Bank Market Power and Monetary Policy Transmission: Evidence from a Structural Estimation
银行市场力量和货币政策传导：结构估计的证据

We examine the quantitative impact of bank market power on the transmission of monetary policy through the banking system. This transmission channel is potentially important, given three decades of consolidation in the banking industry that has softened competitive pressure. Moreover, recent research offers qualitative evidence that bank market power affects the pass-through of monetary policy to the supply of loans (Scharfstein and Sunderam (2016), Drechsler, Savov, and Schnabl (2017)). Yet, market power is not the only friction in the banking system that influences pass-through. For example, traditional analysis of monetary policy transmission focuses on regulatory constraints, such as bank reserve or capital requirements, as the central frictions that influence monetary policy transmission (Bernanke and Blinder (1988), Kashyap and Stein (1995)). However, the qualitative nature of this evidence leaves open the question of the relative magnitude of traditional versus market-power transmission channels.
我们研究了银行市场力量对货币政策通过银行体系传导的定量影响。鉴于银行业三十年来的整合，竞争压力有所减弱，这种传导渠道可能非常重要。此外，最近的研究提供了定性证据，即银行市场力量影响货币政策传导到贷款供应（Scharfstein 和 Sunderam（2016 年），Drechsler，Savov 和 Schnabl（2017 年））。然而，市场力量并非是影响传导的银行体系中唯一的摩擦。例如，传统的货币政策传导分析侧重于监管约束，如银行准备金或资本要求，作为影响货币政策传导的中心摩擦（Bernanke 和 Blinder（1988 年），Kashyap 和 Stein（1995 年））。然而，这些证据的定性性质使传统与市场力量传导渠道的相对重要性问题尚未解决。

To do so, we use data on U.S. banks to estimate a dynamic banking model with three frictions: regulatory constraints, financial frictions, and imperfect competition. The estimation allows our data to discipline the model parameters and thus identify the relative magnitudes of these three frictions. We find that bank market power plays an important role in determining the degree of monetary policy transmission. In terms of magnitude, the effects of bank market power are comparable to those of bank capital regulation, while the effects of bank reserve requirements are limited.
为此，我们使用美国银行的数据来估计一个具有三种摩擦的动态银行模型：监管约束、金融摩擦和不完全竞争。估计使我们的数据约束模型参数，从而确定这三种摩擦的相对大小。我们发现银行市场力量在决定货币政策传导程度方面发挥着重要作用。在数量上，银行市场力量的影响与银行资本监管相当，而银行准备金要求的影响有限。

In further analysis, we first show that banks face nontrivial costs when they access external financial markets. These frictions play a pivotal role in connecting banks' deposit market power to their lending decisions, as external financing costs link banks' sources and uses of funds. In addition, these frictions help explain the differential sensitivity of lending to the policy rate between big and small banks.
在进一步分析中，我们首先展示银行在接入外部金融市场时面临非微不足道的成本。这些摩擦在将银行的存款市场力量与其放贷决策联系起来方面发挥着关键作用，因为外部融资成本将银行的资金来源和用途联系在一起。此外，这些摩擦有助于解释大型银行和小型银行对政策利率的放贷敏感性之间的差异。

Second, we show that bank market power interacts with capital regulation to reverse the effect of monetary policy when the federal funds rate is very low. Specifically, we find that when the federal funds rate is below 0.9%, further cuts in the policy rate can be contractionary. Moreover, we find external validation of this reversal rate by showing in a simple regression framework that the relation between bank capital and interest rates switches sign at a threshold predicted by the model.
其次，我们展示银行市场力量与资本监管相互作用，当联邦基金利率非常低时，可以扭转货币政策的效果。具体来说，我们发现当联邦基金利率低于 0.9%时，进一步降低政策利率可能是紧缩的。此外，我们通过在简单回归框架中展示银行资本与利率之间的关系在模型预测的阈值处转换符号来验证这种逆转率。

To provide intuition for these results, we elaborate on the model. In a dynamic industry equilibrium, imperfectly competitive banks act as intermediaries between borrowers and depositors. Banks' lending decisions are dynamic for two reasons: financial frictions that induce precautionary capital accumulation and a maturity mismatch between short-term deposits and long-term loans. In this setting, monetary policy moves the federal funds rate. Because banks are not price takers in deposit or loan markets, they choose the extent to which they pass rate movements through to depositors and borrowers. The magnitude of this pass-through depends on the tightness of regulatory constraints, the severity of financial frictions, and the intensity of competition.
为了解释这些结果，我们详细阐述了模型。在动态行业均衡中，不完全竞争的银行充当借款人和存款人之间的中介。银行的放贷决策有两个动态原因：金融摩擦导致预防性资本积累，以及短期存款和长期贷款之间的到期不匹配。在这种情况下，货币政策会影响联邦基金利率。由于银行在存款或贷款市场中不是价格接受者，他们选择将利率变动程度传递给存款人和借款人。这种传递的幅度取决于监管约束的严格程度、金融摩擦的严重程度和竞争的强度。

