Obelix vs. Asterix: Size of US commercial banks and its regulatory challenge

Abstract

Big banks pose substantial costs to society in the form of increased systemic risk and government bailouts during crises. So the question is: Should regulators limit the size of banks? To answer this question, regulators need to assess the potential costs of such regulations. If big banks enjoy substantial scale economies (i.e., average costs get lower as bank size increases), limiting the size of banks through regulations may be inefficient and likely to reduce social welfare. However, the literature offers conflicting results regarding the existence of economies of scale for the biggest US banks. We contribute to this literature using a novel approach to estimating nonparametric measures of scale economies and total factor productivity (TFP) growth. For US commercial banks, we find that around 73 % of the top one hundred banks, 98 % of medium and small banks, and seven of the top ten biggest banks by asset size exhibit substantial economies of scale. Likewise, we find that scale economies contribute positively and significantly to their TFP growth. The existence of substantial scale economies raises an important challenge for regulators to pursue size limit regulations.

Appendices

Appendix 1: TFP growth decomposition

Starting with the standard definition of TFP change in Eq. (7) and adding $\dot{TFP}$ to both sides of Eq. (5) we have:

$$\begin{aligned}&\dot{ {TFP}}+\beta _0(\cdot )+\sum _{k=1}^{K-1} \beta _k(\cdot ) \dot{\widetilde{W}_k} + \sum _{q=1}^Q \gamma _q(\cdot ) \dot{Y}_q + \sum _{p=1}^P \varphi _p(\cdot ) \nabla _t Z_p +u \nonumber \\&\quad \equiv \dot{ \widetilde{C}} + \sum _{q=1}^Q R_q \dot{Y}_q - \sum _{k=1}^K S_k \dot{X}_k \end{aligned}$$

(10)

Using the definitions $\dot{\widetilde{C}}=\dot{C} - \dot{W}_K$ and $\dot{C}= \sum _{k=1}^K S_k \dot{W}_k + \sum _{k=1}^K S_k \dot{X}_k$,the right-hand-side of Eq. (10) can be expressed as:

$$\begin{aligned} \dot{ \widetilde{C}} + \sum _{q=1}^Q R_q \dot{Y}_q - \sum _{k=1}^K S_k \dot{X}_k \equiv \sum _{k=1}^K S_k \dot{W_k} + \sum _{q=1}^Q R_q \dot{Y}_q - \dot{W_K} \end{aligned}$$

(11)

Since $\sum _{k=1}^K S_k = 1$, $\dot{W}_k = \dot{\widetilde{W}_k} +\dot{W}_K$, $\forall k=1,\ldots ,K-1$, and $\dot{\widetilde{W}_K} = 0$,

$$\begin{aligned} \sum _{k=1}^K S_k \dot{W_k} + \sum _{q=1}^Q R_q \dot{Y}_q - \dot{W_K} \equiv \sum _{k=1}^{K-1} S_k \dot{\widetilde{W}_k} + \sum _{q=1}^Q R_q \dot{Y}_q \end{aligned}$$

(12)

Using this result, the relationship in Eq. (10) can be expressed as:

$$\begin{aligned} \dot{{ TFP}} \equiv -\beta _0(\cdot ) +\sum _{q=1}^Q (R_q - \gamma _q(\cdot )) \dot{Y}_q + \sum _{k=1}^{K-1} (S_k - \beta _k(\cdot )) \dot{\widetilde{W}_k} - \sum _{p=1}^P\varphi _p(\cdot ) \nabla _t Z_p - u\,\, \end{aligned}$$

(13)

Appendix 2: Model specifications and estimation

Following Li et al. (2002) and Li and Racine (2007), the local-constant estimator for $\varPsi (z)$ in Eq. (9) is expressed as:

$$\begin{aligned} \hat{\varPsi }(z)=\left[ \sum _{i=1}^{N}\sum _{t=1}^T {\mathcal {X}}_{it}{\mathcal {X}}'_{it} {\mathcal {K}}\left( \frac{{\mathcal {Z}}_{it}-z}{h}\right) \right] ^{-1} \sum _{i=1}^{N}\sum _{t=1}^T {\mathcal {X}}_{it}{\mathcal {Y}}_{it}{\mathcal {K}} \left( \frac{{\mathcal {Z}}_{it}-z}{h}\right) \end{aligned}$$

(14)

where $N$ and $T$ denotes number of banks and time periods, respectively, $h$ is a $(K+Q+P)$ vector with each element a selected bandwidth for each $z$ variable and ${\mathcal {K}}(\cdot )$ is the product Gaussian kernel function.³⁴

Note that if the kernel function is absent, then the SPSC estimator reduces to its OLS counterpart. The SPSC model also nests the partially linear model proposed by Robinson (1988) as a special case, which makes only the intercept an unknown smooth function of the ${\mathcal {Z}}$ variables.

