A dynamic efficiency model using data envelopment analysis Jati K. Sengupta * Department of Economics University of California, Santa Barbara California Abstract Changes in productivity due to technological progress or regress are incorporated in models of data envelopment analysis (DEA) in two ways: either through a network technology or in terms of a capacity variable or capital input embodying technical progress. The second way is analyzed here in terms of time-varying capital inputs due to the changes in input prices over time. This provides an optimal control theoretic view of the time-path of capital inputs which minimizes a discounted sum of total input costs for each DMU. The cost of risk aversion and the adjustment cost of input price fluctuations may then be minimized in an extended DEA model framework. These extensions of the DEA model are analyzed here and illustrated by an empirical application to international airlines data. Keywords: production efficiency and allocative efficiency; dynamic efficiency; capital utilization Professor of Economics and Operations Research, Department of Economics, University of California, Santa Barbara, CA 9306-920; Fax (805) 893-3797; E-mail <sengupta@econ.ucsb.edu> * The author is sincerely thankful to two anonymous referees for their constructive suggestions. The usual disclaimer applies.
A dynamic efficiency model using data envelopment analysis. Introduction Data envelopment analysis (DEA) is a new technique developed in operations research and management science over the last two decades for measuring productive efficiency. This is a nonparametric technique based only on the observed input output data of firms or decision making units (DMUs) and it does not require any data on the input and output prices. Due to this flexible feature it has been widely applied to the public sector enterprises, e.g., schools and colleges, state hospitals and clinics and also public utilities, which are nonprofit organizations. Recently private sector units have also applied this technique, particularly in the multi-input multi-output case and compared it with the usual parametric forms of production functions such as the Cobb-Douglas or the translog functions. Charnes et al. [] have recently surveyed the state of the art and its numerous applications and Sengupta [2] has considered some of its dynamic and stochastic extensions. Two types of efficiency measures are usually distinguished at the firm level in production economics. One is technical or production efficiency, which measures the firm s success in producing maximum output from a given set of inputs. The other is the price or allocative efficiency, which measures a firm s success in choosing an optimal set of inputs with a given set of input prices. Most DEA models are applied to estimate the technical efficiency of public sector DMUs, which does not require using any price data. However even for public sector units, e.g., educational units some price data such as salaries and material costs are usually available and determined in competitive markets. Hence it is more useful to consider allocative efficiency considerations in DEA models. Two other considerations are important here. One is the choice
2 of the decision variables, e.g., the allocative efficiency models seek to determine the optimal levels of inputs as the decision variables, whereas the technical efficiency model treats the observed inputs and outputs as given and tests if each DMU achieves it maximum possible level of output for given inputs. Secondly, some inputs like capital have output effects spread over several periods ahead and hence considerations of intertemporal cost minimization acquire strategic importance here. Our object here is two-fold. We consider first a dynamic efficiency model involving intertemporal cost minimization in the framework of a DEA model. Secondly, we consider the implications of the intertemporal cost frontier, when the input prices are not completely known, i.e., they are subject to stochastic errors. 2. The dynamic models Two types of dynamic models are developed here. One treats all inputs varying over time and then incorporates an intertemporal planning horizon into the DEA model. This model has the flexible feature in allowing a firm to compute the time path of optimal input usages over a period of time. The second type of model distinguishes between the current and capital inputs and then minimizes a discounted stream of costs for both current and capital outlays in the framework of a DEA model. This model has the flexibility that it provides a dynamic cost frontier, where the short-run and the steady state optimal solutions can be separately analyzed. ote that the dynamics of production technology considered here through changes in capital and other capacity-related inputs is not of the form of network technology which is analyzed by Fare and Grosskopf [3]. Their approach is more suitable for activity analysis type models where outputs from earlier periods can be used as an input for this period. Our approach