These frictions in our model map into four channels of monetary policy transmission. The first is the bank reserve channel, in which a high federal funds rate raises the opportunity cost of holding reserves, thus contracting deposit creation (Bernanke and Blinder (1988), Kashyap and Stein (1995)). The second is the bank capital channel, in which a high federal funds rate reduces bank capital because of a balance-sheet maturity mismatch and thus constrains banks' capacity to lend (Bolton and Freixas (2000), Van den Heuvel (2002), Brunnermeier and Sannikov (2016)). The third is the deposit market power channel, in which a high federal funds rate allows banks to charge higher markups on deposits, thus leading to a contraction in deposits and loans (Drechsler, Savov, and Schnabl (2017)). The fourth is the loan market power channel, in which banks reduce markups to mitigate the effects of monetary tightening on loan demand (Scharfstein and Sunderam (2016)).
我们模型中的这些摩擦映射为货币政策传导的四个渠道。第一个是银行准备金渠道，高联邦基金利率提高了持有准备金的机会成本，从而收缩存款创造（Bernanke 和 Blinder（1988 年），Kashyap 和 Stein（1995 年））。第二个是银行资本渠道，高联邦基金利率由于资产负债表到期不匹配而减少银行资本，从而限制了银行的贷款能力（Bolton 和 Freixas（2000 年），Van den Heuvel（2002 年），Brunnermeier 和 Sannikov（2016 年））。第三个是存款市场力量渠道，高联邦基金利率使银行能够对存款收取更高的溢价，从而导致存款和贷款收缩（Drechsler，Savov 和 Schnabl（2017 年））。第四个是贷款市场力量渠道，银行降低溢价以减轻货币紧缩对贷款需求的影响（Scharfstein 和 Sunderam（2016 年））。

To gauge the quantitative importance of these transmission channels, we estimate our model using data on U.S. commercial banks from 1994 to 2017. Our estimation combines methods used in the industrial organization literature (Berry, Levinsohn, and Pakes (1995), Nevo (2001)) with those used in the corporate finance literature (Hennessy and Whited (2005), Bazdresch, Kahn, and Whited (2018)). We begin by using demand estimation techniques to obtain the elasticities of loan and deposit demand to interest rates. We then plug these estimates into our model and use simulated minimum distance (SMD) to obtain estimates of parameters that quantify financial frictions and operating costs. The sequential use of these two techniques represents a methodological advance that enables us to consider a rich equilibrium model that would otherwise be intractable to estimate.
为了衡量这些传输渠道的数量重要性，我们使用 1994 年至 2017 年美国商业银行数据来估计我们的模型。我们的估计结合了工业组织文献中使用的方法（Berry，Levinsohn 和 Pakes（1995 年），Nevo（2001 年））以及公司融资文献中使用的方法（Hennessy 和 Whited（2005 年），Bazdresch，Kahn 和 Whited（2018 年））。我们首先使用需求估计技术来获得贷款和存款需求对利率的弹性。然后，我们将这些估计值插入我们的模型中，并使用模拟的最小距离（SMD）来获得量化金融摩擦和运营成本的参数估计。这两种技术的顺序使用代表了一种方法论上的进步，使我们能够考虑一个否则难以估计的丰富均衡模型。

We use counterfactual experiments to assess the relative importance of each transmission channel. We start with a model with all frictions as estimated and then subtract each friction one at a time. We find that eliminating reserve requirements leaves the sensitivity of lending to the federal funds rate nearly unchanged. Eliminating either capital regulation or deposit market power reduces this sensitivity, while eliminating loan market power raises it.
我们使用反事实实验来评估每个传输渠道的相对重要性。我们从一个模型开始，其中所有摩擦都被估计，然后逐个减去每个摩擦。我们发现，消除准备金要求几乎不会改变贷款对联邦基金利率的敏感性。消除资本监管或存款市场力量会降低这种敏感性，而消除贷款市场力量则会提高它。

These counterfactuals also show that rate cuts can be contractionary when rates are already low. Low rates depress bank profits by reducing bank deposit market power, as competition from cash intensifies. Lower profits then tighten the capital constraint and reduce lending. This result helps explain sluggish bank lending growth observed in the ultra-low interest rate environment after the 2008 financial crisis.
这些反事实情况也表明，当利率已经很低时，降息可能会收缩。低利率通过降低银行存款市场份额来抑制银行利润，因为现金的竞争加剧。然后，利润减少会加紧资本约束并减少贷款。这一结果有助于解释 2008 年金融危机后超低利率环境中观察到的银行贷款增长缓慢。