Following Li and Racine (2010), we employ the most commonly used least-squares cross-validation (LSCV) method, which is a fully automatic data-driven approach, to select the bandwidth vector $h$, i.e.,

$$\begin{aligned} CV_{lc}(h)=\min _{h} \, \sum _{i=1}^{N} \sum _{t=1}^{T} \left[ {\mathcal {Y}}_{it} - {\mathcal {X}}'_{it}\hat{\varPsi }_{-it}({\mathcal {Z}}_{it})\right] ^{2} M({\mathcal {Z}}_{it}) \end{aligned}$$

(15)

where $CV_{lc}(h)$ determines the cross-validation bandwidth vector $h$ for local constant estimator, ${\mathcal {X}}'_{it}\hat{\varPsi }_{-it}({\mathcal {Z}}_{it})$ is the leave-one-out local-constant kernel conditional mean, and $0 \le M(\cdot ) \le 1$ is a weight function that serves to avoid difficulties caused by dividing by zero. Unlike other methods proposed in the recent literature (e.g. Feng and Serletis 2010 and Wheelock and Wilson 2011) the SPG model can be easily estimated using widely available statistical software. For instance, the bandwidths for the ${\mathcal {Z}}$ variables and the smooth coefficients can be estimated using the $NP$ package in R (Hayfield and Racine 2008).³⁵

Now, assuming that $f(\cdot )$ in Eq. (3) is translog, the estimating equation for the PL model is:

$$\begin{aligned} f(\cdot )= & {} \alpha _0 + \sum _{k=1}^{K-1} \theta _k \ln \widetilde{W}_k + \frac{1}{2} \sum _{k=1}^{K-1} \sum _{m=1}^{K-1} \theta _{km} \ln \widetilde{W}_k \ln \widetilde{W}_m \nonumber \\&+ \sum _{q=1}^Q \alpha _q \ln Y_q + \frac{1}{2} \sum _{q=1}^Q \sum _{o=1}^Q \alpha _{qo} \ln Y_q \ln Y_o \nonumber \\&+ \sum _{q=1}^Q \sum _{k=1}^{K-1} \delta _{qk} \ln Y_q \ln \widetilde{W}_k + \alpha _t \, t + \frac{1}{2} \alpha _{tt} \, t^2 + \sum _{q=1}^Q \lambda _{qt} \ln Y_q \, t \nonumber \\&+ \sum _{k=1}^{K-1} \delta _{kt} \ln \widetilde{W}_k \, t + \sum _{p=1}^P \phi _p Z_p\nonumber \\&+ \frac{1}{2} \sum _{p=1}^P \sum _{l=1}^P \phi _{pl} Z_p Z_l + \sum _{p=1}^P \sum _{k=1}^{K-1} \rho _{pk} Z_p \ln \widetilde{W}_k + \sum _{p=1}^P \sum _{q=1}^{Q} \tau _{pq} Z_p \ln Y_q \nonumber \\&+ \sum _{p=1}^P \phi _{pt} Z_p \, t + u \end{aligned}$$

(16)

After estimating Eq. (16), the functional coefficients for the PL model are computed using:

$$\begin{aligned} \frac{\partial f}{\partial \ln \widetilde{W}_k}=\beta _k(\cdot )&=\theta _k+\sum _{m=1}^{K-1} \theta _{km} \ln \widetilde{W}_m + \sum _{q=1}^Q \delta _{qk} \ln Y_q + \delta _{kt} t\nonumber \\&\quad + \sum _{p=1}^P \rho _{pk} Z_p, \; \forall k=1,\ldots ,K-1 \end{aligned}$$

(17)

$$\begin{aligned} \frac{\partial f}{\partial \ln Y_q}=\gamma _q(\cdot )&=\alpha _q+ \sum _{o=1}^Q \alpha _{qo} \ln Y_o + \sum _{k=1}^{K-1} \delta _{qk} \ln \widetilde{W}_k + \lambda _{qt} \, t \nonumber \\&\quad + \sum _{p=1}^P \tau _{pq} Z_p, \; \forall q=1,\ldots ,Q \end{aligned}$$

