# Lerner Index (Banks)#

## Introduction#

The Lerner Index is used to measure the market power of banks. It is defined as the markup of price over marginal cost, relative to the price. Mathematically, the Lerner Index $$L$$ is given by:

(1)#$L = \frac{P - MC}{P}$

where:

• $$P$$ is the price, often calculated as total bank revenue over assets.

• $$MC$$ is the marginal cost, which is usually estimated using econometric techniques.

Note

Elzinga and Mills (2011) “The Lerner Index of Monopoly Power: Origins and Uses” is a good read.

The marginal cost ($$MC$$) is often estimated using advanced econometric models, such as the translog cost function. The translog cost function allows for more flexibility in capturing the complexities of banking operations compared to simpler functional forms.

The translog cost function $$\ln C(Q, W)$$ is often used to model the cost $$C$$, e.g., total operating expenses, as a function of output $$Q$$ and a vector of input prices $$W$$. The translog cost function can be represented as:

(2)#\begin{split}\begin{align} \ln C(Q, W) &= \beta_0 + \beta_1 \ln Q + \frac{1}{2} \beta_2 (\ln Q)^2 \\\\ &+ \sum_{i=1}^{n} \gamma_i \ln W_i + \frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} \theta_{ij} \ln W_i \ln W_j \\\\ &+ \sum_{i=1}^{n} \phi_i \ln Q \ln W_i \end{align}\end{split}

Tip

Please refer to my post Translog Production and Cost Functions for details.

Simply put, the use of translog cost function as specified in (2) allows for approximating potentially very complex cost function, hence complex produciton function via duality.

After estimating this cost function, the marginal cost $$MC$$ with respect to the output $$Q$$ can be obtained by differentiating the cost function $$C(Q, W)$$ with respect to $$Q$$:

(3)#$MC = \frac{\partial C(Q, W)}{\partial Q} = \frac{C}{Q} \left( \beta_1 + \beta_2 \ln Q + \sum_{i=1}^{n} \phi_i \ln W_i \right)$

The output $$Q$$ is typically the total assets. [1]

In the banking literature (e.g., Berger, Klapper, and Turk-Ariss (2009)), it is common to have three input factors: labour, funds, and fixed capital, so that we have

1. $$W_1$$ is the input price of labour, measured as the ratios of personnel expenses to total assets

2. $$W_2$$ is the input price of funds, measured as the interest expenses to total deposits

3. $$W_3$$ is the input price of fixed capital, measured as other operating and administrative expenses to total assets

The above specification allows only one output $$Q$$. Some studies allow banks to be multi-product and hence have several outputs. Here I do not discuss them. A suggested reading is Koetter, Kolari, and Spierdijk (2012).

Moreover, Koetter, Kolari, and Spierdijk (2012) propose the adjusted Lerner Index.

(4)#$\text{adjusted } L = \frac{\pi + TC - MC \times Q}{\pi + TC}$

where:

• $$\pi$$ is the bank’s profit before tax

• $$TC$$ is the bank’s total operating costs

Koetter, Kolari and Spierdijk (2012) have argued that the conventional approach of computing the Lerner index assumes both profit efficiency (optimal choice of prices) and cost efficiency (optimal choice of inputs by firms). As a result, the estimated price-cost margins do not correctly measure the true extent of market power. The argument reflects a distinction pointed out by Lerner (1934) himself, who states that “for practical purposes we must read monopoly power not as potential monopoly, but as monopoly in force” (p. 170; italics as in the original). In other words, the Lerner index measures actual (exercised) market power, while Koetter, Kolari, and Spierdijk (2012) are interested in measuring potential market power.

### Alternative specification#

Of course, one we choose a different functional form for the cost function and obtain margainal cost $$MC$$ accordingly. For example, Clerides, Delis, and Kokas (2015) and Deli, Delis, Hasan, and Liu (2019) use a log-linear cost function model.

A log-linear cost function is fundamentally a simplification of translog cost function in that if uses a first-order Taylor expansion instead of second-order Taylor expansion. Refer to my post Translog Production and Cost Functions for the derivation of cost function.

## Estimation of Translog Cost Function#

Tip

See my post Translog Cost Function Etimation for a detailed discussion.

The Translog cost function can be estimated using either Ordinary Least Squares (OLS) or Stochastic Frontier Analysis (SFA), depending on the assumptions about the error term.