Our paper contributes to the literature on the role of banks in transmitting monetary policy (Bernanke and Blinder (1988), Kashyap and Stein (1995), Van den Heuvel (2002), Scharfstein and Sunderam (2016), Brunnermeier and Sannikov (2016), Drechsler, Savov, and Schnabl (2017)). We are the first to structurally estimate a dynamic banking model to quantify various transmission channels. Prior to our work, little was known about the relative importance of these channels, as this type of quantitative exercise is difficult to undertake using reduced-form methods. Moreover, previous studies usually consider these transmission channels in isolation. For example, Scharfstein and Sunderam (2016) and Drechsler, Savov, and Schnabl (2017) study market power in the loan and deposit markets separately. However, little is known about the interactions between channels. Thus, an important contribution of this paper is to provide a unified framework within which to study these interactions.
我们的论文为银行在传递货币政策方面的文献做出了贡献（Bernanke 和 Blinder（1988 年），Kashyap 和 Stein（1995 年），Van den Heuvel（2002 年），Scharfstein 和 Sunderam（2016 年），Brunnermeier 和 Sannikov（2016 年），Drechsler，Savov 和 Schnabl（2017 年））。我们是第一个结构估计动态银行模型以量化各种传输渠道的研究。在我们的工作之前，对于这些渠道的相对重要性几乎一无所知，因为使用简化形式的方法进行这种类型的定量分析是困难的。此外，以往的研究通常将这些传输渠道单独考虑。例如，Scharfstein 和 Sunderam（2016 年）以及 Drechsler，Savov 和 Schnabl（2017 年）分别研究了贷款和存款市场中的市场力量。然而，对于渠道之间的相互作用知之甚少。因此，本文的一个重要贡献是提供一个统一框架，以便研究这些相互作用。

Second, our paper is related to the theoretical literature on the effects of negative interest rate policies (Brunnermeier and Koby (2016), Eggertsson, Juelsrud, and Wold (2017), Campos (2019), Wang (2019)). While these studies are insightful, they typically treat the banking sector with a high level of abstraction. In contrast, we provide a model that is sufficiently realistic to be directly mapped onto microeconomic data. Our paper also contributes to the empirical literature on negative interest rate policy (Demiralp, Eisenschmidt, and Vlassopoulos (2017), Basten and Mariathasan (2018), Heider, Saidi, and Schepens (2019)). These studies show that negative policy rates can have perverse effects on bank lending. Our results suggest that such perverse effects can start to occur even before the policy rate turns negative because a near-zero policy rate results in a compression of banks' deposit spreads. Therefore, in countries such as the United States where the policy rate has never gone negative, the banking sector could nevertheless be hurt by an ultra-low-rate monetary policy.
其次，我们的论文与负利率政策效应的理论文献相关（Brunnermeier 和 Koby（2016 年），Eggertsson，Juelsrud 和 Wold（2017 年），Campos（2019 年），Wang（2019 年））。虽然这些研究很有见地，但它们通常以高度抽象的方式处理银行业。相比之下，我们提供了一个足够现实的模型，可以直接映射到微观经济数据上。我们的论文还为负利率政策的实证文献做出了贡献（Demiralp，Eisenschmidt 和 Vlassopoulos（2017 年），Basten 和 Mariathasan（2018 年），Heider，Saidi 和 Schepens（2019 年））。这些研究表明，负利率政策可能对银行贷款产生逆向效应。我们的结果表明，即使在政策利率变负之前，这种逆向效应也可能开始出现，因为接近零的政策利率会导致银行存款利差的压缩。因此，在像美国这样政策利率从未变负的国家，银行业仍可能受到超低利率货币政策的伤害。

Third, our paper is related to the literature on external financial frictions. Romer and Romer (1990) argue that banks can easily replace deposits with external financing, so shocks to deposits are unlikely to affect bank lending. In contrast, Kashyap and Stein (1995) argue that external financing is costly for banks, so the quantity of deposits matters for bank lending. Our study contributes to this debate, as our structural estimation approach allows us to infer the degree of bank financing costs from the relative size of their deposit taking and external borrowing. We find that this cost is economically significant and that frictions related to bank balance sheets play an important role in the transmission of monetary policy.
第三，我们的论文与外部金融摩擦的文献有关。Romer 和 Romer（1990）认为银行可以轻易用外部融资替代存款，因此存款冲击不太可能影响银行贷款。相比之下，Kashyap 和 Stein（1995）认为外部融资对银行来说是昂贵的，因此存款数量对银行贷款很重要。我们的研究为这一争论做出了贡献，因为我们的结构估计方法使我们能够从银行吸收存款和外部借款的相对规模推断出银行融资成本的程度。我们发现这种成本在经济上是显著的，并且与银行资产负债表相关的摩擦在货币政策传导中发挥着重要作用。

Finally, our paper contributes to the structural industrial organization literature on the banking system (Egan, Hortaçsu, and Matvos (2017), Buchak et al. (2018), Xiao (2020), Egan, Lewellen, and Sunderam (2022)). While this work usually features a static industry equilibrium, we introduce the dynamic adjustment of banks' balance sheets to study the role of maturity transformation and financial frictions. Our paper is also tangentially related to recent work that uses dynamic banking models to study optimal capital regulation (Begenau (2018), Begenau and Landvoigt (2021), Landvoigt, Van Nieuwerburgh, and Elenev (2021)). In particular, Corbae and D'Erasmo (2021) develop a dynamic dominant-fringe model to study the quantitative impact of regulatory policies on bank risk-taking and market structure. Our paper stands apart from this literature in that we emphasize the role of imperfect competition in monetary transmission. Moreover, our approach is more empirical in nature, as we estimate, rather than calibrate, all of our model parameters.
最后，我们的论文对银行体系的结构产业组织文献做出了贡献（Egan，Hortaçsu 和 Matvos（2017 年），Buchak 等（2018 年），肖（2020 年），Egan，Lewellen 和 Sunderam（2022 年））。虽然这项工作通常涉及静态行业均衡，但我们引入了银行资产负债表的动态调整，以研究到期转换和金融摩擦的作用。我们的论文也与最近使用动态银行模型研究最佳资本监管的相关工作有关（Begenau（2018 年），Begenau 和 Landvoigt（2021 年），Landvoigt，Van Nieuwerburgh 和 Elenev（2021 年））。特别是，Corbae 和 D'Erasmo（2021 年）开发了一个动态主导-边缘模型，以研究监管政策对银行风险承担和市场结构的定量影响。我们的论文与这一文献有所不同，我们强调了在货币传输中的不完全竞争的作用。此外，我们的方法更具实证性，因为我们估计而不是校准我们的模型参数。

