in σ0 has significantly more negative impacts on L macro variables than σ0 . A large increase in the spread and initial bank defaults occur because the shock ϵH 0 causes a rise in financial dispersion. The most vulnerable banks are hit hard by the shock and default. The spread increases by 64 basis points following the increase in ϵH 0 , but it rises by only 2 basis points when ϵL 0 increases. It is important to note that about two-thirds of the difference in the response of the spread is explained by the difference in the excess loan premium. The significant decline in loan quantity and the increase in the excess loan premium imply that a rise in financial dispersion results in a nontrivial contraction in the supply of loans. On the other hand, the fact that the increase in ϵL 0 reduces only the loan quantity while the excess loan premium remains stable indicates that it shrinks both loan demand and supply, but the magnitudes are relatively small. Figure 15 supports this interpretation by showing the actual equilibrium paths of the loan market in the model. The loan market equilibrium path in response to the increase in ϵH 0 (left panel) indicates that there is initially a significant reduction in the supply of loans (loan quantity falls and the required return on loans increases), followed by gradual declines in the demand for loans (both loan quantity and the required return on loans decrease). This suggests that the model can generate a substantial Figure 15. Loan market equilibrium paths Notes: The label “S.S.” represents an equilibrium of the loan market in the steady state. The arrows indicate the loan market equilibrium paths after dispersion shocks. 41 reduction in the supply of loans, which can be interpreted as “financial shocks” or “credit shocks” in the context of the findings of Jermann and Quadrini (2012) and Khan and Thomas (2013), when ϵH L 0 increases. By contrast, the increase in ϵ0 reduces only the loan quantity, which is possible when both loan supply and demand contract by a similar amount. Overall, this shows that the innovation ϵH t plays an important role during financial recessions—characterized by a large increase in the spread and a contraction in loan quantity—while the innovation ϵL t contributes to non-financial recessions. Another reason behind the larger macro impact of the increase in ϵH 0 is that it results in a greater increase in the default rate of entrepreneurs, as it affects those who are already risky and more vulnerable. It is important to note, however, that the difference in entrepreneurs’ default rates between the two cases is not the main source of amplification, and its impact is relatively small. This point can be more clearly illustrated by comparing the responses of the baseline model with those of the model with unconstrained banks, which are also shown in Figure 13. Compare first the responses of the two models when ϵH 0 increases (the black lines with dots versus the blue lines with triangles). The increases in the default rates of entrepreneurs in the two models are comparable. Nonetheless, the baseline model generates significantly larger effects on macro and financial variables. This comparison illustrates the amplification effects of the asym- metric bank distress mechanism. The concentration of losses due to entrepreneurs’ higher default rate on vulnerable bankers leads the most fragile banks to default, thereby raising the funding cost for all surviving banks, which in turn may induce additional defaults. Thus, a kind of multiplier effect arises.25 In addition, a higher ϵH 0 results in significant deleveraging in the banking sector due to increased uncertainty, which reduces the supply of funds and thereby raises the excess loan pre- mium sharply. This explains why the spread increases more, and macro variables fluctuate more in the baseline model than in the model with unconstrained banks. Compare next the two models when ϵL 0 rises (the black-solid lines versus the blue-dashed lines). In contrast to the responses to the increase in ϵH 0 , which raises financial dispersion, the two models behave quite similarly. If anything, the overall