A generalized ineﬃciency model with input and output dependence

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Introduction
In this paper we consider technical inefficiency as a function of inputs and outputs.The dominant paradigm in the literature is to use environmental or contextual variables as determinants of technical inefficiency (Battese and Coelli, 1988, 1995, Battese and Broca, 1997, and Wang, 2002).Such determinants are useful when one wishes to argue that inefficiency may depend on the context, regulation, etc.However, it is known that achieving higher efficiency levels is costly in terms of resources and, therefore, in terms of inputs and / or outputs ( Bogetoft and Hougaard, 2003).Using inputs and outputs as determinants of inefficiency opens up new problems and is by no means trivial as in the case of incorporating environmental or contextual variables.The reason is that most behavioral assumptions (cost minimization, profit or revenue maximization) require that inputs and / or outputs are selected optimally: When inefficiency depends on inputs and / or outputs, the decision making unit (DMU) optimization problem changes, as the first order conditions must take into account the dependence of inefficiency on the endogenous variables of the optimization problem.
Our paper is related to Hampf (2017), who used sequential definitions of the production technology, to decompose cost inefficiency into rational and residual inefficiency as well as inefficiency caused by technical change.Related literature includes Fukuyama and Matousek (2018), Aparicio, Mahlberg, Pastor, and Sahoo (2014), Kapelko and Oude Lansink (2017), Kapelko, Oude Lansink, and Stefanou (2014), Tran and Tsionas (2016), Tsionas and Mamatzakis (2019), etc.In addition, Hampf (2017) provided lower bounds for unobserved adjustment costs based on unexploited cost reductions due to rational inefficiency.Specifically, based on the decomposition into technical and allocative rational inefficiency, lower bounds for the radial and the non-radial adjustment costs can be estimated.Hampf (2017) suggested that parametric models could also be used, as in Park and Lesourd (2000).This study is in this direction of research, although we provide a completely new parametric model.

Preliminaries
Suppose x ∈ K + is a vector of inputs, and we have a production function of the form: where u ≥ 0 represents technical inefficiency and v is statistical noise.To describe inefficiency in a general input -output space we need to assume u = u(x, y).To see this consider Figure 1.From a point A it is possible to move to points B 1 , B 2 or B 3 .Point B 2 corresponds to a vertical movement (direction 2) corresponding to no change in the inputs.This change is rarely possible and points like B 1 or B 3 are more likely to occur in practice.This, however, requires simultaneous changes in both inputs and outputs.
In addition, to move from A to any other point like B 3 requires real resources in terms of both inputs and outputs, viz.
it is costly to adjust inputs and outputs.A move from point A to B 1 (direction 1) could occur when adjustment costs are high and inputs need to be reduced: In this case, however, output can be increased (direction 1) when, for example, managerial practices improve.This increase in output may be also subject to adjustment costs when management (and, therefore, its cost) is unobserved.These adjustment costs are used to evaluate the feasibility of exploiting cost reductions caused by inefficiency.
Point A could be interpreted as last period's input-output combination and, in turn, moving to points like B 1 , B 2 or B 3 , amounts to selecting a particular movement in the input-output space.Such movements (with the exception of "2") are, however, costly and, in practice, the costs are unobserved.Moreover, it is not necessary to assume that such movements result in efficiency improvements and, in general, it is clear that inefficiency should depend on both inputs and outputs.Suppose that we write: where f (ln x) = ln F (x).We should notice that the distribution of y conditional on x depends also on y through u(x, y).This may seem to prevent further analysis of the model as y appears on both sides of (2).Technically, what is required to obtain the distribution of y conditional on x is the Jacobian of transformation: so that the distribution has density: Inefficiency dependence on both inputs and outputs can be motivated based on Figure 1 where inefficiency changes require changes in both inputs and outputs.These changes are, however, costly, in real life since increasing or decreasing inputs (and the associated changes in outputs) is, generally speaking, costly.A typical example is when labor or capital have to be increased or decreased: Increases take time and are costly while decreases may be subject to labor laws or disinvestment which is also time-consuming as well as costly.In particular, less utilization of equipment services implies costs of depreciation among others like maintenance and repair and similar costs are in place when equipment services have to be increased.Labor service utilization can be increased only with time and obvious additional costs per unit.These considerations imply that, in general, input-output changes are costly and resemble a situation close to the one in Figure 1.An alternative is to use a directional distance function, see Chambers, Chung, andFäre (1996), andChung, Färe, andGrosskopf (1997).Deriving optimal directions has received some attention, see Peyrache and Daraio (2012) who recommend sensitivity analysis to different direction vectors, and Färe, Grosskopf, and Whittaker (2013) who propose to estimate the optimal directions by maximizing inefficiency with respect to the direction vectors.This line of research has been explored in Atkinson, Primont, and Tsionas (2018).The methodology presented here is different in several respects.First, we can estimate directly adjustment costs.Second, directions are implicitly defined as the movement from point A to points like B 1 , B 2 or B 3 can be estimated.Third, these directions are endogenized in the model via the adjustment costs for each input and output.Fourth, technical inefficiency is directly a function of inputs, outputs and control or contextual variables, so that effects of input-output changes on inefficiency can be estimated directly.This introduces statistical challenges which are successfully resolved.To the best of our knowledge, there are no models that can incorporate the influence of input-output-contextual variables on technical inefficiency, simultaneously.