The paper proceeds as follows. Section I describes our data. Section II presents the model. Section III discusses the estimation procedure and results. Section IV presents the results of our counterfactuals. Section V examines subsample heterogeneity and model robustness. Section VI concludes.
本文的内容如下。第一部分描述了我们的数据。第二部分介绍了模型。第三部分讨论了估计过程和结果。第四部分展示了我们的对照实验结果。第五部分考察了子样本的异质性和模型的稳健性。第六部分总结。

I. Data and Stylized Facts
I. 数据和事实化的事实

Our main data set is the Consolidated Reports of Condition and Income (Call Reports). This data set provides quarterly bank-level balance sheet information for U.S. commercial banks. It includes deposit and loan amounts, interest income and expense, loan maturities, salary expenses, and fixed-asset-related expenses. We merge the Call Reports with the Federal Deposit Insurance Corporation (FDIC) Summary of Deposits, which provides branch-level information on each bank since 1994 at an annual frequency. The sample period is 1994 to 2017.
我们的主要数据集是《资产和收入综合报告》（Call Reports）。这个数据集提供了美国商业银行的季度银行层面资产负债表信息。它包括存款和贷款金额、利息收入和支出、贷款到期日、工资支出以及固定资产相关支出。我们将《资产和收入综合报告》与联邦存款保险公司（FDIC）存款摘要合并，该摘要提供了自 1994 年以来每家银行的分行级信息，以年度频率。样本期为 1994 年至 2017 年。

We also use several other data sources. First, we retrieve publicly listed bank returns from the Center for Research in Security Prices (CRSP) and link these returns to bank concentration measures using the link table provided by the Federal Reserve Bank of New York. Banking industry stock returns are from Kenneth French's website. We collect Federal Open Market Committee (FOMC) meeting dates from the FOMC meeting calendar. Finally, we obtain the following time series from the Federal Reserve Economic Data (FRED) database: National Bureau of Economic Research (NBER) recession dates, the effective federal funds rate, two- and five-year Treasury yields, the aggregate amount of corporate bonds issued by U.S. firms, and the aggregate amounts of cash, Treasury bonds, and money-market mutual funds held by households. Details regarding the construction of our variables are in Table I.
我们还使用了其他几个数据来源。首先，我们从安全价格研究中心（CRSP）检索公开列出的银行回报，并使用纽约联邦储备银行提供的链接表将这些回报与银行集中度指标进行关联。银行业股票回报来自肯尼斯·弗伦奇（Kenneth French）的网站。我们从联邦公开市场委员会（FOMC）会议日历中收集 FOMC 会议日期。最后，我们从联邦储备经济数据（FRED）数据库获取以下时间序列：美国经济研究局（NBER）的衰退日期，有效的联邦基金利率，两年和五年期国债收益率，美国公司发行的公司债券总额，以及家庭持有的现金、国债和货币市场互惠基金的总额。有关我们变量构建的详细信息请参见表 I。

Table II provides summary statistics for our sample. Three patterns are of note. First, mean deposit and loan market shares for the U.S. national market lie near the 90^th percentile, indicating a very skewed distribution of market shares in which a few large banks dominate the market. Second, we see little variation in the number of employees per branch, but we see high variance and skewness in the number of branches per bank. This skewness is consistent with the skewness in market shares, as the number of branches is highly correlated with bank size. Third, we find that average loan maturity is 3.429 years.
表 II 提供了我们样本的摘要统计数据。有三个模式值得注意。首先，美国全国市场的平均存款和贷款市场份额接近 90％分位数，表明市场份额分布非常倾斜，少数大银行主导市场。其次，我们看到每家分行的员工数量变化不大，但我们看到每家银行的分行数量存在高方差和偏度。这种偏斜与市场份额的偏斜一致，因为分行数量与银行规模高度相关。第三，我们发现平均贷款期限为 3.429 年。