effects are smaller in the baseline model. The responses of bank leverage and default indicate that the asymmetric bank distress mechanism does not operate in this case. There is no default in the banking sector, and hence no weakening of bankers’ ability to leverage. Bankers in the baseline model indeed dampen the effect of the shock 25 This is a potential source of nonlinearity and asymmetry in the model. The magnitude of the amplification effect depends to a large degree on the size of defaults that shocks initially generate, implying that the effect is nonlinear. The effect of a decrease in dispersion (a lower ϵH 0 ) does not amplify because it does not involve changes in the default rate of banks, indicating that the effect is asymmetric. Thus, using nonlinear global methods may be required to characterize the model dynamics more accurately. Many studies have emphasized the importance of nonlinear dynamics associated with financial frictions (see, e.g., Mendoza, 2010; He and Krishnamurthy, 2013; Brunnermeier and Sannikov, 2014; and Boissay, Collard, and Smets, 2016). 42 by absorbing losses due to higher default by entrepreneurs. Recall, however, that entrepreneurs bear all the risk in the model with unconstrained banks and therefore reduce their net worth more. Nonetheless, the overall difference is negligible. The important finding in this comparison is that the two models behave similarly when the asymmetric bank distress mechanism is not operating. The mechanism in the model amplifies shocks to the economy only when financial dispersion rises (i.e., in response to higher ϵH t ). 5.4. Counterfactual Exercises Another key quantitative exercise is conducted to evaluate the importance of the asymmetric bank distress amplifier. First, the behavior of economic aggregates in the model and the data is compared to assess the model’s ability to account for observed dynamics. Second, the role of the asymmetric bank distress amplifier during financial recessions is explored by comparing the baseline model against the model with unconstrained banks. First, three series of model-implied shocks to aggregate productivity and the dispersion of idiosyncratic productivity for both types of entrepreneurs are extracted using an inversion filter. Guerrieri and Iacoviello (2017) and Cuba-Borda, Guerrieri, Iacoviello, and Zhong (2019) describe how the inversion filter can be used to compute the log-likelihood function when estimating a model with occasionally binding constraints. Since the model is calibrated, the inversion filter is used solely to extract the implied paths of the three shocks, given the model solution with calibrated parameters. The solution of the model is given by Xt = P(Xt−1 , ϵt )Xt−1 + D(Xt−1 , ϵt ) + Q(Xt−1 , ϵt )ϵt H L ′ where Xt is a vector collecting all of the model’s variables, ϵt = (ϵA t , ϵt , ϵt ) is a vector of inno- vations, and the matrix P, the vector D, and the matrix Q are functions of the lagged state vector and the current innovations, respectively. Assuming no measurement error, the vector of observed series Yt is simply Yt = HXt , where H is the matrix that selects the variables corresponding to the data. Therefore, the three shocks can be recovered by recursively solving for ϵt starting from the model’s steady state. To guarantee that HQ(Xt−1 , ϵt ) is locally invertible, the same number of observed variables as innovations in the vector ϵt is used, and the chosen variables are closely related to the innovations: output, non-financial dispersion, and financial dispersion. The observed time series for output is real gross domestic product (FRED GDPC1), in logs and detrended using a one-sided HP filter with a smoothing parameter of 100,000. The observed time series for non-financial dispersion and financial dispersion are the cross-sectional standard deviation of quarterly log stock returns 43 Figure 16. Fit and extracted shocks using the inversion filter Notes: The black solid line is the data. The red dot-line presents the model. The top panels show the variables in log deviation from the steady state. The middle panels plot the exogenous variables in levels and the bottom panels display the innovations. for non-financial firms and the cross-sectional interquartile range of quarterly log stock returns for financial firms, respectively. These are the same time series used for the calibration in Section 5.1. Both series are computed as log deviations from their respective sample means. The corre- sponding variables in the model are Yt , Std(ω | ω ≥ ω ep ¯t k ) Rt ℓt−1 θ∈{H,L} πθ 1 − Γ(¯ ωtep θ , σt ) , and q (p75 ¯t t ,ω ep H , σt L , σt ) − q ( pt 25 ,ω ¯tep H , σt L , σt ˜ t ℓb ) R t −1 . 