Capital Requirements, Risk Choice, and Liquidity 1. Provision in a Business-Cycle Model 2

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1 Capital Requirements, Risk Choice, and Liquidity 1 Provision in a Business-Cycle Model 2 Juliane Begenau 3 Stanford GSB January 14, Abstract 5 This paper develops a dynamic general equilibrium model to quantify the effects 6 of bank capital requirements. Households preferences for liquid assets imply a liq- 7 uidity premium on deposits. The banking sector supplies deposits and has excessive 8 risk-taking incentives. I show that the scarcity of deposits created by an increased 9 capital requirement can reduce the cost of capital for banks and increase bank lend- 10 ing. A higher capital requirement also increases banks monitoring incentives, which 11 improves the efficiency of banks activities. Under reasonable parametrizations, the 12 marginal benefit of a higher capital requirement related to this channel significantly 13 exceeds the marginal cost, indicating that U.S. capital requirements have been subop- 14 timally low. 15 Keywords: Capital Requirement, Bank Regulation, Bank Lending, Demand for Safe 17 Assets 18 JEL codes: E44, G21, G I am deeply indebted to Monika Piazzesi, Martin Schneider, and Pablo Kurlat for their invaluable guidance and patience. I also thank my discussants Frederic Boissay, Laurent Clerc, Michael Kiley, Dmitriy Sergeyev, Javier Suarez, and Skander Van den Heuvel and Saki Bigio, Theresa Kuchler, Moritz Lenel, David Scharfstein, Emil Siriwardane, Erik Stafford, Jeremy Stein, and Adi Sunderam for helpful comments. Numerous seminar and conference participants provided insightful comments. I gratefully acknowledge support from the Macro Financial Modeling Group dissertation grant from the Alfred P. Sloan Foundation and the Kohlhagen Fellowship Fund and the Haley-Shaw Fellowship Fund of the Stanford Institute for Economic Policy Research (SIEPR). First draft: November Correspondence: Juliane Begenau, Stanford Graduation School of Business, Knight Way, E264, Stanford, CA 94305, telephone: ; begenau@stanford.edu.

2 More equity... would restrict their [banks ] ability to provide loans to the 20 rest of the economy. This reduces growth and has negative effects for all. 21 Josef Ackermann, former CEO of Deutsche Bank, November 20, Introduction 23 What is the optimal capital requirement for banks? Bank regulators and some academics 24 argue that a low capital requirement allows banks to accept too much risk. Bank man- 25 agers and opposing academics counter that too high of a capital requirement increases 26 banks funding costs, thereby lowering the supply of credit to the economy. Motivated by 27 this debate, this paper analyzes which level of capital requirement mitigates bank risk- 28 taking without substantially reducing useful bank services, such as the credit supply and 29 liquidity provision. 30 To do this, I develop a general equilibrium model with banks that allows me to ana- 31 lyze and quantify this important trade-off. In the model, banks matter because of three 32 features. First, a subset of production in the economy depends on bank financing. Sec- 33 ond, banks issue risk-free deposits, which households value as safe and liquid assets. I 34 model households preference for deposits with a utility function that is increasing in de- 35 posits. Third, government subsidies to banks create excessive risk-taking incentives for 36 banks. I model these subsidies as a reduced-form transfer function that is increasing in 37 size, leverage, and losses in the banking sector. The calibrated model matches important 38 data moments in the U.S. economy and in the U.S. banking sector over the business cycle. 39 The model yields two key insights. First, I find that the optimal capital requirement in 40 the model is 12.4% of risky assets. This level weighs lower liquidity provision in the form 41 of deposits against higher and more stable consumption. Second, I show that a higher 42 capital requirement can increase the supply of bank credit. This finding is the result of 43 a general equilibrium effect. A higher capital requirement leads to a lower supply of 44 1

3 coveted deposits and thus increases households willingness to hold deposits at a lower 45 deposit rate. As a result, an increase in capital requirements can reduce banks funding 46 costs and thus increase bank lending. 47 At first glance, the potential positive effect of a higher capital requirement on bank 48 lending seems counterintuitive and inconsistent with the available empirical evidence. 49 The conventional wisdom that a higher capital requirement negatively affects lending is 50 based on the assumption that banks funding rates, such as deposit rates, do not mate- 51 rially respond to regulatory changes in the capital requirement. The empirical evidence 52 relies on this assumption to identify the effect of a higher capital requirement on lending. 53 For example, studies as Peek and Rosengren (1995) exploit capital requirement variations 54 that affect only a subset of banks as opposed to a sector-wide change that could have 55 moved equilibrium interest rates. Otherwise, the response in banks loan supply could 56 have been driven by other factors, such as changes in the loan demand. Thus, by de- 57 sign, empirical studies identify a partial equilibrium response (i.e., holding equilibrium 58 funding rates constant) to a higher capital requirement. 59 To see why capital requirements can cause an increase in lending in my model, it is 60 helpful to discuss the thought experiment that motivates the conventional wisdom. Con- 61 sider a simple model with banks that have two sources of funding: equity and deposits. 62 Assume that deposits are a cheaper source of funding. Thus, banks fund themselves with 63 as many deposits as allowed by the capital requirement, imposed by regulators, and their 64 marginal cost of funding is given by the weighted average of the costs of equity and de- 65 posits (with the weights determined by the capital requirement). As usual, banks optimal 66 lending decision equates the marginal benefit of lending with the marginal cost. 67 Now suppose that a higher bank capital requirement forces banks to increase their 68 equity-capital ratio (i.e., equity to risk-weighted assets), while everything else is assumed 69 to remain unchanged, including the deposit rate. This change increases the weight on 70 relatively expensive equity financing, thereby driving up banks cost of capital and thus 71 2