In Figure 1, we show the prices that banks charge for their deposits and loans. Panel A depicts a time-series kernel regression of the average quarterly U.S. bank deposit spread on the federal funds rate, where the deposit spread is the difference between the federal funds rate and the deposit rate. Because this spread measures the price that banks charge for their depository services, in the absence of market power, one would expect to observe constant deposit spreads that equal the marginal cost of providing deposits. However, we find a positive relation between deposit spreads and the federal funds rate, which steepens when the rate is close to zero. This relation implies that banks charge higher prices for their depository services as the federal funds rate rises. Intuitively, if banks have market power, a higher federal funds rate allows them to increase profits by raising markups above marginal costs because depositors find cash costly to hold (Drechsler, Savov, and Schnabl (2017)).
在图 1 中，我们展示了银行对其存款和贷款收取的价格。面板 A 展示了美国银行平均季度存款利差随联邦基金利率的时间序列核回归，其中存款利差是联邦基金利率和存款利率之间的差异。由于这种利差衡量了银行对其存款服务收取的价格，在没有市场力量的情况下，人们会期望观察到等于提供存款的边际成本的恒定存款利差。然而，我们发现存款利差与联邦基金利率之间存在正相关关系，当利率接近零时，这种关系变得更为陡峭。这种关系意味着随着联邦基金利率的上升，银行对其存款服务收取更高的价格。直观地说，如果银行拥有市场力量，较高的联邦基金利率使它们能够通过将标记价格提高到边际成本之上来增加利润，因为存款人发现持有现金成本高昂（Drechsler，Savov 和 Schnabl（2017））。

Panel B of Figure 1 contains an analogous kernel regression of the average U.S. bank loan spread on the federal funds rate, where the loan spread is the difference between the loan rate, adjusted for loan loss provisions, and the five-year Treasury yield. We find that the loan spread falls as the federal funds rate rises. This pattern is consistent with Scharfstein and Sunderam (2016), who show that banks lower markups on loans in the face of rising rates to mitigate the effects of falling loan demand. In sum, Figure 1 suggests that market power creates wedges between the federal funds rate and the rates at which banks borrow and lend. Furthermore, the sizes of these wedges depend on the level of interest rates.
图 1 的 B 面板包含了美国银行贷款利差与联邦基金利率的类似核回归，其中贷款利差是贷款利率与五年期国债收益率之间的差异，调整了贷款损失准备金。我们发现，随着联邦基金利率的上升，贷款利差下降。这种模式与 Scharfstein 和 Sunderam（2016）一致，他们表明银行在面对上升利率时降低贷款的加价，以缓解贷款需求下降的影响。总之，图 1 表明市场力量在联邦基金利率与银行借贷利率之间产生了差距。此外，这些差距的大小取决于利率水平。

II. Model II. 模型

While this evidence suggests interesting equilibrium interactions between bank market power and monetary policy, it does not reveal any underlying mechanisms behind these patterns in the data. To understand this evidence further, we consider an infinite-horizon bank industry equilibrium model with three sectors: households, firms, and banks. In the model, households and firms solve static discrete-choice problems in which they choose from several saving and financing vehicles. Banks act as intermediaries between households and firms by taking short-term deposits from households and providing long-term loans to firms.1
尽管这些证据表明银行市场力量和货币政策之间存在有趣的均衡互动，但并未揭示这些数据模式背后的任何基本机制。为了进一步理解这些证据，我们考虑一个具有三个部门的无限期银行行业均衡模型：家庭、企业和银行。在这个模型中，家庭和企业解决静态离散选择问题，他们可以从多种储蓄和融资工具中进行选择。银行充当家庭和企业之间的中介，从家庭那里接受短期存款，并向企业提供长期贷款。

The richness of the model lies in the banking sector, as several frictions imply that monetary policy affects the extent of intermediation that banks provide. First, competition in the deposit and loan markets is imperfect, so banks strategically choose deposit and loan rates to maximize profits. Second, banks are subject to regulation. Reserve regulation links the opportunity cost of taking deposits to the prevailing federal funds rate. Capital regulation incentivizes banks to optimize their lending intertemporally, with an eye to preserving excess equity capital as a buffer against future capital inadequacy. Third, access to nondeposit external financing is more costly than taking deposits. This friction implies that shocks to the quantity of deposits are transmitted to the supply of loans, as banks cannot costlessly replace deposits with other borrowing. These frictions are important because, in their absence, banks are simply pass-through entities and bond market interest rates summarize monetary policy.
模型的丰富性在于银行业，因为几个摩擦意味着货币政策影响银行提供中介服务的程度。首先，存款和贷款市场的竞争不完全，因此银行战略性地选择存款和贷款利率以最大化利润。其次，银行受监管。准备金监管将接受存款的机会成本与当前的联邦基金利率联系起来。资本监管激励银行在时间上优化他们的贷款，以保留过剩的股本作为未来资本不足的缓冲。第三，获得非存款外部融资的成本比接受存款更高。这种摩擦意味着存款数量的冲击传导到贷款供应，因为银行无法无成本地用其他借款替代存款。这些摩擦很重要，因为在它们不存在的情况下，银行只是传递实体，债券市场利率总结了货币政策。

B. Firms B. 公司

There is a mass K of firms, where we again drop the time subscript. Each firm wants to borrow one dollar, so aggregate borrowing demand is K. Firms can borrow by issuing long-term bonds or by taking out long-term bank loans. We assume that each bank is a differentiated lender, given factors such as geographic location and industry expertise. Letting each option be indexed by j, the firms' choice set is given by $$\mathcal {A}^l=\lbrace 0,1,\ldots ,J,J+1\rbrace $$, where option $$J+1$$ represents bonds, options $$1,\ldots ,J$$ represent loans from each bank, and option 0 is the option to not borrow at all.