26 The innovation to aggregate productivity, ϵA t , is closely associated with output, and innovations to the standard deviations of idiosyncratic pro- ductivity for entrepreneurs, ϵH L t and ϵt , are tightly linked to non-financial and financial dispersion. H L In particular, the difference between σt and σt determines the level of financial dispersion. The top panels of Figure 16 report output and financial and non-financial dispersion in the data 26 ¯ ep ) and bankers (i.e., those To be consistent with the data, only nonbankrupt entrepreneurs (i.e., those with ω > ω ¯ ) are included when computing non-financial and financial dispersion. Thus, p75 = p with p > p ep ¯ep + 0.75(1 − p¯ep ) 25 and p = p ¯ + 0.25(1 − p ep ep ¯ ). 44 and in the model over the period 1985:Q1–2014:Q4. By construction, the model-generated series, computed in log deviation from the steady state, exactly match the observed data. Output falls during three recessions, non-financial dispersion increases in all three recessions, and financial dispersion is elevated only during the two financial recessions. The middle panels of Figure 16 plot the exogenous variables in levels, and the bottom panels display the extracted series for ϵt . As expected, spikes in ϵL H t are noticeable during the 2001 recession, while spikes in ϵt are pronounced L during the 2001 and 2007–2009 recessions. Overall, these patterns suggest that σt contributed to H the recessions of 2001 and 2007–2009, while σt played a critical role during the recessions of 1990 and 2007–2009. Next, the shocks extracted above are fed back into the model to solve for the paths of all other variables. Figure 17 compares the trajectories of the main variables against the observed data. The model does a good job of accounting for fluctuations in investment. It generates a sharp drop in investment during the 2007–2009 financial recession and captures investment booms during upturns. The model also successfully replicates the procyclical dynamics of loans and spreads. In particular, the spread falls continuously before the 2007–2009 financial recession and then suddenly jumps, as observed in the data. While the peak level of the spread in the model is lower than in the data, the trough-to-peak jump is comparable: 547 basis points in the model versus 607 basis points Figure 17. Data versus model-generated time series Notes: The black solid line is the data. Investment and loans are in logs and detrended using a one-sided HP filter with a smoothing parameter of 100,000. Spread and entrepreneur’s default rates are expressed as a deviation from their respective sample means. Bank’s leverage and default rates are in levels. Investment and loans in the model are in log deviation from their steady state values, and other variables are shown in the same fashion as the data. 45 in the data over the period 2007:Q2–2009:Q2. The spread shows a wide discrepancy between the model and the data during the 1990 financial recession. The model predicts a large increase in the spread, reflecting the rise in financial disper- sion during that period, but the spread in the data remains stable. This difference arises because the model is designed to capture the spread on bonds and loans for both non-financial corporate and non-corporate firms, whereas the data reflect only the average spread on senior unsecured bonds issued by non-financial corporate firms (the GZ spread constructed by Gilchrist and Zakrajšek, 2012). The model also accounts reasonably well for bank leverage and defaults of both banks and en- trepreneurs. Banks take on more leverage during booms and deleverage during