4 decreasing the level of lending. With fewer loans to fund, banks issue fewer deposits, and 72 thus both the supply of loans and deposits fall. In this thought experiment, the deposit 73 rate is assumed to be unaffected by the decrease in the supply of deposits. A higher bank 74 capital requirement leads therefore to an unequivocal decrease in the supply of loans. 75 Now consider a modification of this thought experiment that allows the deposit rate to 76 endogenously respond to the supply of deposits. Assume households derive utility from 77 holding bank deposits like in my model. This has two effects. First, deposits are a cheaper 78 funding source than equity for banks, because households are willing to forgo interest on 79 deposits in exchange for a convenience yield on deposits. Second, the convenience yield 80 is decreasing in the amount of deposits because households value deposits less the more 81 abundant they are in the economy. Thus, when a higher capital requirement causes a drop 82 in the supply of deposits, the convenience yield of deposits increases and the deposit rate 83 decreases. Which effect dominates in equilibrium the initial increase in banks equity 84 financing cost caused by the higher weight on equity financing or the subsequent decrease 85 in the deposit rates is a quantitative question. When the deposit rate channel dominates, 86 banks cost of capital decreases, thereby increasing their optimal level of lending. This 87 discussion thus underscores the importance of quantifying the effects of a higher capital 88 requirement in a general equilibrium framework. 89 To quantitatively evaluate the trade-offs associated with tighter capital requirements, I 90 build on a standard business-cycle model with aggregate shocks, a representative house- 91 hold, and two production sectors: a bank-independent sector and a bank-dependent sec- 92 tor. Both production sectors produce the same consumption good. The bank-dependent 93 production sector is owned and operated by a representative banking sector. This means 94 that banks invest a portion of their assets on behalf of bank-dependent firms in a de- 95 creasing returns to scale production technology. Banks have also access to a monitoring 96 technology that increases the average return of bank-dependent production and lowers 97 its exposure to aggregate shocks. They can choose the share of production they wish to 98 3

5 monitor. Monitoring is costly, as modeled by a standard convex monitoring cost function. 99 Banks are funded with deposits and equity. The Modigliani-Miller capital structure 100 irrelevance principle does not hold for two reasons. First, aside of consumption, house- 101 holds derive utility from holding deposits and thus are willing to pay a convenience yield. 102 Second, the banking sector receives a government subsidy. As a consequence, banks 103 choose as much leverage as they are allowed to choose by regulation. For tractability, 104 I model the subsidy as a reduced-form transfer function that is increasing in size, lever- 105 age, and losses in the banking sector. That this transfer function is not microfounded is 106 a weakness of my model, but Appendix A.1.3. shows that the reduced-form government 107 subsidy has similar properties as the implied subsidy in a microfounded model. 108 I match the model to quarterly U.S. data from the National Income and Product Ac- 109 counts (NIPA) and the Federal Deposit Insurance Corporation (FDIC) for The 110 welfare effects mainly depend on two parameters: the sensitivity of the subsidy function 111 to leverage and the elasticity of households deposit demand. I infer the subsidy s sen- 112 sitivity to leverage from banks first-order condition for bank leverage, together with the 113 subsidy estimates provided by Gandhi and Lustig (2015). The deposit demand elastic- 114 ity determines how much households dislike supply-shock-driven variations in the de- 115 posit consumption ratio. I calibrate this parameter by targeting the volatility of that ratio 116 and attribute all the observed volatility to supply shocks. Hence, my calibrated elasticity 117 is likely to be a lower bound of the true elasticity in the economy. 118 The model matches key balance sheet and income statement moments from banks, 119 together with macroeconomic aggregates. Moreover, its dynamics are consistent with 120 many business-cycle moments in the U.S. data that my calibration does not target. For 121 example, it is consistent with the procyclicality and volatility of banks balance sheet and 122 income statement variables. That it also captures the correlations between NIPA and 123 balance sheet variables makes it particularly suitable for studying the macroeconomic 124 effects of a higher bank capital requirement

6 The government subsidy distorts banks decisions, which gives rise to a welfare im- 126 proving bank capital requirement. I define the optimal capital requirement in the model 127 as the one that maximizes the welfare of the representative household, as measured by 128 the discounted expected lifetime utility from consumption and bank deposits. To find it, I 129 simulate the model at the calibrated capital requirement level of 9%. Then I calculate the 130 transition path and the new equilibrium for a range of new capital requirement levels. A 131 capital requirement level of 12.4% maximizes welfare. 132 An increase in the capital requirement from the calibrated level of 9% to the optimal 133 level of 12.4% reduces the supply of deposits by 86 basis points (bps), but increases con- 134 sumption by 33-bps and decreases the volatility in consumption by 18%. Bank lending 135 increases by 2.35%, which increases the output of the bank-dependent sector by 11%. The 136 general equilibrium effect associated with an increase in bank lending and thus deposits 137 reduces the deposit rate by 76-bps, leading to an 84-bps fall in banks cost of capital. In 138 the model, consumption increases because of two effects. First, output in the economy is 139 higher, because more lending leads to an increase in banks production. Second, a higher 140 capital requirement leads to more and smoother consumption, because banks increase 141 monitoring. The reason for this is simple. When banks decrease leverage because of the 142 higher capital requirement, the value of the government subsidy declines. Smaller gov- 143 ernment subsidies are paid out in states in which banks make losses. This, in turn, reduces 144 incentives for banks to monitor at suboptimal levels, leading to an increase in monitoring 145 efforts. More monitoring increases the average return on bank-dependent production, 146 which leads to higher consumption. Volatility in the economy decreases because higher 147 monitoring efforts mean that banks lower the fraction of risky bank activities that have a 148 relatively high exposure to the aggregate shock. 149 While the assumption of households preference for deposits and the reduced-form 150 subsidy function lead to a tractable model, it also makes the welfare results potentially 151 sensitive to functional forms and parametrizations. For this reason, I analyze the sensi