The model
Suppose x ∈ K + is a vector of inputs, y ∈ M + is a vector of outputs and z ∈ J is a vector of control or environmental variables.
Feasible combinations of inputs and outputs are described by: The firm maximizes profits: where w ∈ K + is a vector of input prices and p ∈ M + and is a vector of output prices.We represent the technology using an output distance function (ODF, Fare and Primont, 1995, p. 11): It is non-decreasing, positively linearly homogeneous, increasing and convex in y, and decreasing and quasi-concave in x.
Alternatively, we can write: where u ≥ 0 represents technical inefficiency.In this paper we assume that technical inefficiency may be a function of inputs, outputs and the control variables: The problem of the firm instead of ( 6) is: x,y,z) . (11) The adjustment cost coefficients (for outputs and inputs) are expressed in terms of prices of outputs and inputs.In (11), y o , x o and z o are the previous period values of outputs, inputs and the controls.There are adjustment costs c y m for each output, c x x for each input, and c z j for each control.If a variable can be adjusted without cost we can simply set the respective adjustment cost to zero.If it is beyond the control of the form we can set it to infinity or a very large number.The first order conditions to the problem are as follows: We can express these conditions in equivalent form as follows: In turn, we have: To eliminate the Lagrange multiplier λ we have: where pm = p m /p 1 , wk = w k /w 1 , and cz j = c z j /c z 1 .This is a system of M + K + J − 3 equations in M + K + J endogenous variables.The two of the three missing equations are provided by that is we have: where we set w 1 = c z 1 = 1 without loss of generality, for normalization.
The last equation is given by (9).The system consisting of ( 9), ( 15) and ( 17) can be written in the form: where Y = [y , x , z ] , F is a vector function in M +K+J , c is the vector of adjustment costs, and θ ∈ Θ ⊆ d is the vector of parameters in the ODF.If we have panel data, the adjustment coefficients are firm specific, and technical inefficiency is firm-specific and time-varying we have: where The first element of Σ is σ 2 v and other elements in the first row and column of Σ are zero.Before proceeding we need to parametrize technical inefficiency and the ODF.For the ODF we assume a translog functional form.This can be written as: where For technical inefficiency we assume: i.e., it follows a truncated normal distribution, where µ it is also a translog: γ jj ln Ỹit,j ln Ỹit,j .
Estimation relies on Maximum Simulated Likelihood (MSL) whose details are presented in the Technical Appendix.
Quantities of interest like returns to scale, technical efficiency, technical efficiency change and productivity growth can be estimated as usual (see notes to Figure 4).Given the truncated normal specification in ( 22), we can define inefficiency as: , where E denotes (conditional) expectation.From Kumbhakar and Lovell (2000, p. 86) we have: (at the estimated parameters), and σ 2 v is the uppermost left element of Σ.