3 involves the concept of a dynamic production function, where output depends on both current inputs and the investment in capital inputs. Since we are interested in overall efficiency over time, when the expected future prices of inputs prevail, the current and capital inputs are the decision variables as in a typical model of optimal control. This is a forward looking approach for specifying an optimal path of the dynamic capital inputs, which may be compared with the observed path when the latter is available. Consider first a static DEA model for determining the technical efficiency of a reference unit with m inputs and s outputs: Min θ λ, θ subject to X λ θ X j j k Y Y jλ j k () λ = ; λ 0, θ 0 j j where the reference unit is DMU k with the observed input output data (X k,y k ), where X k and Y k are column vectors with m and s elements respectively. This is called an input oriented model in the DEA literature [3]. Here the reference unit is compared with the other units in the cluster. * Let λ* = ( λ j) and θ* be the optimal solutions of the above DEA model. Then the optimal value θ* provides a measure of technical efficiency (TE), e.g., let θ* = and the first two sets of inequalities in () hold with equality, then DMU k is 00 percent at the TE level. If θ* is positive but less than one, then it is not technically efficient. Overall efficiency (OE j ) of a DMU or firm j however combines both TE j and the allocative efficiency (AE j ) as follows:
4 OE j = TE j AE j ;,2,..., (2) Recently Sueyoshi [4] has applied these concepts in DEA framework. For testing overall efficiency of the reference unit DMU k one sets up the linear programming (LP) model as follows: Min q x x,λ s.t. X jλ j x (3) Y λ Y j j k λ = ; x 0; λ 0 j Here input price vector q with a prime denoting its transpose represents the competitive market price and x is an m-element input vector optimally decided by DMU k along with the weights λ j. Let λ* and x* be the optimal solutions of the LP model (3) above. Then the minimal cost is * given by ck = q x *, whereas the observed cost of the reference unit k is c k = q X k. Hence OE k is determined as OE k = c k * /ck = q x*/q X k and TE k = θ* AE k = OE k /TE k = c k * (θ*ck ) -
5 ow consider a dynamic extension of the overall efficiency model (3), where the reference unit DMU k or the firm k uses a quadratic loss function to choose the sequence of decision variables x(t) = (x i (t)) over a planning horizon. The objective now is to minimize the expected present value of a quadratic loss function subject to the constraints of (3) as follows: t Min L = E ρ q t x t + d t Wd t + z t Hz(t x( t), λ( t) t t= [ ( ) ( ) ( / 2)( ( ) ( )) ( / 2 )( ( ) ))] s.t. X ( t) λ ( t) x( t) (4) j j Y ( t) λ ( t) Y ( t) j j k λ ( t) = ; x( t) ; λ( t) 0 j where ρ is a known discount factor and the vectors d(t) = x(t) - x(t-) and z(t) = x(t) - x$( t ) are deviations with W and H being diagonal matrices representing the weights. The quadratic part of the objective function may be interpreted as adjustment costs, the first component being the cost of fluctuations in input usages and the second being a disequilibrium cost due to the deviations from the desired path denoted by x$( t ). This interpretation is very similar to the optimal production scheduling model known as the HMMS model developed by Holt, Modigliani, Muth and Simon [5], where the first component is the smoothness objective and the second the goal programming objective. ote that the target levels $x (t) of the goals may be time-varying, so that the deviation reflects the gap or inconsistency from the normal levels as desired by the management. The various dynamic implications of adjustment costs in the DEA model have
6 been discussed by Sengupta [2,6] elsewhere. On using the Lagrange multiplier µ(t) = (µ i (t)) for the first constraint and assuming an interior optimal solution with x i (t) > 0, the optimal intertemporal path of inputs x i (t) may be specified as follows: * α i x i (t) = * wi x i (t-) + * ρwi x i (t+) + * hi $x i (t) - q i (t) + µ i ( t ); i =, 2,..., m (5) where asterisk denotes optimal values, α i = w i + ρ w i + h i and it is assumed that future expectations are realized, i.e., E t (x i (t+)) = x i (t+). This last assumption is also called the rational expectations hypothesis, implying a perfect foresight condition. This type of hypothesis has been frequently used in recent macrodynamic models in economics. If this condition is not fulfilled, then the term x * i ( t + ) would be replaced by * Et xi ( t + ) and this may lead