7 tivity of the welfare results to the parameters governing the subsidy function and house- 153 holds preferences for deposits. This analysis shows that reducing or increasing the elas- 154 ticity of deposit demand or the sensitivity of the subsidy function to banks leverage, 155 size, or losses leave the fundamental effects and trade-offs of a higher capital require- 156 ments in the model unaltered. My model also assumes that banks are the only providers 157 of safe and liquid assets. This is akin to assuming that the nonbank supply of these assets 158 is unaffected by changes in the capital requirement, such as for safe assets supplied by 159 the government (e.g., government bonds). I also show robustness of the welfare result 160 to a change in the supply of government bonds that banks can invest in. Across all ex- 161 periments, the resultant optimal capital requirement levels are close to the optimal level 162 implied by my benchmark calibration, ranging within 3 percentage points of my optimal 163 benchmark estimate of 12.4%. 164 Related literature 165 This paper connects to the banking literature on optimal capital regulation and to recent 166 work in macro-finance. Relative to nonfinancial firms, banks have unusually high lever- 167 age ratios. A strand of the theoretical banking literature has argued that this is optimal 168 because bank debt is safe and liquid (Gorton and Pennacchi (1990), Diamond and Rajan 169 (2001), Gorton, Lewellen, and Metrick (2012), and DeAngelo and Stulz (2015)). Under 170 this view, increasing the bank capital requirement and thus reducing bank leverage ratios 171 could be costly. By contrast, other authors (e.g., Admati, DeMarzo, Hellwig, and Pflei- 172 derer (2012)) have argued that a key benefit of increasing banks capital requirements is 173 that it can lead to less risk-taking by banks. My model captures both this cost and this 174 benefit, and thus makes it possible to determine the optimal capital requirement that bal- 175 ances the cost and benefit of imposing a higher level Bernanke and Gertler (1989) and Kiyotaki and Moore (1997) laid the foundation for Hanson, Stein, and Kashyap (2010) and Kashyap, Rajan, and Stein (2008) argue for higher capital requirements by citing the tax advantage of debt and competitive pressure over cheap funding sources as the leading source for banks high leverage. Harris, Opp, and Opp (2014), Malherbe (2015), and Bahaj and Malherbe (2018) study the effects of capital requirements in stylized models. 6

8 macroeconomic models with financial frictions. Bernanke, Gertler, and Gilchrist (1999) 178 and Christiano, Motto, and Rostagno (2010) have incorporated credit market imperfec- 179 tions into tractable New Keynesian models. I build on this work to develop a tractable 180 macroeconomic framework focusing on the effects of capital requirements. 181 This paper is more closely related to work that quantifies the effects of capital require- 182 ments and leverage constraints (see, for instance, Christiano and Ikeda (2013), Martinez- 183 Miera and Suarez (2014), Van Den Heuvel (2008), Nguyen (2014), De Nicolò et al. (2014), 184 Clerc et al. (2014), and Corbae and D Erasmo (2018)). Van Den Heuvel (2008) is one of the 185 first to use a quantitative general equilibrium growth model with households liquidity 186 demand. Unlike this paper, his work quantifies only the costs of a higher capital require- 187 ment on welfare in terms of a reduction in deposits and therefore leaves the question of 188 the optimal level to future research. A common feature of previous work is that a higher 189 capital requirement reduces financial fragility, but it also reduces the amount of lending, 190 which results in a lower gross domestic product (GDP). In my model, the effects on risk- 191 taking and lending activities are still present, but I also incorporate the consequences of 192 a change in the supply of bank deposits on the deposit rate in a setting in which house- 193 holds derive utility from deposits. When bank deposits become scarce, households are 194 willing to hold them even if they pay a low interest rate, because their convenience yield 195 increases. This lowers banks cost of capital, leading to more not less lending in the 196 economy. 197 The idea that the demand for safe and liquid assets drives down yields and thus can 198 have important effects on the banking sector is at the center of the Bernanke (2005) sav- 199 ings glut hypothesis, first formalized by Caballero, Farhi, and Gourinchas (2008) and sub- 200 sequently discussed by Caballero and Krishnamurthy (2009), Mendoza, Quadrini, and 201 Rios-Rull (2009), and Gorton, Lewellen, and Metrick (2012), among others. My model 202 captures this idea and analyzes its importance for optimal bank capital regulation in a 203 quantitative setting

9 Finally, this paper presents a first step toward quantifying the optimal capital require- 205 ment in a model in which banks have excessive risk-taking incentives and are important 206 for liquidity provision. Begenau and Landvoigt (2018) build on this paper to study the 207 response of the shadow banking system to a higher capital requirement, which is beyond 208 the scope of the current paper Model 210 This section presents the model and outlines the key assumptions. I use a general equilib- 211 rium model cast in discrete time, with an infinite horizon. The economy produces a single 212 good c using two production technologies, each operated by a different sector. These two 213 sectors are a bank-independent sector (sector f ) and a bank-dependent sector (sector B) Bank-independent sector 215 Firms in the bank-independent sector are competitive and rent capital k f and labor H f 216 from households to operate a Cobb-Douglas production function. The capital stock in the 217 bank-independent sector is owned by households Bank-independent sector production technology The firms in the bank-independent219 sector produce output with a Cobb-Douglas technology: 220 y f t = Z f t ( ) k f α ( ) t 1 H f 1 α t, (1) where Z f t is the productivity level at time t, k f t 1 is the capital stock installed at t 1, α is 221 the share of capital, and H f t is the number of hours worked. Productivity is stochastic: 222 log Z f t = ρ f log Z f t 1 + σ f ɛ f t, (2) 8

