In the empirical literature, the technical efficiency of the banks is measured using either the production or the intermediation approaches. The production approach considers banks as a producer of services like deposits and loans with labor and capital as primary inputs. Intermediation approach considers banks as an agent to convert the deposits and other funds into loans and other assets. The debate on the ``deposit dilemma'' in the literature acknowledges whether the deposits (or raised funds) should be considered as an input or an output in measuring the banking efficiency. However, many studies adopting the conventional DEA method used deposits as an input as well as an output without discussing the issue of ``deposit dilemma.'' Some studies using two-stage network model have used the deposits as output in the first stage of the production and as an input in the second stage of the production. This study applies the intermediation approach of Berger and Mester [4].

First, we use a standard two-stage network DEA of Kao and Hwang [17] to derive the system and divisional efficiency scores for the actual data considering all the outputs in stages 1 and 2 as desirable outputs. Subsequently, the undesirable exogenous output from stage 1 is introduced in the general two-stage network DEA framework explained below to derive the system and its divisional efficiency scores. We define the efficiency gap (gain/loss) of that undesirable output as the difference between the system efficiency scores of the standard and the general models.

### Two-stage DEA

For *N* number DMUs (decision making units) DMUj*, j* = 1 … *N*, is based on the CRS (constant returns-to-scale) production process with *i*th input and *r*th output X_{ij}, *i* = 1 … *m* and *Y*_{rj}*, r* = 1 … *s*, Kao and Hwang [17] proposed a two-stage network DEA model derived as the following linear program:

$$ \begin{aligned} E_{k} = \hbox{max} \mathop \sum \limits_{r = 1}^{s} u_{r} Y_{rk} \quad {\text{s}} . {\text{t}} .\quad \mathop \sum \limits_{i = 1}^{m} v_{i} X_{ik} = 1, \hfill \\ \mathop \sum \limits_{r = 1}^{s} u_{r} Y_{rj} - \mathop \sum \limits_{i = 1}^{m} v_{i} X_{ij} \le 0,\quad \mathop \sum \limits_{p = 1}^{q} w_{p} Z_{pj} - \mathop \sum \limits_{i = 1}^{m} v_{i} X_{ij } \le 0, \hfill \\ \mathop \sum \limits_{r = 1}^{s} u_{r} Y_{rj} - \mathop \sum \limits_{p = 1}^{q} w_{p} Z_{pj} \le 0,\quad {\text{for}}\quad \, j = 1, \ldots ,n;\quad u_{r} ,v_{i} ,w_{p} \ge \varepsilon , \, r = 1, \ldots ..,s;\quad i = 1, \ldots \ldots .,m; \, p = 1, \ldots ,q. \hfill \\ \end{aligned} $$

(1)

After the optimal multipliers \( u_{r}^{*} , v_{i}^{*} \) and \( w_{p}^{*} \) are solved, the system and divisional efficiencies are calculated as \( E_{k} = \sum\nolimits_{r = 1}^{s} {u_{r} Y_{rk} } ,E_{k}^{1} = \sum\nolimits_{p = 1}^{q} {w_{p} Z_{pk} /} \sum\nolimits_{i = 1}^{m} {v_{i} X_{ik } } \) and \( E_{k}^{2} = \sum\nolimits_{r = 1}^{s} {u_{r} } Y_{rj} /\sum\nolimits_{p = 1}^{q} {w_{p} Z_{pj} } \); this implies \( E_{k} = E_{k}^{1} *E_{k}^{2} \). Kao and Hwang [17] highlighted the problem of non-unique optimal multipliers solved from (1); they give a solution based on the set of multipliers which provide the largest \( E_{k}^{1} \) while keeping the total efficiency score at *E*_{k} derived from (1) for the objective \( E_{k}^{1} = \hbox{max} \sum\nolimits_{p = 1}^{q} {w_{p} Z_{pk} } \, {\text{s}} . {\text{t}} .\,\sum\nolimits_{i = 1}^{m} {v_{i} X_{ik} = 1} \)\( \sum\nolimits_{i = 1}^{m} {v_{i} X_{ik} = 1} \), by adding one more constraint \( \sum\nolimits_{r = 1}^{s} {u_{r} Y_{rk} } - Ek\sum\nolimits_{i = 1}^{m} {v_{i} X_{ik} = 0} \) and keeping all the other constraints in (1). After \( E_{k}^{1} \) is calculated, the efficiency of the second stage is obtained as: \( E_{k}^{2} = E_{k} /E_{k}^{1} \).