For tractability, we assume that both bonds and bank loans are long term and that a fraction η of the outstanding balance is due every year. Thus, if the borrower obtains one dollar, this debt has a maturity of $$\frac{1}{\eta }$$ years, on average. If a firm obtains a loan from bank j, it will be charged a fixed interest rate of $$r^{l}_{j}$$. If a firm issues long-term bonds, the interest rate is given by an expected default cost, δ, plus the expected weighted average of future federal funds rates, $$f_t$$, which is given by

(5)

Each of the firm's financing options is characterized by a rate, $$r_{j}^{l}$$, and a vector of product characteristics, $$x_{j}^{l}$$, capturing the convenience of using each of the financing options.

The firm then chooses the best option to maximize its profits,
公司随后选择最佳选项以最大化其利润

$$\begin{equation} \max _{j\in \mathcal {A}^l}\pi _{i,j}=\alpha _{i}^{l}r_{j}^{l}+\beta ^{l}x_{j}^{l}+\xi _{j}^{l}+\epsilon _{i,j}^{l}, \end{equation}$$(6)

where $$\pi _{i,j}$$ is the profits of firm i from choosing option j, and
是公司 i 选择选项 j 的利润$$\alpha _{i}^{l}$$ is the sensitivity to the interest rate
对利率的敏感性$$r_{j}^{l}$$, which follows a uniform distribution with mean
遵循均匀分布，均值$$\alpha ^l$$ and standard deviation $$\sigma ^l_\alpha$$. The sensitivities to nonrate characteristics,
对非利率特征的敏感性。$$x_{j}^{l}$$, are given by
, 由给定$$\beta ^{l}$$, and $$\xi _{j}^{l}$$ is the unobservable product-level demand shock. We let $$\epsilon _{i,j}^{l}$$ represent an idiosyncratic shock when firm i borrows from bank j. The optimal choice of firm i is given by the indicator function
其中 $$\pi _{i,j}$$ 是公司 i 选择选项 j 的利润， $$\alpha _{i}^{l}$$ 是对利率 $$r_{j}^{l}$$ 的敏感度，其遵循均匀分布，均值为 $$\alpha ^l$$ ，标准差为 $$\sigma ^l_\alpha$$ 。非利率特征的敏感度 $$x_{j}^{l}$$ 由 $$\beta ^{l}$$ 给出， $$\xi _{j}^{l}$$ 是不可观测的产品级需求冲击。当公司 i 从银行 j 借款时，我们让 $$\epsilon _{i,j}^{l}$$ 代表特异性冲击。公司 i 的最佳选择由指示函数给出。

$$\begin{equation} \mathbb {I}^l_{i,j}= {\begin{cases} 1,& \text{if }\; \pi _{i,j}\ge \pi _{i,k}, \text{ for }\; k\in \mathcal {A}^{l}\\ 0, & \text{otherwise}. \end{cases}} \end{equation}$$(7)

We aggregate the optimal choices across all the firms to compute the loan market share of each bank j. Assuming that $$\epsilon _{i,j}^{l}$$ follows a generalized extreme value distribution with a cumulative distribution function given by $$F(\epsilon )=\exp (-\exp (-\epsilon ))$$, we can derive the standard logit market share, $$s_{j}^{l}$$, as
我们汇总所有公司的最佳选择，计算每家银行 j 的贷款市场份额。假设 $$\epsilon _{i,j}^{l}$$ 遵循广义极值分布，其累积分布函数为 $$F(\epsilon )=\exp (-\exp (-\epsilon ))$$ ，我们可以推导出标准 logit 市场份额 $$s_{j}^{l}$$ 。

$$\begin{equation} \begin{aligned} &s_{j}^{l}{\left(r_{j}^{l}|f\right)}\equiv \int \mathbb {I}^l_{i,j}dF{\left(\epsilon \right)} \\ &=\sum _{i=1}^{I}\mu _{i}^l\frac{\exp {\left(\alpha _{i}^{l}r_{j}^{l}+\beta ^{l}x_{j}^{l}+\xi _{j}^{l}\right)}}{\exp {\left(\alpha _{i}^{l}(\overline{f}+\overline{\delta })+\beta ^{l}x_{J+1}^{l}+\xi _{J+1}^{l}\right)}+\exp {\left(\beta ^{l}x_{n}^{l}+\xi _{n}^{l}\right)}+\sum _{m=1}^{J}\exp {\left(\alpha _{i}^{l}r_{m}^{l}+\beta ^{l}x_{m}^{l}+\xi _{m}^{l}\right)}}, \end{aligned} \end{equation}$$(8)