Data and empirical results
The data is the same as in Malikov, Kumbhakar, and Tsionas (2016).We use data on commercial banks from Call Reports available from the Federal Reserve Bank of Chicago and include all FDIC insured commercial banks with reported data for 2001:I-2010:IV.We focus on a selected subsample of relatively homogeneous large banks, viz.those with total assets in excess of one billion dollars (in 2005 U.S. dollars) in the first three years of observation.The data sample is an unbalanced panel with 2,397 bank-year observations for 285 banks.We have the following desirable outputs: consumer loans (y 1 ), real estate loans (y 2 ), commercial and industrial loans (y 3 ) and securities (y 4 ).These output categories are the same as those in Berger andMester (1997, 2003).Following Hughes andMester (1998, 2013), we include off-balance-sheet income (y 5 ) as output.The undesirable output is total non-performing loans (NPL).The variable inputs are labor, i.e., the number of full-time equivalent employees (x 1 ), physical capital (x 2 ), purchased funds (x 3 ), interest-bearing transaction accounts (x 4 ) and non-transaction accounts (x 5 ).
We also include financial (equity) capital (EQ) as an additional input to the production technology.However, due to the unavailability of the price of equity capital, we follow Berger andMester (1997, 2003) and Feng and Serletis (2009) in modeling EQ as quasi-fixed.This is in line with Hughes andMester's (1993, 1998) in that banks may use equity as a source of funds.We derive the prices of variable inputs (w 1 through w 5 ) by dividing total expenses on each input by the respective input quantity.
In Figure 2 we report sample distributions of adjustment cost parameters and inefficiency effects.Adjustment cost parameters are lowest for labor (x 1 ), followed by purchased funds (x 3 ), interest-bearing transaction accounts (x 4 ) and physical capital (x 2 ) which has the largest adjustment coefficient, which makes intuitive sense on prior grounds as equipment and capital, in general, cannot be adjusted easily.The reason why this makes sense is because the model captures at least some part of what is going on in the real world with input and output adjustments.This is, conventionally, ignored and interest focuses on inefficiencies without crediting the firm with rational (profit-maximizing) adjustments in the input-output space.However, ignoring these adjustments overestimates (irrational) inefficiency and does not allow for inefficiency reductions realized through changes in outputs and inputs.
Inefficiency effects in the form of elasticities ( ∂ log E(u) ∂ log Y ) are, for the most part, positive for all outputs and negative for all inputs with the exception of x 2 .Output adjustment costs are highest for NPLs and lowest for y 4 (securities) with other adjustment costs lying somewhere in between.For selected inputs and outputs, sample distributions by percentile are shown in Figure 3.It is important to mention that these elasticities serve as bank-specific directional parameters in the input-output space as in Figure 1.For the most part, and based also on the results reported in Figure 3, the directions are positive for inputs and negative for outputs suggesting an adjustment process (relative to Figure 1) that is more complicated relative to the straightforward northwest or northeast directions.With more than one input and output this adjustment process is necessarily more complex.
The results of Figure 3 can be useful when bank-specific directions are sought on a quantile-based basis.To understand the nature of adjustment cost parameters better we convert the c y and c x into percentages in terms of the corresponding input and output values.The results are reported in Table 1.From these results we see that output adjustment costs are 4-5% but input adjustment costs are more variable ranging from 3% to just under 10%.The adjustment costs due to NPLs average to 8.77% (s.d.2.33%).Since we have 285 banks with total assets more than a billion USD, these adjustment costs are sizable, giving in turn a justification for inefficiency which is about 15% on the average.These real costs in other words, imply that relative movements in the input-output space are costly and inefficiency cannot be reduced at will.In a certain sense this is why the presence of inefficiency is "rational" and is justified by the technology or cost / profit structure of US commercial banks.From the policy point of view it seems relatively unlikely that inefficiency and adjustment costs are due to the "quiet life hypothesis" which states that commercial banks can trade some inefficiency for the monopolist's "quiet life".Inefficiency appears to be due to the impossibility of certain adjustments in the input-output space or inertia due to the technology and profit-maximizing behavior.This is why an argument like "reduce inefficiency first and bail out later" may not work for commercial banks as the amount of reductions in inefficiency is rather limited by the technology and profit-maximizing behavior.Surely, some amount of inefficiency can be reduced directly but for the most part inefficiency is just the way things are given the technology and behavioral assumptions about commercial banks in the US.
The distribution of weights w s at the MSL parameter estimates θ is reported in Figure A1 of Technical Appendix (left panel).In the right panel of Figure A1 we report 20 representative (kernel) densities of the weights corresponding to specific observations (i, t) across all simulations at the MSL parameter estimates θ.In Figure 4 x 2 x 3 x 4 x 5 Notes: All adjustment cost parameters are relative to x5 (non-transaction accounts).We have the following desirable outputs: consumer loans (y1), real estate loans (y2), commercial and industrial loans (y3) and securities (y4).These output categories are the same as those in Berger andMester (1997, 2003).Following Hughes andMester (1998, 2013), we include off-balance-sheet income (y5) as output.The undesirable output is total non-performing loans (NPL).The variable inputs are labor, i.e., the number of full-time equivalent employees (x1), physical capital (x2), purchased funds (x3), interest-bearing transaction accounts (x4) and non-transaction accounts (x5).We also include financial (equity) capital (e) as an additional input to the production technology.Notes: All adjustment cost parameters are relative to x5 (non-transaction accounts).We have the following desirable outputs: consumer loans (y1), real estate loans (y2), commercial and industrial loans (y3) and securities (y4).These output categories are the same as those in Berger andMester (1997, 2003).Following Hughes andMester (1998, 2013), we include off-balance-sheet income (y5) as output.The undesirable output is total non-performing loans (NPL).The variable inputs are labor, i.e., the number of full-time equivalent employees (x1), physical capital (x2), purchased funds (x3), interest-bearing transaction accounts (x4) and non-transaction accounts (x5).We also include financial (equity) capital (e) as an additional input to the production technology. of ( 21) with respect to time.Efficiency change is EC it = ln(r it /r i,t−1 ).Productivity growth is P G it = e y,it + EC it .Efficiency change is positive for some banks and negative for others, and the same is true for productivity growth.Moreover, technical inefficiency ranges from 5% to 35%.