downturns, consis- tent with the data. Although the model does not fully capture the post-crisis decline observed in the data, it does exhibit a substantial drop. The divergence between the model and the data after 2012 reflects the continued deleveraging by banks due to tightened regulations following the 2007–2009 financial recession. Regarding the bank default rate, some banks default during financial reces- sions, but the default rate remains close to zero during booms and non-financial recessions, in line with the data. The countercyclical behavior of the entrepreneurs’ default rate is also consistent with the data: entrepreneurs in the model default more during recessions due to greater dispersion in idiosyncratic productivity. Importantly, the bank leverage and default series indicate significant amplification driven by asymmetric bank distress. Recall that the asymmetric bank distress mechanism in the model am- plifies economic downturns through two channels: the bank deleveraging channel and the bank default channel. The model can generate sharp declines in investment and loans, along with dra- matic increases in the spread during the 2007–2009 financial recession, through these two channels. As in the data, banks in the model deleverage substantially, and some banks default, because losses from entrepreneur defaults are asymmetrically distributed and concentrated among banks in the lower percentiles during the 2007–2009 financial crisis. The procyclicality of bank leverage and loan quantity is consistent with the main finding of Adrian and Shin (2014), which emphasizes that shifts in financial intermediaries’ leverage drive fluctuations in credit availability over the business cycle. To illustrate the above point further, consider a comparison between the baseline model and the model with unconstrained banks. The only difference between the two models is that banks in the latter are financially unconstrained: they are homogeneous, perfectly diversify idiosyncratic risk, and are not exposed to aggregate risk. There are still, however, two types of entrepreneurs sub- ject to different levels of idiosyncratic productivity dispersion. Therefore, the quantitative effects 46 Figure 18. The role of the asymmetric bank distress amplifier Notes: Output and investment are in log deviation from their steady state values, and spread is expressed in terms of percentage points deviation from steady state. of the asymmetric bank distress amplification mechanism can be illustrated by feeding the same innovations into both models and comparing the responses of the main variables. Figure 18 displays the responses of output, investment, and the spread in the two models. As expected, macro variables such as output and investment in the two models behave similarly except during financial recessions when the asymmetric bank distress mechanism kicks in. In particular, the differences are pronounced during the 2007-2009 financial recession. Output and investment in the baseline model drop by 1.5 percentage points and 16.9 percentage points more, respectively.27 The spread in the baseline model increases by 234 basis points more at its peak but 92 basis points lower just before the crisis begins. In addition, the spread fluctuates clearly more in the baseline model than in the other model. This is because the movement of the spread in the latter model is driven only by fluctuations in the default risk premium. In the baseline model, the excess loan premium also changes depending on the bank’s borrowing conditions on top of the default risk premium. The comparison of the two models shows that the model with asymmetric bank dis- tress doubles the decline in investment and produces 2.5 time the rise in spreads, highlighting the quantitative importance of the asymmetric bank distress mechanism. In addition, the model with 27 The effect on output is relatively weak compared to the effects on other variables because the mechanism operates mainly through a reduction in investment, which is small relative to the capital stock. For this reason, the fluctuation in output is still dominated by productivity shocks. Introducing working capital financing is one way of modeling a direct link between the credit and output and strengthening the effect (Mendoza, 2010). 