where $$\mu _{i}^l$$ is the fraction of type i firms and $$\overline{f}$$ is the long-term bond interest rate. The numerator represents the utility from borrowing from bank j. Similarly, the first term in the denominator, $$\exp (\alpha _{i}^{l}(\overline{f}+\overline{\delta })+\beta ^{l}x_{J+1}^{l}+\xi _{J+1}^{l})$$, represents the utility from issuing bonds, and the second term,
其中 $$\mu _{i}^l$$ 是第 i 类公司的比例， $$\overline{f}$$ 是长期债券利率。分子代表从银行 j 借款的效用。类似地，分母中的第一项 $$\exp (\alpha _{i}^{l}(\overline{f}+\overline{\delta })+\beta ^{l}x_{J+1}^{l}+\xi _{J+1}^{l})$$ 代表发行债券的效用，第二项代表。$$\exp (\beta ^{l}x_{n}^{l}+\xi _{n}^{l})$$, is the utility of not borrowing. The demand function for loans is then given by the market share multiplied by the total loan market size,
其中 $$\mu _{i}^l$$ 是第 i 类公司的比例， $$\overline{f}$$ 是长期债券利率。分子代表从银行 j 借款的效用。类似地，分母中的第一项， $$\exp (\alpha _{i}^{l}(\overline{f}+\overline{\delta })+\beta ^{l}x_{J+1}^{l}+\xi _{J+1}^{l})$$ ，代表发行债券的效用，第二项， $$\exp (\beta ^{l}x_{n}^{l}+\xi _{n}^{l})$$ ，代表不借款的效用。贷款的需求函数由市场份额乘以总贷款市场规模给出，

$$\begin{equation} B_{j}{\left(r_{j}^{l}{\left| f \right.} \right)}=s_{j}^{l}{\left(r_{j}^{l}{\left| f \right.} \right)}K. \end{equation}$$(9)

E. Bank's Problem and Equilibrium
E. 银行的问题和平衡

Figure 2 shows the sequence of events in each time period. The bank enters the period and observes both the federal funds rate $$f_t$$ and the realization of the fraction of defaults $$\delta _t$$. Next, banks interact with households and firms by setting the loan and deposit spreads. The loan and deposit demand functions then dictate the amount of deposits from households and the amount of loans to firms. Depending on the extent of these activities, the banks adjust their reserves, holdings of government securities, and nonreservable borrowing. Finally, banks collect profits at the end of the period and distribute dividends to shareholders.

As noted above, loan and deposit demand depend on the rates offered by all banks in the economy. Accordingly, when each bank chooses its own deposit and loan rates, $$r_{t}^d$$ and $$r_{t}^l$$, as well as nonreservable borrowing $$N_t$$ and investment in government securities $$G_{t}$$, it rationally considers the choices made by other banks in both current and future periods. As such, all of a bank's optimal choices depend on the composition of the banking sector, that is, the cross-sectional distribution of bank states, which we denote by $$\Gamma _t$$. Letting $$P^\Gamma$$ denote the probability law governing the evolution of $$\Gamma _t$$, we can express the evolution of $$\Gamma _t$$ as
如上所述，贷款和存款需求取决于经济中所有银行提供的利率。因此，当每家银行选择其自己的存款和贷款利率，以及不可储备借款和投资政府证券时，它会理性地考虑其他银行在当前和未来时期所做的选择。因此，银行的所有最佳选择都取决于银行业的构成，即银行状态的横截面分布，我们用 $$\Gamma _t$$ 表示。让 $$P^\Gamma$$ 表示统治 $$\Gamma _t$$ 演变的概率法则，我们可以表达 $$\Gamma _t$$ 的演变。

$$\begin{equation} \Gamma _{t+1} = P^\Gamma (\Gamma _t). \end{equation}$$(23)

In every period, after observing the federal funds rate
在每个时期，观察联邦基金利率后$$f_t$$ and the random fraction of defaulted loans
和违约贷款的随机比例$$\delta _t$$, banks choose the optimal policy to maximize the expected discounted cash dividends to shareholders,
银行选择最优政策，以最大化预期折现现金股利给股东

$$\begin{equation} V(f_{t},\delta _{t},L_{t},E_{t}\big\vert \Gamma _t)=\underset{\lbrace r_{t}^l, r_{t}^{d}, G_t, N_t, R_t, C_{t+1}\rbrace }{\max } \frac{1}{1+\gamma }{\left\lbrace C_{t+1}+\mathbb {E} V(f_{t+1},\delta _{t+1},L_{t+1},E_{t+1}\big\vert \Gamma _{t+1})\right\rbrace} , \end{equation}$$(24)

$$\begin{equation*} s.t.(4),(9),(10),(11),(12),(16),(17),(18),(19),(20),(21),(22),(23). \end{equation*}$$

We define equilibrium in this economy as follows.
我们将这个经济中的平衡定义如下。

Definition 1.A stationary equilibrium occurs when:
定义 1.当发生静态平衡时：

• 1. All banks solve the problem given by (24), taking as given the other banks' choices of loan and deposit rates.
  所有银行解决了（24）给出的问题，假设其他银行的贷款和存款利率已经确定。
• 2. All households and firms maximize their utilities given the list of rates put forth by banks.
  所有家庭和企业都会根据银行提出的利率列表来最大化他们的效用。
• 3. In each period, the deposit and loan markets clear.
  在每个时期，存款和贷款市场都会达到平衡。
• 4. The probability law governing the evolution of the industry, $$P^\Gamma$$, is consistent with banks' optimal choices.