### General two-stage DEA

The general two-stage network DEA model of Kao [16] gives more freedom to alter the internal structure of the production process before achieving the final output. It differs from the basic two-stage network DEA model of Kao and Hwang [17] in terms of additional outputs and input measures. The production process is as follows: (1) inputs of stage 1 produce exogenous outputs (undesirable outputs) and intermediate products (desirable outputs), (2) dropping of undesirable outputs produced in stage 1 while carrying forward the desirable outputs along with additional inputs^{Footnote 2} into stage 2, (3) this produces final outputs in stage 2.

The general two-stage network DEA model is given as the following linear program:

$$ \begin{aligned} & E_{0}^{s} = \hbox{max} \mathop \sum \limits_{r = 1}^{{s^{\left( 1 \right)} }} U_{r} Y_{r0}^{{\left( {\text{undesirable}} \right)}} + \mathop \sum \limits_{{r = s^{\left( 1 \right)} + 1}}^{s} U_{r} Y_{r0}^{{\left( {\text{final}} \right)}} ;\quad {\text{s}} . {\text{t}} .\, \mathop \sum \limits_{i = 1}^{m} V_{i} X_{i0} = 1 \\ & \mathop \sum \limits_{r = 1}^{{s^{\left( 1 \right)} }} U_{r} Y_{rj}^{{\left( {\text{undesirable}} \right)}} + \mathop \sum \limits_{g = 1}^{h} W_{g} Z_{gj} - \mathop \sum \limits_{i = 1}^{m} V_{i} X_{ij} \le 0,\quad j = 1 \ldots ..n \\ & \mathop \sum \limits_{{r = s^{\left( 1 \right)} + 1}}^{s} U_{r} Y_{rj}^{{\left( {\text{final}} \right)}} - \mathop \sum \limits_{g = 1}^{h} W_{g} Z_{gj} \le 0\quad j = 1 \ldots ..n \\ & U_{r} ,V_{i} ,W_{g} \ge 0,\quad r = 1 \ldots ..s,\quad i = 1 \ldots \ldots m,\quad g = 1 \ldots ..h \\ \end{aligned} $$

(2)

For expressing simply, we removed the non-Archimedean number ε. At optimality, the system and division efficiencies are calculated as:

$$ \begin{aligned} E_{k}^{s} & = \mathop \sum \limits_{r = 1}^{{s^{\left( 1 \right)} }} U_{{\ddot{r}}} Y_{r0}^{{\left( {\text{undesirable}} \right)}} + \mathop \sum \limits_{{r = s^{\left( 1 \right)} + 1}}^{s} U_{{\ddot{r}}} Y_{r0}^{{\left( {\text{final}} \right)}} \\ E_{k}^{1} & = {{\left( {\mathop \sum \limits_{r = 1}^{{s^{\left( 1 \right)} }} U_{{\ddot{r}}} Y_{r0}^{{\left( {\text{undesirable}} \right)}} + \mathop \sum \limits_{g = 1}^{h} W_{{\ddot{g}}} Z_{g0} } \right)} \mathord{\left/ {\vphantom {{\left( {\mathop \sum \limits_{r = 1}^{{s^{\left( 1 \right)} }} U_{{\ddot{r}}} Y_{r0}^{{\left( {\text{undesirable}} \right)}} + \mathop \sum \limits_{g = 1}^{h} W_{{\ddot{g}}} Z_{g0} } \right)} {\mathop \sum \limits_{i = 1}^{m} V_{{\ddot{i}}} X_{i0} }}} \right. \kern-0pt} {\mathop \sum \limits_{i = 1}^{m} V_{{\ddot{i}}} X_{i0} }} \\ E_{k}^{2} & = {{\mathop \sum \limits_{{r = s^{\left( 1 \right)} + 1}}^{s} U_{{\ddot{r}}} Y_{r0}^{{\left( {\text{final}} \right)}} } \mathord{\left/ {\vphantom {{\mathop \sum \limits_{{r = s^{\left( 1 \right)} + 1}}^{s} U_{{\ddot{r}}} Y_{r0}^{{\left( {\text{final}} \right)}} } {\mathop \sum \limits_{g = 1}^{h} W_{{\ddot{g}}} Z_{g0} }}} \right. \kern-0pt} {\mathop \sum \limits_{g = 1}^{h} W_{{\ddot{g}}} Z_{g0} }} \\ \end{aligned} $$