47 asymmetric bank distress can generate reasonable dynamics of financial variables such as leverage and default in the banking sector, which is infeasible with standard financial accelerator models. 6. Conclusion This paper argues that the defining feature of a financial crisis is asymmetric bank distress. The proposed model demonstrates that such distress amplifies economic downturns through two key channels. First, the concentration of losses on a subset of banks leads to costly bank failures, reducing the aggregate net worth of the banking sector. Second, it weakens banks’ leverage capac- ity by heightening uncertainty about repayment, stemming from asymmetric information between banks and depositors. This contraction in loan supply drives up spreads and triggers a sharp de- cline in investment. Quantitative analysis reveals that this amplification mechanism played a sig- nificant role in shaping the observed fluctuations in macroeconomic and financial variables during the Great Recession. Feeding identical shocks into a model without this mechanism—akin to the standard Bernanke-Gertler framework—produces only about half the decline in investment and a 60% smaller increase in spreads, underscoring the mechanism’s quantitative importance. The mechanism proposed in this paper contributes to the literature on financial amplification, which seeks to explain how small shocks can trigger large crises. Mendoza (2010) shows that Fish- erian collateral constraints significantly amplify the effects of standard-sized real shocks when they become binding, helping to explain sudden stops in emerging markets. Gorton and Ordoñez (2014) argue that the 2007–09 financial crisis was not triggered by a large shock, and emphasize the puzzle of why small shocks sometimes lead to major crises. The model presented in this paper demon- strates that the impact of an identical shock can vary greatly depending on how it is distributed across financial intermediaries and the state of the economy. When a shock disproportionately af- fects a subset of highly leveraged, vulnerable intermediaries, it can lead to widespread defaults and heightened uncertainty about the financial sector—thereby opening the door to a large crisis. Although this paper does not directly address the housing market, the low- and high-type en- trepreneurs can be interpreted as representing prime and subprime mortgages, respectively. The mechanism proposed in this paper can be applied to the 2007–09 financial crisis as follows. Losses from subprime mortgages were concentrated among a few intermediaries, such as Lehman Broth- ers and Bear Stearns, triggering a systemic crisis through fire sales—interpreted here as a drop in capital prices—and a sudden, systemic deleveraging of the financial sector. Several extensions of this analysis offer promising directions for future research. First, the framework can be embedded into a standard New Keynesian DSGE model to study the lending 48 and risk-taking channels of monetary policy.28 Second, the model can be extended to analyze optimal bank capital and liquidity regulations by introducing institutional heterogeneity among intermediaries—so that only a subset are subject to regulation and covered by deposit insurance— and by including government-issued safe liquid assets. Third, the assumption that bank types are uncorrelated across time and banks can be relaxed. In reality, correlations exist, meaning informa- tion about one bank today may reveal information about its future condition or about other banks. The core idea of this paper still holds as long as some uncertainty about bank types is preserved. When some banks are revealed to be distressed, uncertainty about the health of other banks will still rise. Importantly, relaxing this assumption would allow the model to explore why some banks take more risk than others by generating meaningful ex-ante heterogeneity. Fourth, asymmetric information between regulators and depositors about bank types can be introduced