Concluding remarks
In this paper we propose a generalized (distance function) model where inefficiency depends on both inputs and outputs.
Statistical challenges are involved as, to put things in a simplified matter, in a production function model, inefficiency depends on both inputs and outputs.The challenges from dependence on output are successfully resolved.Moreover, we recognize that inefficiency can be rational due to adjustment costs involved in the re-structuring in the input-output space.The model is successfully applied to a data set of large U.S. banks.Limitations of the model, and thus possible avenues for future research include, among others, conversion of parts of the model in semiparametric or nonparametric components.The semiparametric component may lift the parametric specification in either the form of the ODF, the inefficiency specification part or possibly both.This involves identification problems and may not be easy in practice.A fruitful avenue for future research may also involve more general adjustment cost processes, possibly in a nonparametric way, which would enrich our understanding of inefficiency and associated changes in the input-space of the firm.From the policy viewpoint, and to keep in mind for future applications, our inefficiency estimates take into account and, therefore, they are robust to rational, profit-maximizing, changes in inputs and outputs.It would be interesting to come up with new measures of rational inefficiency where overall inefficiency is decomposed into rational and residual (or "irrational").One possible way of doing this is to follow Hampf (2017)   it and, in turn, examining whether at least 30% of these weights are non-zero.As these weights depend on θ we first choose α so that the weights w s are not degenerate for the initial values of θ.
Next we examine them for degeneracy at the final estimates.If they are degenerate we proceed with another draw for α and we repeat the same process until non-degeneracy is obtained.After a number of iterations of this procedure final weights were non-degenerate and we satisfied ourselves that their distribution was not too far away from a uniform, using visual means.The final distribution could be described well by a beta distribution with parameters 3.42 and 1.75, approximately.The simulated likelihood is optimized using a subspace-search algorithm.2Initial conditions are obtained from a version of the model without adjustment costs and without dependence of technical inefficiency on x, y and z.Finally, the function: ,y,z) , (A.8) must satisfy the standard theoretical properties of ODF (non-decreasing, positively linearly homogeneous, increasing and convex in y, and decreasing and quasi-concave in x) along with f (x, y, z) ≤ 0 ∀(x, y, z) ∈ T .Since D(x, y, z) is in levels but Ee −u(x,y,z)     has a different functional form, the separate components of the function in (A.8), in principle, can be identified.Without the expectation, both D(x, y, z) and e −u(x,y,z) are quadratic functions in x, y, z and no separate identification is possible if both are translog (or another common functional form for that matter).To examine identification in practice, we consider minus the Hessian of the log likelihood: As this depends on the data Y , we average across all observations.Finally, we report parameter estimates of the model in Table A1 for the ODF parameters and Table A2 for the frontier 213 parameters..

Figure 1 :
Figure 1: Inefficiency depending on inputs x and outputs y B3 y B2 3 2

Figure 2 :
Figure 2: Sample distributions of adjustment cost parameters and inefficiency effects

Figure
Figure 4: Aspects of the model = 1, ..., S is not degenerate in the sense that most weights are zero and only a few are close to unity.In practice, we consider w s = (nT ) , c; Y ) ∂(θ, c)∂(θ, c) , (A.9) a positive-definite matrix near the maximum of the log likelihood function.The determinant of the Hessian should be non-zero if we have identification, so, we can consider the smallest eigenvalue of H (say λ min ) which should be bounded away from zero.

Figure A. 1 :Figure
Figure A.1: Distribution of importance weights from Maximum Simulated Likelihood Left panel: The distribution of weights w s at the MSL parameter estimates θ.Right panel: 20 representative kernel densities of the weights corresponding to specific observations (i, t) across all simulations at the MSL parameter estimates θ.
examine identification in practice, we consider minus the Hessian of the log likelihood:H = − ∂ 2 ln L (θ,c;Y ) ∂(θ,c)∂(θ,c) , a positive definite matrix near the maximum of the log likelihood function.The determinant of the Hessian should be non-zero if we have identification so we can consider the smallest eigenvalue of H (say λmin) which should be bounded away from zero.As this depends on the data Y , we can average across all observations.

Table 1
and define residual inefficiency as deviation from its conditional expectation.Although this is certainly possible it would be much more interesting to build residual inefficiency directly into the model.