10 where ɛ f t is drawn from a multivariate normal distribution Bank-independent sector problem I abstract from financial frictions in the bank- 224 independent sector and assume that bank-independent firms maximize profits: 225 π f t = max k f t 1, H f t y f t r f t k f t 1 w f t H f t, where r f t is the return households receive for renting out capital k f t 1, and w f t is the wage 226 households receive for labor hours H f t. Because of perfect competition, firms make no 227 profits in equilibrium Capital accumulation 229 A common capital market is used for capital in bank-dependent production and for that 230 in bank-independent production. Capital in sector j { f, B} depreciates at a rate of δ j 231 and accumulates according to 232 k j t = ij t + (1 δ j) k j t 1. Adjustments to the stock of capital are costly. When investment exceeds the replacement 233 of depreciated capital, investors incur a proportional capital adjustment cost of 234 ϕ j ( i j t k j t δ j ) 2 k j t, (3) where ϕ j is the sector-specific adjustment cost parameter. 235 Capital in the bank-independent sector is owned by households. Capital in the bank- 236 dependent sector is owned by banks

11 2.3 Banking sector 238 Banks play two roles in this economy: First, they produce a good that households con- 239 sume. Second, households value bank deposits. Banks are owned by households and 240 maximize shareholder value by generating cash flows that are discounted using the stochas-241 tic discount factor of households. 242 There exists a measure of one continuum of infinitely lived, ex ante identical banks. 243 Because all banks are ex ante identical and the shock to the bank-dependent sector is an 244 aggregate shock, banks choices are perfectly correlated, and I can speak of a representa- 245 tive bank that takes prices as given. That is, I can think of the sum of banks described 246 here as the aggregate banking sector. For this reason, I describe the problem of banks as 247 if it is the problem of the representative bank Bank-dependent sector production technology and monitoring I assume that a 249 subset of production takes place in the bank-dependent sector. The bank-dependent pro- 250 duction technology y B t has decreasing returns to capital. Z B,bad t is the stochastic produc- 251 tivity level of banks production technology when banks do not monitor. In t 1, banks 252 can choose to monitor a share mt 1 B of production in t. In this case, the productivity level 253 of the monitored production share is Z B,good t. For each i {good, bad}, Z B,i t follows 254 log Z B,i t = µ B,i + ρ B,i log Z B,i t 1 + σb,i ɛ B t, (4) with mean µ B,i, autocorrelation ρ B,i. The shock ɛ B t is common to all banks and drawn 255 jointly with ɛ f t from 256 ɛ f t ɛ B t N 0 0, 1 σ f B σ f B 1, where σ f B is the covariance between ɛ f t and ɛb t. For simplicity, I assume ρb,good = ρ B,bad

12 I assume that monitoring increases the value and efficiency of production, because 258 the mean of Z B,good t is higher and the standard deviation of Z B,good t is lower compared 259 with Z B,bad t. This means µ B,good > µ B,bad and σ B,good < σ B,bad. These assumptions are 260 consistent with the evidence. For example, Nini, Smith, and Sufi (2012) present evidence 261 that suggest a positive role of creditors in firm performance. Dell Ariccia and Marquez 262 (2006) and Dell Ariccia, Igan, and Laeven (2012) show that tighter lending standards im- 263 prove bank profitability. It follows that monitoring gives banks a risk choice: a lower m B t 264 increases banks exposure to the common shock in t The cost of monitoring is governed by 266 ( ) G mt B = φ ( ) 2 mt B, 2 with φ > 0. The variable monitoring intensity m B t and convex monitoring cost assump- 267 tion are standard in the banking literature (e.g., Besanko and Kanatas (1993); Stein (2003); 268 Tirole (2006), p. 360; see also Appendix Section A.1.1. for more discussion and references). 269 Given the monitoring share choice m B t 1 and capital kb t 1, banks produce output yb t 270 with a decreasing returns to scale technology: 271 ( yt B = ( ) ) mt 1 B ZB,good t + 1 mt 1 B Z B,bad t }{{} : Zt B ( k B t 1) v, (5) where v < Discussion of assumptions 273 The bank dependence of one production sector reflects that banks provide funds to bor- 274 rowers who have limited access to capital markets (e.g., see the discussion in Freixas and 275 Rochet (1998) and in Donaldson et al. (2018)). The idea that some agents need lenders 276 (banks) to realize production projects has significant empirical support (e.g., James (1987); 277 Petersen and Rajan (1994); Berger and Udell (1995); Hadlock and James (2002); Denis and

13 Mihov (2003)). Theoretically, a bank dependence emerges because of asymmetric infor- 279 mation problems between borrowers and lenders that banks are especially well equipped 280 to solve (e.g., Sharpe (1990); Diamond (1984); Holmstrom and Tirole (1997)). This allows 281 banks to choose the risk and scale of their activities. Following Brunnermeier and San- 282 nikov (2014), I assume that the bank-dependent production sector is owned and operated 283 by banks. This implies that banks choose the riskiness and scale of bank-dependent pro- 284 duction directly. 285 Diminishing returns to capital capture the following intuitive idea. Given differences 286 in the risk profiles of borrowers and monitoring costs, there exist more or less profitable 287 borrowers. Holding monitoring intensity fixed, the first dollars of bank loans go to the 288 most profitable borrowers. The remaining potential borrowers are less and less profitable 289 investments, because they require increasingly more monitoring, default more, or are 290 increasingly less productive (see, e.g., Dell Ariccia, Igan, and Laeven (2012) for empirical 291 evidence on the decreasing creditworthiness of the marginal borrower). 292 I separate the investment amount from the investment quality to capture that more 293 lending is not necessarily economically valuable. Allowing banks to choose both m b t and 294 k B t is thus necessary to study whether capital requirements lead to valuable investments Banks balance sheet The balance sheet equates risky assets kt 1 B and riskless assets 296 b t 1 (issued by the government) to bank deposits s t 1 and equity e t 1 : 297 k B t 1 + b t 1 = s t 1 + e t Banks profits At the beginning of the period t, the aggregate shocks ɛ B t and ɛ f t are 298 realized. Banks enter the period with capital k B t 1, government securities b t 1, bank de- 299 posits s t 1, equity e t 1, and a monitoring intensity mt 1 B. Banks receive operating income 300 from the bank-dependent production technology and investment income from riskless 301 government bonds. Bank expenses are interest payments on deposits. Therefore, profits