In OLS, the Translog cost function is estimated by minimizing the sum of squared residuals. The error term is assumed to be normally distributed and captures random noise.

In SFA, the error term is decomposed into two parts: one capturing random noise and another capturing inefficiency.

• OLS is simpler but does not separate inefficiency from random noise.

• SFA is more complex but provides additional insights into inefficiency.

Important

In this current implementation, I use OLS to estimate the translog cost function with time fixed effects and standard error clustering by firm. Support for a SFA estimation via MLE may be added in the future.

Another important note is that we need to impose the homogeneity of degree 1 constraint, which basically states that if all input prices are scaled by a scalar $$s$$, the cost would scale by the same $$s$$.

This is achieved by normalizing the total cost $$C$$ and input prices by $$W_1$$ (we can choose $$W_2$$ or $$W_3$$ instead). After normalization, we will have scaled $$\tilde{C}=\frac{C}{W_1}$$, $$\tilde{W}_2=\frac{W_2}{W_1}$$ and $$\tilde{W}_3=\frac{W_3}{W_1}$$.

Tip

Again, see Translog Cost Function Etimation for a detailed discussion and proof.

### Time fixed effect#

With three inputs and one output, the model to estimate is then

(5)#\begin{split}\begin{align} \ln \tilde{C}_{it} &= \beta_0 + \beta_1 \ln Q_{it} + \frac{1}{2} \beta_2 (\ln Q_{it})^2 \\\\ &+ \gamma_2 \ln\tilde{W}_{2,it} + \gamma_3 \ln\tilde{W}_{3,it} + \phi_3 \ln Q_{it} \ln\tilde{W}_{2,it} + \phi_3 \ln Q_{it} \ln\tilde{W}_{3,it} + d_{t} + \varepsilon_{it} \end{align}\end{split}

where $$d_t$$ is the time fixed effects.

The marginal cost for bank $$i$$ at time $$t$$ is then obtained as

(6)#$MC = \frac{\partial C}{\partial Q} = \frac{C}{Q}\frac{\partial \ln C}{\partial \ln Q} = \frac{C}{Q}\frac{\partial (\ln \tilde{C} + \ln W_1) }{\partial \ln Q} = \frac{C}{Q}\frac{\partial \ln \tilde{C} }{\partial \ln Q}$

so

(7)#$MC_{it} = \frac{C_{it}}{Q_{it}} \left( \hat{\beta}_1 + \hat{\beta}_2 \ln Q + \hat{\phi}_2 \ln \tilde{W}_2 + \hat{\phi}_3 \ln \tilde{W}_3\right)$

This $$MC_{it}$$ is used to compute the Lerner Index.

Model (5) adds year fixed effects to (2) and can be estimated with robust standard errors by bank to capture the specificities of each firm (e.g., Berger, Klapper, and Turk-Ariss (2009), Beck, De Jonghe, and Schepens (2013), among others)

### Trend#

Alternatively, we can incorporate trend instead, similar to Koetter, Kolari, and Spierdijk (2012),

(8)#\begin{split}\begin{align} \ln \tilde{C}_{it} &= \beta_0 + \beta_1 \ln Q_{it} + \frac{1}{2} \beta_2 (\ln Q_{it})^2 \\\\ &+ \gamma_2 \ln\tilde{W}_{2,it} + \gamma_3 \ln\tilde{W}_{3,it} + \phi_3 \ln Q_{it} \ln\tilde{W}_{2,it} + \phi_3 \ln Q_{it} \ln\tilde{W}_{3,it} \\\\ &+ \eta_0 trend + \eta_1 trend^2 + \eta_2 trend \ln Q_{it} + \omega_{2} trend \ln\tilde{W}_{2,it} + \omega_{3} trend \ln\tilde{W}_{3,it} + \varepsilon_{it} \end{align}\end{split}

such that the marginal cost is

(9)#$MC_{it} = \frac{C_{it}}{Q_{it}} \left( \hat{\beta}_1 + \hat{\beta}_2 \ln Q + \hat{\phi}_2 \ln \tilde{W}_2 + \hat{\phi}_3 \ln \tilde{W}_3 + \eta_2 trend\right)$

## API#

Note

To be implemented.

class frds.measures.LernerIndex[source]#

Lerner Index (Banks)

__init__() None[source]#