(3)

Even though the intermediate products Zg are not reflected directly in calculating the system efficiency \( E_{k}^{s} \), they have been considered for in deriving the division efficiencies \( E_{k}^{1} \) and \( E_{k}^{2} \) which will affect \( E_{k}^{s} \) through the multiplier *U*_{r} and *V*_{r}. For decomposing the system efficiency into divisional efficiencies done through the following equation \( E_{k}^{s} = \left[ {\omega^{1} E_{k}^{1} + (1 - \omega^{1} } \right] \times \left[ {\omega^{2} E_{k}^{2} + \left( {1 - \omega^{2} } \right)} \right] \) where \( \omega^{1} = {{\sum\nolimits_{i = 1}^{m} {V_{{\ddot{i}}} X_{i0} } } \mathord{\left/ {\vphantom {{\sum\nolimits_{i = 1}^{m} {V_{{\ddot{i}}} X_{i0} } } {\left( {\sum\nolimits_{i = 1}^{m} {V_{{\ddot{i}}} X_{i0} } } \right)}}} \right. \kern-0pt} {\left( {\sum\nolimits_{i = 1}^{m} {V_{{\ddot{i}}} X_{i0} } } \right)}} \) and

$$ \omega^{2} = {{\mathop \sum \limits_{g = 1}^{h} W_{{\ddot{g}}} Z_{g0} + V_{{\ddot{i}}} X_{i0} } \mathord{\left/ {\vphantom {{\mathop \sum \limits_{g = 1}^{h} W_{{\ddot{g}}} Z_{g0} + V_{{\ddot{i}}} X_{i0} } {\left( {\mathop \sum \limits_{i = 1}^{m} V_{{\ddot{i}}} X_{i0} + \mathop \sum \limits_{r = 1}^{{s^{\left( 1 \right)} }} U_{{\ddot{r}}} Y_{r0}^{{\left( {\text{undesirable}} \right)}} } \right)}}} \right. \kern-0pt} {\left( {\mathop \sum \limits_{i = 1}^{m} V_{{\ddot{i}}} X_{i0} + \mathop \sum \limits_{r = 1}^{{s^{\left( 1 \right)} }} U_{{\ddot{r}}} Y_{r0}^{{\left( {\text{undesirable}} \right)}} } \right)}} $$

### Data

We used data from 46 banks of India (26 public sector banks, 20 private sector banks) for the period from 2014 to 2016, mainly collected from the Statistical Tables Relating to Banks in India published by the Reserve Bank of India (www.rbi.org.in). These 46 banks represent more than 85% of banking business across India. Also, there are 43 foreign and 51 cooperative banks that hold less than 15% of the banking business. Therefore, we are not included foreign banks and cooperative banks in our analysis. The shareholding pattern of these banks was collected from the latest Annual Reports of the respective banks to corroborate any effects on the efficiency gap. For the analysis, we applied the same definitions and variables used by Gulati and Kumar [13], Kumar and Gulati [18] and Sharma and Dalip [25]. For the standard two-stage network DEA, we considered fixed assets (*x*_{1}), employees (*x*_{2}), and loanable funds (*x*_{3}) (deposits plus borrowings) as inputs along with advances (*z*_{1}) and investments (*z*_{2}) as intermediaries and net interest income (*y*_{1}) and non-interest income (*y*_{2}) as final outputs. For the general two-stage network DEA, we separated net NPA (*y*^{undesirable}) from the advances (*z*_{1}) and used it as an undesirable output in the model with the rest of the advances (\( z_{1}^{*} \)) modeled as an intermediary along with investments (*z*_{2}). Figure 1 gives the flowchart of the general two-stage DEA.