14 are 303 π t = y B t δ B k B t 1 }{{} production income + r gov t 1 b t 1 }{{} interest inc. r t 1 s }{{ t 1. } interest exp Market imperfections in the banking sector Banks are subject to Basel-III-type 304 capital requirements. The Basel III accords stipulate that banks must back a specific per- 305 centage of risk-weighted assets with equity. Safe assets, such as government bonds, have 306 a risk weight of 0%. The capital requirement is 307 e t ξk B t, where ξ determines the amount of equity e t needed to back risky assets k B t. 308 Banks incur dividend adjustment costs that arise when dividend payouts deviate from 309 the target level d. This introduces intertemporal rigidities consistent with the empirical 310 evidence (e.g., Lintner (1956); Dickens, Casey, and Newman (2002)). Following Jermann 311 and Quadrini (2012), the payout cost function takes the following form: 312 f (d t ) = κ 2 ( dt d ) 2, where κ governs the size of this cost. 2 Outside the steady state, dividend adjustment 313 costs introduce intertemporal rigidities into the balance sheet that make banks choices 314 of equity dependent on the current level of equity. This is consistent with the evidence 315 in Adrian and Shin (2011). The stickiness of equity can be derived either from a debt 316 overhang problem (e.g., Admati, DeMarzo, Hellwig, and Pfleiderer (2012)) or equity is- 317 suance costs. Paying out too much in dividends also can be costly because of an increasing 318 marginal tax rate on equity distributions (see Hennessy and Whited (2007)). 319 Finally, I assume that the benefits for banks from a government guarantee are captured The introduction of dividend adjustment costs neither qualitatively nor materially quantitatively affects the results of the model. However, they improve the model s fit of second moments. Given log consumption volatility, second-order moments do not play a large role in the welfare results. 13

15 by a reduced-form transfer function: 321 TR ( k B t 1, e t 1 + π t k B t 1 ) = ω 1 k Bt 1 exp ( ω 2 ( e t 1 + π t k B t 1 )), (6) where ω 1 and ω 2 are positive constants. It is increasing in banks size kt 1 B and decreasing 322 in banks after-profit capitalization (e t 1 + π t ) /kt 1 B, the inverse of leverage. 323 Discussion of assumption 324 In my model, I take the existence of government guarantees that result in subsidies to 325 the banking sector as given. Equation (6) captures the benefits from government subsi- 326 dies, which range from the tax advantage of debt to any government actions that help 327 stabilize banks. Modeling the benefits from government guarantees as a reduced-form 328 subsidy function is a weakness of my model. In Appendix Section A.1.3., I show how this 329 functional form can be derived from first principles. 330 I chose a reduced-form approach because an endogenous formulation of government 331 guarantees to banks within a quantitative general equilibrium framework is only tractable 332 without aggregate risk (e.g., Nguyen (2014)) or if bank size is not a state variable (see Be- 333 genau and Landvoigt (2018)). Both features are essential for this study. Aggregate risk 334 is important in the model, because I want to analyze how capital requirements change 335 banks risk-taking behavior vis-a-vis policy relevant aggregate risk (i.e., m B ). The as- 336 sumption that bank size is not a state variable implies that banks risk choice is indepen- 337 dent of bank size. This assumption is problematic, because government guarantees are 338 increasing in bank size as shown by Gandhi and Lustig (2015), meaning that banks risk 339 choice is size dependent. In addition, banks of different sizes often follow very different 340 business models that affect their exposure to aggregate shocks. For example, large banks 341 tend to be less capitalized, invest more in trading activities, and have less stable deposit 342 funding. In other words, the two assumptions (no aggregate risk and no size state vari- 343 able) required to endogenize the subsidy in equation (6) would not allow me to quantify

16 the trade-off of a higher capital requirement regarding banks risk-taking, lending, and 345 liquidity provision choice. 346 The first-order purpose of a government subsidy for banks in the model is to al- 347 low banks to trade-off the benefits of the subsidy against the costs of more risk-taking. 348 This trade-off depends on the sensitivity of the subsidy to changes in bank capitalization ( ) 349, which also would be the case in a model with an endogenous subsidy. This 350 e t 1 +π t k B t 1 sensitivity can be estimated using the first-order condition of banks, data on banks prof- 351 its and banks leverage, and the subsidy estimate by Gandhi and Lustig (2015). Section discusses the calibration in more detail. Table 10 shows how the results vary when the 353 parameters of the subsidy function change Banks problem In period t, banks choose capital k B t, monitoring intensity mb t, 355 equity e t, government bonds b t, and deposits s t, in order to maximize the present value 356 of the stream of future cash flows to shareholders. These cash flows are called dividends. 357 Banks use the cash flows from profits π t and the government subsidy TR ( ) to finance 358 additions to next period s equity capital e t e t 1, the capital adjustment and monitoring 359 costs, as well as dividend payouts to shareholders. Due to dividend payout costs, the 360 necessary cash flow to payout d t is d t + f (d t ). Thus, dividends are 361 d t = e t 1 e t + π t f (d t ) + TR ( i B ) 2 ϕ t b kt B δ B kt B }{{} capital adjustment costs ( kt 1 B, e t 1 + π t kt 1 B φ ( ) 2 ) mt B. } 2 {{} monitoring costs The banks problem is written recursively. Defining ẽ = e + π as equity after profits is 362 useful in stating the problem. The vector ε summarizes the two aggregate shocks ɛ f and 363 ɛ B. Banks state variables are denoted by ε, the aggregate state vector X (described in 2.6), 364 equity after profits ẽ ( e, k B, m B, ε, X ), and k B. Banks have unlimited liability: if ẽ < 0, they