to study policy implications of stress testing during crises. Revealing information about distressed banks to the public may enhance financial stability by protecting healthy banks from contagion-driven runs but may also introduce financial instability by accelerating the failure of exposed banks. Finally, pre- cautionary motives are not considered due to the limitations of the linear solution method and the restriction on active equity financing. As emphasized by Mendoza (2010), precautionary behavior can make crises low-probability events nested within normal business cycles. Incorporating such motives would yield important insights for the design of monetary and macroprudential policies with financial stability considerations. 28 Many studies incorporate the Bernanke-Gertler financial accelerator into New Keynesian DSGE models, as in BGG and CMR (e.g., Del Negro, Giannoni, and Schorfheide, 2015; Del Negro, Hasegawa, and Schorfheide, 2016; Carrillo, Mendoza, Nuguer, and Roldán-Peña, 2021). The framework proposed in this paper can be incorporated similarly, as it requires only minor modifications to the financial friction equations. See Appendix C. 49 References Adrian, T. and H. S. Shin (2014). Procyclical leverage and value-at-risk. The Review of Financial Studies, 27(2), 373–403. Afanasyeva, E. and J. Güntner (2020). Bank market power and the risk channel of monetary policy. Journal of Monetary Economics, 111, 118–134. Akinci, O. and A. Queralto (2022). Credit spreads, financial crises, and macroprudential policy. 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Proof of Proposition 1 ¯ ω ∂ ∂ ω ) − µGσθ (¯ (Γσθ (¯ ω )) = ω ) + (1 − µ) ¯ 1 − F θ (¯ ω ωf θ (ω ) dω ∂ω¯ ∂ω¯ 0 ω ) 1 − µω = 1 − F (¯ θ ¯ h (¯ ω) θ ω )/ 1 − F θ (¯ ω ) ≡ f θ (¯ where hθ (¯ ω ) is the hazard rate. Observe that limω ¯ hθ (¯ ¯ →0 ω ω ) = 0 and limω ω ) = ∞. Moreover, ∂ ω ¯ hθ (¯ ¯ →∞ ω ¯ hθ (¯ ¯ > 0 because the hazard rate of any monotonically ω )/∂ ω increasing transformation of the normal distribution satisfies ∂ωh(ω )/∂ω > 0. This proves that ∂ ¯ ∗ such that there exists a global maximizer ω ω ) − µGσθ (¯ (Γσθ (¯ ω )) > 0 for ω ¯ ∗. ¯<ω ∂ω¯ A.2. Proof of Proposition 2 ¯ ω ω ) − µGσθ (¯ Γσθ (¯ ω ) + (1 − µ) ¯ 1 − F (¯ ω) = ω θ ωf θ (ω ) dω 0 2 2 ¯+ log ω 0 . 5σθ ¯ + 0 . 5σθ log ω ¯ 1−Φ =ω + (1 − µ)Φ − σθ σθ σθ 2 ¯ + 0 . 5σθ log ω Define z ≡ . Then, σθ ∂ 1 1 1 1 ω )) = −ωϕ(z ) ω ) − µGσθ (¯ (Γσθ (¯ − 2 log(¯ ω ) − (1 − µ)ϕ(z − σθ ) + 2 log(¯ ω) ∂σθ 2 σθ 2 σθ <0 A.3. Proof of Unique Solution for Ex-post Participation Constraint of Deposit Contract ˜ Y (0) − µb L(0) ℓb ≤ Rf (ℓb − 1) because p Consider the case in which R ˜ Y (0) − ¯ is simply zero if R µb L(0) ℓb > Rf (ℓb − 1). ¯ p ∂ ∂ (Y (¯ p) − µb L(¯ p)) = p)(1 − p q (¯ ¯) + (1 − µb ) q (p) dp = q ′ (¯ p)(1 − p ¯) − µb q (¯ p) ¯ ∂p ¯ ∂p 0 Note that I assume 54 ω , σL ) − µG(¯ Γ(¯ ω , σL ) 2+a > ω , σH ) − µG(¯ Γ(¯ ω , σH ) 1+a This assumption implies that σH is sufficiently larger than σL and ensures that q ′ (0) > q (0). There- fore, lim q ′ (¯ p)(1 − p p) = q ′ (0) − µb q (0) > 0 ¯) − µb q (¯ ¯→0 p lim q ′ (¯ ¯) − µb q (¯ p)(1 − p p) = −µb q (0) < 0 ¯→1 p and ∂ ′ p)(1 − p [q (¯ p)] = q ′′ (¯ ¯) − µb q (¯ ¯) − (1 + µb )q ′ (¯ p)(1 − p p) < 0 ¯ ∂p because Rk k q ′ ( p) = h ′ ( p) ω , σH ) − Γ(¯ ω , σH ) − µG(¯ Γ(¯ ω , σL ) − µG(¯ ω , σL ) >0 R k−1 (− ) (−) (+) Rk k q ′′ (p) = h′′ (p) ω , σH ) − Γ(¯ ω , σH ) − µG(¯ Γ(¯ ω , σL ) − µG(¯ ω , σL ) <0 R k−1 (+) (−) (+) ¯∗ such that Therefore there exists a p ∂ (Y (¯ p)) = q ′ (¯ p) − µb L(¯ p) ⪌ 0 p)(1 − p ¯) − µb q (¯ for ¯∗ ¯⪋ p p ¯ ∂p Since the banker’s share 1 − Y (¯ p) decreases in p ¯, the banker would never choose p ¯∗ . Through- ¯> p out this paper, I assume that the aggregate bankers’ net worth is sufficient to absorb any losses on loans to prevent the whole banking system from collapsing, implying p∗ ) ℓb ≥ Rf (ℓb − 1) p∗ ) − µb L(¯ ˜ Y (¯ R This proves there exists exactly one point that satisfies the participation constraint of the deposit ¯∗ ]. contract on the interval [0, p 55 Appendix B. Other Sectors of the Model B.1. Households The representative household consumes the consumption good (Ct ), lends to bankers (Dt ) at gross f risk-free interest rate (Rt ), and supplies labor (Lt ) to firms at real wage (wt ) to maximize expected utility ∞ Ct1−σ L1+η E0 βt − ψL t , t=0 1−σ 1+η subject to a sequence of budget constraints, C t + Dt ≤ w t L t + R t f −1 Dt−1 + Πt ∀t. E0 denotes the expectations operator conditional on time-0 information, β ∈ (0, 1) is the discount factor, σ is the coefficient of relative risk aversion, 1/η is the elasticity of labor supply, ψL is the labor weight in the utility function, and Πt are profits paid by capital producers to be defined below. Denoting UC and UL the marginal utility of consumption and leisure, the household’s optimization conditions are written by: wt = −UL (Ct , Lt )/UC (Ct , Lt ), (B.1) f 1 = Rt Et Mt,t+1 , (B.2) where Mt,t+1 ≡ βUC (Ct+1 , Lt+1 )/UC (Ct , Lt ) is the household’s stochastic discount factor. B.2. Firms A representative firm produces consumption goods using the following production function: k h e b Yt = At Ktα Lα e α b α t ( Lt ) ( Lt ) , where αk + αh + αe + αb = 1, Yt is aggregate output of consumption goods, Kt is the aggregate capital stock, Lt , Le b t , and Lt denote the labor supplied by households, entrepreneurs, and bankers, respectively.29 At is a stationary shock to total factor productivity (TFP) following an exogenous 29 Following BGG and Carlstrom and Fuerst (1997), entrepreneurs are assumed to supply labor services to ensure that each entrepreneur always has a nonzero level of net worth, because the financial contracting problem is not well defined otherwise. The same is assumed for bankers for the same reason. Note that αe and αb are assumed to be 56 AR(1) process to be specified later. Firms hire labor and rent capital in perfectly competitive markets to maximize profits. Profits are zero at equilibrium because the production function displays constant returns to scale. Compe- tition in the factor market implies that the firm’s optimality conditions equal marginal products of inputs with their prices: r t = α k Y t / Kt , (B.3) w t = α h Yt / L t , (B.4) e wt = α e Yt / Le t, (B.5) b wt = α b Yt / Lb t, (B.6) e b where rt denotes the real rental rate of capital and wt , wt , and wt are the real wage of households, entrepreneurs, and bankers, respectively. For simplicity, Le b t = Lt = 1 . B.3. Capital producers Capital producers purchase consumption goods (It ) from firms and the non-depreciated capital stock (1 − δ )Kt from entrepreneurs, where δ is the depreciation rate. Then they transform these two into new capital Kt+1 facing adjustment costs, and then sell the new stock to entrepreneurs at the unit price Qt .30 Thus, the profit-maximization problem of the capital producers is given by max Qt Kt+1 − It − Qt (1 − δ )Kt It subject to the law of motion of the capital stock 2 ϕk I t Kt+1 = (1 − δ )Kt + It − −δ Kt , (B.7) 2 Kt where ϕk governs the sensitivity of the price of capital to variations in the investment capital ratio. At equilibrium, the unit price of capital in terms of the consumption good, Qt , can be derived from the Euler equation for investment very small so that this modification does not have any significant effects on the results. 30 As in BGG, capital is homogeneous and entrepreneurs must sell all the non-depreciated capital stock and repurchase all the new capital stock in the same period so that agency problems apply to the entire capital stock, not just marginal investment. 57 −1 It Qt = 1 − ϕk −δ . (B.8) Kt B.4. Equilibrium A competitive equilibrium is defined by a set of allocations {Ct , Cte , Ctb , Lt , It , Kt+1 , Bt , Dt , t } and prices {rt , wt , wt , wt , Qt , Rt , Rt , Rt , Rt , Rt , Rt } that satisfy Equa- ℓt , ω ¯t ¯ t+1 , ω ep b , ℓt , p ¯ep ¯t+1 , p e b k ˜ b d f tions (25)-(33), (35)-(42), and (B.1)-(B.8), and the exogenous processes for technology and disper- sion that follow Equations (43), (44), and (45), respectively. Appendix C. Financial Frictions in Log-linear Form It is useful to express the key equations that incorporate financial frictions in the model in log- linear form to highlight the main differences relative to the literature. All variables in the following equations are