One of the state variables for the banks' problem ($$\Gamma _t$$) is an object whose dimension depends on the number of banks in the economy. This dimensionality poses a challenge to solving the banks' problem numerically. To simplify the model solution, we consider a low-dimensional approximation of $$\Gamma _t$$. Specifically, we postulate that all information about $$\Gamma _t$$ that is relevant to banks' optimization can be summarized by the contemporaneous federal funds rate, $$f_t$$. Accordingly, we define the equilibrium average loan and deposit rates, $$\overline{r}^{l,i}(f_t)$$ and $$\overline{r}^{d,i}(f_t)$$, respectively, as
银行问题的一个状态变量（ $$\Gamma _t$$ ）是一个对象，其维度取决于经济中的银行数量。这种维度性对于数值解决银行问题构成了挑战。为了简化模型解决方案，我们考虑 $$\Gamma _t$$ 的低维近似。具体来说，我们假设与银行优化相关的 $$\Gamma _t$$ 的所有信息都可以通过同时的联邦基金利率 $$f_t$$ 来总结。因此，我们分别定义平衡平均贷款利率和存款利率 $$\overline{r}^{l,i}(f_t)$$ 和 $$\overline{r}^{d,i}(f_t)$$ 。

$$\begin{equation} \exp (\alpha _i^d\overline{r}^{d,i}(f_t)+ \beta ^d x^d + \xi ^d) \equiv \mathbb{E}{\left[\exp (\alpha _i^d r^d_{j,t}+ \beta ^d x^d + \xi ^d)|f_t\right]}, \end{equation}$$(25)

$$\begin{equation} \exp (\alpha _i^l\overline{r}^{l,i}(f_t)+ \beta ^l x^l + \xi ^l) \equiv \mathbb{E}{\left[\exp (\alpha _i^l r^l_{j,t} + \beta ^l x^l + \xi ^l)|f_t\right]}, \end{equation}$$(26)

where $$\overline{r}^{l,i}(f_t)$$ and $$\overline{r}^{d,i}(f_t)$$ summarize the choices of other banks, thereby allowing each bank to derive its choices of deposit and loan rates.
其中 $$\overline{r}^{l,i}(f_t)$$ 和 $$\overline{r}^{d,i}(f_t)$$ 总结了其他银行的选择，从而使每家银行能够得出其存款和贷款利率的选择。

In solving the model, which we describe in detail in Section I of the Internet Appendix, we ensure that
在解决我们在互联网附录的第一部分中详细描述的模型时，我们确保$$\bar{r}^{l,i}(f_t)$$ and  和$$\bar{r}^{d,i}(f_t)$$ are consistent with equilibrium bank choices by iterating over their values until we reach convergence. To check the accuracy of our solution, we regress the simulated evolution of the aggregate deposit and loan rates on the perceived law of motion based on banks' beliefs. The R²s for these approximations are over 95% for the deposit market and 97% for the loan market. Thus, although the banks do not consider the full distribution,
与平衡银行选择一致，通过迭代它们的值直至达到收敛。为了检验我们解决方案的准确性，我们将模拟的总存款和贷款利率的演变回归到基于银行信念的感知运动定律上。这些近似值的 R ² 为存款市场超过 95%，贷款市场为 97%。因此，尽管银行没有考虑到完整的分布，$$\Gamma _t$$, when making their decisions, their forecasting errors are small. This accuracy stems from two mechanisms in the model. First, without any financial or regulatory constraints, banks have static optimal deposit and loan rates. For example, in the lending market, the optimal level of loans is set to equalize expected marginal interest income and funding costs, which is a function of the current federal funds rate only. Therefore, the static optimal rate depends only on the federal funds rate, not on other aggregate moments. Second, taking the loan market as an example, although banks can deviate from the static optimum by charging higher loan spreads, this behavior is limited by competition from other banks. Thus, banks that deviate from the static optimum introduce only modest distortions into the other banks' rate forecasts.
在解决我们在互联网附录的第一部分中详细描述的模型时，我们通过迭代 $$\bar{r}^{l,i}(f_t)$$ 和 $$\bar{r}^{d,i}(f_t)$$ 的值直到收敛，确保它们与均衡银行选择一致。为了检验我们解决方案的准确性，我们将模拟的总存款和贷款利率的演变回归到基于银行信念的感知运动定律上。这些近似值的 R ² 分别为存款市场超过 95％，贷款市场超过 97％。因此，尽管银行在做决策时没有考虑完整的分布 $$\Gamma _t$$ ，但他们的预测误差很小。这种准确性源于模型中的两种机制。首先，在没有任何金融或监管约束的情况下，银行具有静态最优存款和贷款利率。例如，在贷款市场中，贷款的最优水平被设定为使预期边际利息收入和资金成本相等，这是仅与当前联邦基金利率有关的函数。因此，静态最优利率仅取决于联邦基金利率，而不取决于其他总体时刻。 其次，以贷款市场为例，尽管银行可以通过收取更高的贷款利差偏离静态最优，但这种行为受到其他银行的竞争限制。因此，偏离静态最优的银行只会对其他银行的利率预测引入轻微扭曲。

III. Estimation III. 估计

In this section, we describe our estimation method, present results, and conduct counterfactuals to measure the relative importance of various banking frictions for monetary policy transmission.
在本节中，我们描述了我们的估计方法，呈现了结果，并进行了反事实分析，以衡量各种银行摩擦对货币政策传导的相对重要性。