17 set d < 0. Banks discount the future with the pricing kernel M (X, ε ) from shareholders. 366 They choose capital k B, government bonds b, bank deposits s, the amount of risk-taking 367 m B, equity after profits ẽ, and dividends d to solve 368 ( ) V B ẽ, k B, X, ε = max [ d + E ε ε M ( X, ε ) ( V B ẽ ( ), k B, X, ε )], (7) k B,b,s,m B,ẽ ( ),d subject to 369 ( d = ẽ e f (d) + TR k B, ( k B ( 1 δ b) ) k B 2 ϕ b k B δ k B φ ( 2 ẽ e, k B, m B, X, ε ) ( = e + π k B, m B, b, s, X, ε ), k B + b = e + s, e ξk B. ẽ k B ) ( m B) 2, E is the expectations operator conditional on information known at t Households 371 Households are identical and live forever. They own capital k f from the bank-independent 372 sector, rent capital k f, and inelastically supply labor H f to its firms. They also own bank 373 equity and determine banks discount factor Preferences Households value consumption c and money-like assets in the form of 375 deposits s that generate utility in the period they are acquired and pay interest in the fol- 376 lowing period. Similar to Christiano, Motto, and Rostagno (2010), here the utility function

18 is defined over consumption and bank deposit s t in a money-in-the-utility specification: 378 s t c t 1 η U (c t, s t ) = log c t + θ 1 η, (8) where θ > 0 is the utility weight on deposits, and η governs the curvature of the deposit- 379 consumption ratio. More consumption raises the marginal utility of liquidity. 380 Discussion of assumption 381 Several rationales can explain liquidity demand. These include exposure to liquidity 382 shocks like in Diamond and Dybvig (1983) and transaction and liquidity costs like in 383 Baumol (1952) and Tobin (1956). Feenstra (1986) shows that transaction-based money 384 demand can be represented by a money-in-the-utility function. 385 Strong empirical evidence supports the existence of a demand for liquidity and a 386 convenience yield on liquid assets (e.g., Gorton, Lewellen, and Metrick (2012); Krishna- 387 murthy and Vissing-Jorgensen (2012); Nagel (2016)). In Appendix Section A.1.1., I discuss 388 this assumption in great detail and provide additional evidence Households problem I define the net worth n t 1 of households after the realiza- 390 tion of shocks ɛ b t and ɛ f t (summarized by the vector ε t ) and after intraperiod decisions 391 (e.g., labor market decisions) have been made as 392 n t 1 = Liquid Asset Wealth + Production Income Taxes. Liquid asset wealth consists of the gross return on s t 1 deposits (chosen last period) 393 Liquid Asset Wealth = (1 + r t 1 ) s t 1. 17

19 Households do not hold bonds. 3 Households receive bank-independent production in- 394 come from labor H f and capital k f t 1 : 395 Production Income = (r f t + 1 δ ) k f t 1 + w f t H f. Given the fixed labor supply H f, the production inputs earn their marginal product. The 396 wage is w f t = (1 α) y f t and the net return on k f t 1 is r f t = αy f t. Taxes T are lump sum. 397 Aside from net worth, households receive stock income from holding Θ t 1 bank shares 398 (chosen last period) that pay dividends d t and are valued at p t per unit of bank shares: 399 Income from Bank Equity = (d t + p t ) Θ t 1. Households value function is written recursively. Variables with superscript indicate 400 the next period s value. The value function depends on n, k f, Θ, the aggregate state 401 vector X, and the realization of shocks ε. Households maximize their value function by 402 choosing consumption c, deposits s, capital k f, bank shares Θ, and net worth n : 403 ( ) V H n, k f, Θ, X, ε = max {c,s,k f,θ,n (X,ε )} U ( c, s ) [ ( + E ε ε βv H n ( X, ε ), k f, X, ε )], (9) 3 Even if households were given the choice, they would not want to invest in bonds. Bonds earn the same interest rate as deposits, because they are risk-free and receive a risk weight of zero in banks capital requirement constraint. But bonds do not provide a liquidity benefit. If the rates were not equal, there would be an arbitrage opportunity. If bonds were cheaper than deposits, banks could issue more deposits to buy bonds. This would drive up bond prices. If bonds were more expensive, banks would not want to hold bonds. 18

20 subject to the budget constraint 404 ( c + s k f ( 1 δ f ) ) k f ϕ f k f δ k f + p (X, ε) Θ = n + [d (X, ε) + p (X, ε)] Θ, }{{} capital adjustment costs (10) and the definition for net worth tomorrow: 405 n ( X, ε ) = (1 + r (X)) s + ( r f ( X, ε ) + 1 δ f ) k f. (11) Households incur capital adjustment costs per unit of capital k f (see equation (3)). The 406 stochastic discount factor in the economy is given by 407 M ( X, ε ε ) = β ( Uc (c (X, ε ), s ) ). U c (c (X, ε), s)) 2.5 Government 408 The government follows a balanced budget rule whereby it maintains debt levels of 409 B t+1 = B t so that 410 TR t + r gov t B t = T t. (12) 2.6 Recursive competitive equilibrium 411 Shocks occur first, and decisions are made subsequently. Then a new period starts again. 412 The state vector X contains the aggregate net worth of banks Ẽ, the aggregate net worth 413 of households N, the aggregate capital stock of households K f, the aggregate capital 414 stock of banks K B, the monitoring intensity M b, and the productivity levels of the bank- 415 independent sector Z f and banks, Z B,good and Z B,bad