(18)

$$\begin{aligned} \frac{\partial f}{\partial Z_p}=\varphi _p(\cdot )&=\phi _p+ \sum _{l=1}^P \phi _{pl} Z_l + \sum _{k=1}^{K-1} \rho _{pk} \ln \widetilde{W}_k + \sum _{q=1}^Q \tau _{pq} \ln Y_q \nonumber \\&\quad + \phi _{pt}\, t, \; \forall p=1,\ldots ,P \end{aligned}$$

(19)

$$\begin{aligned} \frac{\partial f}{\partial t}=\beta _0(\cdot )&=\alpha _t+\alpha _{tt} t + \sum _{q=1}^Q \lambda _{qt} \ln Y_q + \sum _{k=1}^{K-1} \delta _{kt} \ln \widetilde{W}_k+\sum _{p=1}^P \phi _{pt} Z_p \end{aligned}$$

(20)

Plugging Eq. (17)–(20) into Eq. (5) gives:

$$\begin{aligned} \dot{ \widetilde{C}}= & {} \left[ \alpha _t+\alpha _{tt} t + \sum _{q=1}^Q \lambda _{qt} \ln Y_q + \sum _{k=1}^{K-1} \delta _{kt} \ln \widetilde{W}_k+\sum _{p=1}^P \phi _{pt} Z_p \right] \nonumber \\&+ \sum _{k=1}^{K-1} \left[ \theta _k+\sum _{m=1}^{K-1} \theta _{km} \ln \widetilde{W}_m + \sum _{q=1}^Q \delta _{qk} \ln Y_q + \delta _{kt} t+ \sum _{p=1}^P \rho _{pk} Z_p\right] \dot{\widetilde{W}_k} \nonumber \\&+ \sum _{q=1}^Q \left[ \alpha _q+\sum _{o=1}^Q \alpha _{qo} \ln Y_o +\sum _{k=1}^{K-1} \delta _{qk} \ln \widetilde{W}_k + \lambda _{qt} \, t + \sum _{p=1}^P \tau _{pq} Z_p\right] \dot{Y}_q \nonumber \\&+ \sum _{p=1}^P \left[ \phi _p+\sum _{l=1}^P \phi _{pl} Z_l + \sum _{k=1}^{K-1} \rho _{pk} \ln \widetilde{W}_k + \sum _{q=1}^Q \tau _{pq} \ln Y_q + \phi _{pt}\, t\right] \nabla _t Z_p + u\nonumber \\ \end{aligned}$$

(21)

Rearranging Eq. (21) gives the estimating equation for the PG model as follows:

$$\begin{aligned} \dot{ \widetilde{C}}= & {} \alpha _t+\alpha _{tt} t + \sum _{k=1}^{K-1} \theta _k \dot{\widetilde{W}_k} + \sum _{k=1}^{K-1}\sum _{m=1}^{K-1} \theta _{km} \ln \widetilde{W}_m \dot{\widetilde{W}_k} \nonumber \\&+ \sum _{q=1}^Q\alpha _q \dot{Y}_q + \sum _{q=1}^Q \sum _{o=1}^Q \alpha _{qo} \ln Y_o \dot{Y}_q \nonumber \\&+ \sum _{p=1}^P\phi _p\nabla _t Z_p + \sum _{p=1}^P\sum _{l=1}^P \phi _{pl} Z_l \nabla _t Z_p + \sum _{q=1}^Q \lambda _{qt} \left( \ln Y_q+t \dot{Y}_q \right) \nonumber \\&+ \sum _{k=1}^{K-1} \delta _{kt} \left( \ln \widetilde{W}_k + t \dot{\widetilde{W}_k}\right) \nonumber \\&+ \sum _{p=1}^P \phi _{pt} \left( Z_p + t \nabla _t Z_p \right) + \sum _{k=1}^{K-1} \sum _{q=1}^Q \delta _{qk} \left( \ln Y_q \dot{\widetilde{W}_k} + \ln \widetilde{W}_k \dot{Y}_q\right) \nonumber \\&+ \sum _{k=1}^{K-1}\sum _{p=1}^P \rho _{pk} \left( Z_p \dot{\widetilde{W}_k} + \ln \widetilde{W}_k \nabla _t Z_p\right) \nonumber \\&+ \sum _{q=1}^Q \sum _{p=1}^P \tau _{pq} \left( Z_p \dot{Y}_q + \ln Y_q \nabla _t Z_p\right) + u \end{aligned}$$

(22)

After estimating Eq. (22), we compute the functional coefficients for the PG model using Eqs. (17)–(20).