expressed in log deviations from their deterministic steady state. The fundamental in- sight of standard financial accelerator models such as BG and BGG is summarized by the following equation Et r ˆt , +1 − r k f ˆt ˆt = ζℓ ℓ (C.1) where r ˆtk f ˆt is the ˆt +1 is the aggregate return to capital, r is the real risk-free interest rate, and ℓ entrepreneur’s leverage ratio. Equation (C.1) shows how entrepreneurs’ expected external finance premium depends on their leverage. A rise in leverage raises the cost of external funds and thereby decreases investment. In the absence of financial frictions, this relation collapses to Et r ˆtk +1 −r ˆtf = 0. CMR augment the above equation by adding time-varying cross-sectional dispersion to BGG (i.e., risk shocks). Et r ˆt + ζσ Et σ +1 − r k f ˆt ˆt = ζℓ ℓ ˆt+1 , (C.2) ˆt+1 is the cross-sectional dispersion of idiosyncratic productivity of entrepreneurs. This where σ CMR specification remedies one of the drawbacks of previous models, which is the procyclicality of risk premia, and generates countercyclical credit spreads and default. 58 The financial contracts laid out in this paper keep the above insights and modify the mechanism as follows. The equation obtained from the loan contract is similar to Equation (C.2). Et r ˆt + ζσ Et σ +1 − r ˆL,t+1 + ζσH Et σ k ˆt ˆt+1 = ζℓ ℓ L ˆH,t+1 . (C.3) The main differences are that it reflects the fact that bankers in this economy may require a higher f ˆt , than the risk-free rate, r return, r ˆt , and there are two time-varying cross-sectional dispersion terms because there are two types of entrepreneurs. Another equation obtained from the deposit contract shows how the bankers’ external finance premium depends on their leverage: ˆt+1 − r r ˆtf ˆb + φσ Et σ = φ ℓb ℓ ˆL,t+1 + φσH Et σ ˆH,t+1 . (C.4) t L ˆt+1 − r Note that r ˆtf = 0 in the standard financial accelerator models due to the absence of financial frictions between banks and households. It is worth nothing that this relation may collapse to ˆt+1 − r r ˆtf = 0 in this model if banks accumulate a sufficient amount of net worth or bankers’ capacity to leverage is not constrained. Finally, an equation that incorporates financial frictions in this model can be obtained by com- bining Equations (C.3) and (C.4): Et r ˆb + (ζσ + φσ )Et σ ˆt + φℓ ℓ +1 − r ˆL,t+1 + (ζσH + φσH )Et σ k f ˆt ˆt = ζℓ ℓ b t L L ˆH,t+1 . (C.5) This equation shows that entrepreneurs’ expected external finance premium depends not only on their leverage but also on bankers’ leverage and there are two dispersion shocks that have different ˆL.t+1 contribute to recessions but increases in σ implications on it. Rises in σ ˆH,t+1 play an important role during financial recessions. 59 Appendix D. Additional Figures and Tables Figure D.1. Return on assets and total assets of BHCs Notes: The right panel includes BHCs with assets > $5 billions only. Source: FR Y9-C Figure D.2. US non- and financial firm equity returns Source: The Center for Research in Security Prices (CRSP) 60 Table D.1. Number of observations for each country (Worldscope Data) Nonfinancial firms Banks Country iso3 start end years avg min max avg min max Australia AUS 1992 2015 24 902 164 1515 10 9 13 Austria AUT 1994 2015 22 63 50 84 9 7 13 Denmark DNK 1988 2015 28 109 52 141 34 17 47 France FRA 1988 2015 28 524 319 675 34 21 60 Germany DEU 1985 2015 31 440 89 683 21 10 30 Greece GRC 1991 2015 25 171 22 250 10 6 14 Hong Kong SAR, China HKG 1995 2015 21 638 130 1019 12 10 15 India IND 1999 2015 17 1353 309 2345 31 11 40 Indonesia IDN 2000 2015 16 266 206 355 25 14 38 Italy ITA 1985 2015 31 144 35 195 30 11 43 Japan JPN 1985 2015 31 2422 698 3432 96 61 118 Korea, Rep. KOR 1993 2015 23 909 124 1643 16 9 22 Malaysia MYS 1995 2015 21 645 216 845 14 11 19 Norway NOR 1989 2015 27 92 39 142 18 9 26 Philippines PHL 1995 2015 21 124 42 169 13 10 16 Spain ESP 1988 2015 28 96 56 116 15 7 21 Switzerland CHE 1988 2015 28 142 92 178 24 21 35 Taiwan, China TWN 1996 2015 20 1006 190 1642 23 11 32 Thailand THA 1991 2015 25 283 17 473 16 10 34 United Kingdom GBR 1986 2015 30 1172 399 1504 17 10 23 United States USA 1985 2015 31 4034 1812 6367 643 205 991 61