21 Definition. Given an exogenous government debt policy B, 4 a recursive competitive 417 equilibrium is defined by a pricing kernel M (X, ε); prices: w f (X, ε), r f (X, ε), p (X, ε), 418 r (X), and r gov (X); value functions for households V H and banks V B ; households pol- 419 icy functions for consumption P c H labor supply P H f H P b B, deposits Ps H, capital Pk f H ; banks policy functions for capital Pk B B, bank equity shares PΘ H, and 420, monitoring efforts Pm B, bonds 421, deposits Ps B, equity Pe B, and dividends Pd B ; and X the function governing the law of 422 motion for X such that 423 B 1. Given the price system and a law of motion for X: 424 (a) the policy function PB k b,pm b B, Pb B, Ps B, Pe B, and Pd B and the value function for 425 banks V B solve the Bellman equation, defined by equation (7). 426 (b) the policy functions P c H, Ps H, Pk f H, PΘ H, and PH f H and the value function for house- 427 holds V H solve the Bellman equation, defined by equation (9). 428 i. The equilibrium bank stock price satisfies 429 p (X, ε) = E ε [ M ( X, ε ε ) ( p ( X, ε ) + d ( X, ε ))]. 2. w f (X, ε) and r f (X, ε) satisfy the optimality conditions of bank-independent firms For all realization of shocks, the policy functions imply 431 (a) market clearing for 432 i. government bonds: P b B = B 433 ii. deposits: P s B = Ps H Government securities are not a choice variable in this model. The government should optimally set B =, financing the securities with nondistortionary taxes. The question of how to set government securities and the capital requirement jointly is only interesting if the choice of government securities involves a trade-off; however, this question is outside the scope of this paper. 20

22 iii. capital: P k f H = k f and P k B H = k B and 435 k f + k B = i f + i B + ( ) ( 1 δ f k f + 1 δ B) k B, with i f and i B denoting each sector s investment in its capital stock 436 iv. labor: P H f H = H f 437 v. bank shares: Θ = vi. consumption: 439 ( ) ( c = y B + y f + 1 δ f k f + 1 δ B) k B κ ( d 2 d) 2 ( ) k f i ( f ( ϕ f k f δ f i k B B ) 2 ) 1 + ϕ b δb k B φ 2 (m B) 2 (b) consistency with aggregation: n = N, ẽ = Ẽ, k f = K f, and k B = K B The government budget constraint in equation (12) is satisfied The law of motion for X is consistent with the policy functions, rational expecta- 442 tions, and X = X (X). 443 Appendix Section A.2 lists the full set of equilibrium equations The trade-off from a higher capital requirement 445 This section summarizes the effects of a higher capital requirement in the nonstochastic 446 steady state. This is the equilibrium in which Z B,good, Z B,bad, and Z f are constants. 447 Definition. Given an exogenous government debt policy B, a steady-state equilib- 448 rium is defined by constant values for all variables such that the equilibrium definition in 449 Section 2.6 is satisfied

23 3.1 A higher deposit demand decreases the deposit rate 451 That households value bank deposits as modeled by equation (8) implies that deposits 452 deliver a convenience yield. Consequently, it is optimal for banks to increase leverage as 453 much as regulation allows for it, even in the absence of government guarantees. DeAn- 454 gelo and Stulz (2015) show this mechanism in a stylized model. The convenience yield 455 (the left-hand side [LHS] of equation (13)) comes from households first-order condition 456 with respect to deposits and equals the marginal increase in utility from increasing de- 457 posits by $1, keeping the marginal utility of consumption constant: 458 U (c, s) s ( 1/ ) U (c, s) = c ( r e ) r 1 + r e, (13) where U (c, s) / s = θs η c η 1 is the marginal utility of deposits and U (c, s) / c = 459 1/c θs 1 η c η 2 is the marginal utility of consumption. Both are positive. The return 460 on equity is r e, and the return on deposits equals their yield r. As long as households 461 derive value from holding an additional unit of deposits (i.e., the LHS of equation (13) is 462 positive), the cost of equity financing exceeds the cost of debt financing. 463 The convenience yield of deposits varies with the quantity of deposits s. With η > 1, 464 a reduction in s makes deposits more desirable, increasing the convenience yield. This 465 drives down the deposit rate r and increases the spread between r e and r The bank capital requirement is binding 467 In the nonstochastic steady state and for every combination of the parameters, the capi- 468 tal requirement constraint of banks is binding if households derive utility from deposits 469 (positive convenience yield) or if banks receive transfers from the government that en- 470 courage deposit financing over equity financing. This can be seen from the first-order

24 condition of banks with respect to equity: 472 ( ) ( ) µ = re r 1 + r e + ω ẽ 1 + r 2TR 1, k B 1 + r e, (14) where µ is the Lagrange multiplier on the capital requirement constraint in equation (7). 473 With a positive convenience yield on deposits (equation (13)), the capital requirement con- 474 straint is binding (i.e., µ > 0). The marginal transfer (the second term on the right-hand 475 side [RHS] of equation (14)) is also positive and thus tightens the constraint. By increasing 476 equity, banks forgo the convenience yield on deposits and the marginal subsidy The optimal size of the banking sector 478 Because of the decreasing returns to scale in the production technology, the banking sector 479 has an optimal size (i.e., risky assets k B ) that is determined by the first-order conditions 480 for equity (equation (14)) and k B : 481 ) TR (1 k B + ω 2 (1 v) yb k }{{ B + } := TR k B >0 ( 1 + r B δ B) = ξ (1 + r e ) + (1 ξ) (1 + r), (15) }{{} Funding cost of k B where r B vy B /k B. Given equity r e and debt r funding rates, the optimal size of the 482 banking sector trades off the benefits (on the LHS) and the costs of risky assets (on the 483 RHS). The subsidy drives a wedge between the funding costs and the return on risky 484 assets. The funding cost of k B is the weighted average of the return paid to shareholders 485 (1 + r e ) and creditors (1 + r). Holding the RHS constant, the higher the subsidy, the lower 486 the required marginal productivity of risky assets. This implies that banks are larger in 487 a world with subsidies than in a world without, consistent with the evidence in Gandhi 488 and Lustig (2015)

25 3.4 The monitoring choice and excessive risk-taking incentives 490 The monitoring intensity m B determines the production weight on good projects that fea- 491 ture a lower exposure to the aggregate shock and a higher payoff. Thus, m B allows banks 492 to modulate the effective exposure to the aggregate shock ε B and the expected payoff of 493 risky assets. The presence of the subsidy distorts the monitoring choice. The first-order 494 condition with respect to m B, 495 [ φm B = β Z B,good Z B,bad] ( k B) ν ( 1 ω 2 TR/k B), (16) shows that the optimal monitoring choice without the subsidy (ω 2 = 0) maximizes the 496 expected payoff of risky assets. The subsidy creates a wedge that decreases the optimal 497 monitoring intensity, thus reducing the expected return of bank-dependent production 498 and increasing its risk. Intuitively, the subsidy compensates banks during times when 499 profits are low, thereby increasing their willingness to load up on aggregate risk The effects of an increase in the capital requirement 501 The conventional argument against a higher capital requirement is the higher cost of 502 funding loans. This argument implicitly assumes that banks funding costs are invariant 503 to changes in the capital requirement. In my model, this is not the case. From equations 504 (13) and (14), the capital requirement constraint is binding. Increasing ξ leads to, ceteris 505 paribus, higher funding costs for k B (i.e., the RHS of equation (15) goes up) as the weight 506 on equity financing increases. To balance equation (15), the LHS has to increase as well. 507 Ignoring the first term in equation (15), the LHS can only increase when risky assets such 508 as loans generate a higher return. With the standard expression for r B = vz B ( k B) v 1 and 509 v < 1, banks increase the return on risky assets by reducing k B. Thus, banks reduce lever- 510 age by holding equity constant and reducing assets. This summarizes the conventional 511 wisdom about banks lending response to a higher capital requirement

26 In general equilibrium, an additional response can trigger the counterintuitive reduc- 513 tion in funding costs after an increase in the capital requirement. When banks comply 514 with the higher capital requirement by delevering, they reduce their supply of deposits 515 to the economy. Equation (13) determines the response of the deposit rate to a change in 516 deposits. As long as households value deposits more the scarcer they become (η > 1), a 517 reduction in the supply of deposits will drive down the interest rate on deposits. If the 518 reduction in the interest rate is large enough, banks cost of capital falls, giving banks 519 incentives to increase lending. 520 Next, I discuss why a higher capital requirement reduces risk-taking incentives. A 521 reduction in leverage marginally reduces the subsidy from risk-taking in equation (16), 522 causing banks to choose higher levels of m B. Risk-taking via leverage and less monitoring 523 is thus a complementary activity. An increase in m B leads to an increase in the average 524 productivity level of banks investment. More monitoring also lowers banks exposure 525 to aggregate risk, reducing the volatility of bank-dependent output and thus total output. 526 The capital requirement trades off a reduction in liquidity provision via deposits against 527 an increase in consumption and a reduction in volatility fueled by more efficient bank in- 528 vestments. The next section addresses the size of these effects Key Parameters 530 Before explaining how I quantify the model, let s highlight the parameters that matter 531 most for the welfare results. The magnitude of the fall in r depends on the curvature pa- 532 rameter η in the utility function of the households. In addition, the parameter ω 2 matters 533 for the optimal risk choice of banks. A high value for ω 2 implies a larger sensitivity of 534 the subsidy to leverage. In Tables 9 and 10, I will show how changing these parameters 535 affects the welfare results

27 Table 1: MAPPING THE MODEL TO THE DATA Model NIPA and FDIC balance sheet and income statement y B : bank output income interest income on securities k B : bank capital assets securities cash fixed assets y f : firm output NIPA GDP bank output k f : firm capital NIPA capital stock k B c: consumption NIPA consumption s: deposits total bank debt π: profits net income + noninterest expense r: deposit rate total interest expenses/total bank debt e: equity Tier 1 equity This table lists model objects in the left column and their data counterparts in the right column. The FDIC data can be downloaded from 4 Mapping the model to the data 537 I calibrate the model to quarterly U.S. data from 1999 Q1 to 2016 Q4. This period reflects a 538 deregulated banking system that arguably started with the passing of the Gramm-Leach- 539 Bliley Act. The bank-independent sector is mapped to NIPA data, whereas the bank- 540 dependent sector is mapped to aggregate commercial banks and savings institutions data 541 collected by the FDIC. The FDIC collects balance sheet and income statement data from 542 all depository institutions. I convert dollar values into thousands of 2009 U.S. dollars per 543 capita, using St. Louis Fed population numbers. Table 1 summarizes how I map model 544 objects to the data. I discuss the rationale for my choices in the next section Parametrization 546 The calibrated parameters can be divided into three groups. The parameters in the first 547 group (Panel A of Table 2) are directly matched with their data counterpart. I choose the 548 parameters of the second group (Panel B of Table 2) such that the steady-state equilibrium 549 conditions of the model are consistent with the selected data moments. The remaining pa