to errors of forecasting future values. In the updating of optimal decision rules this may generate an additional source of inefficiency. Several implications of this optimal linear decision rule (5) may now be briefly discussed. First of all, if the observed input path X k (t) does not follow the optimal path x*(t) for any t, we have intertemporal inefficiency due to this divergence. This divergence may be cumulative, if it persists over several time points. Secondly, the myopic optimal value x* computed from (3) can be directly compared with the intertemporal optimal path x*(t). Since the static efficiency associated with the myopic input levels x* ignores the potential losses arising over time, it is frequently biased. Hence the static efficiency ranking of DMUs is most likely to be suboptimal in a dynamic setting. Thirdly, one could consider the linear decision rules (5) as a set of linear difference equations and update the optimal decision rules as in the HMMS model of production and inventory decisions. Treadway [7], Kennan [8] and Sengupta [9] have discussed the implications of this optimal decision rule in the theory of adjustment costs when
7 technological change occurs in the expansion of capital inputs. Finally, the cost and production frontiers implicit in the static DEA model (3) may be updated at each time point t by incorporating the intertemporally optimal values, when the target levels $x (t) change. Since the linear decision rule (5) depends linearly on the parameters w i, ρ, h i x$ i ( t) and q i (t), one could explain the source of divergence of static from dynamic efficiency in terms of the changes in these parameter values. For example the higher the future levels of input price q i (t), the lower will be the optimal input demand x * i ( t ). Thus any DMUk has to forecast the future values of the prices and costs q i (t), w i, h i with a low forecasting error so that it may be on the dynamically efficient cost frontier. Otherwise a large error variance may cause large deviations from the expected intertemporal optimal path for x(t). Thus when the agent is risk averse, he may consider the potential loss due to such large error variances and incorporate them in the original objective function of model (4). In this case the objective function gets transformed into more nonlinear than a quadratic. ext we consider a second type of dynamic formulation, when capital inputs are distinguished from the current inputs. For simplicity we assume the first m- inputs to be current and the m-th input as capital. If q m (t) is the price of the capital input, then q m (t) x m (t) is the investment in durable goods in the process. Assuming continuous discounting at an instantaneous rate r, the cost on current account of an initial investment outlay q m (t) x m (t) is r q m x m. Hence the total current cost is c = m qixi + rqmxm i= (6)
8 Minimizing this cost function c in (7) subject to the constraints of model (3) provides a measure of overall efficiency in the short period. If x* is the optimal input vector, then the overall inefficiency of DMU k in the use of capital input is given by OE k (x m ) = * rqmxm rqmxmk = * xm / Xmk (7) In the dynamic case we have to introduce a planning horizon, which may be finite or infinite. In the finite case the decision problem is one of choosing the current and capital inputs so as to minimize the total discounted cost over the horizon 0 < t < T: T m rt c = e q i ( t) xi ( t) + q m( t) xm( t) dt 0 i= s.t. x λ ( t) x ( t); i =, 2,..., m ij j i x λ ( t) x ( t) mj j m y λ ( t) Y ; r =, 2,..., s (8) rj j rk λ j( t) =, x 0; λ( t) 0 However this formulation (8) ignores the investment aspect of capital expansion. Let z m (t) be gross investment and δ is the constant rate of depreciation for the expansion of capital inputs, i.e., &x m (t) = z m (t) - δ x m (t) (9)
9 where the dot denotes the time derivative. In this case the cost of investment c(z m (t)) has to be included as a component of the objective function, where the last term has to be dropped. The final objective function thus becomes: T m rt Max J = -c = - e qi ( t) xi ( t) + c(zm( t)) dt 0 i= s.t. (8) and (9) (0) This model helps to determine the optimal time path z * m ( t ) of investment and hence the optimal time path of capital expansion in terms of { x * m ( t ), 0 < t < T}. This type of model can be easily solved by Pontryagin s maximum principle, whereby we introduce the Hamiltonian function H as: H = e -rt m qi ( t ) xi ( t ) + c ( zm ( t )) + pm ( zm ( t ) δ xm ( t )) i= where p m = p m (t) is the adjoint function. If the optimal path of x m (t) exists, then by Pontryagin principle there must exist a continuous function p m (t) satisfying p& m( t) = ( r + δ ) pm( t) µ () where µ = µ(t) is the Lagrange multiplier associated with the second constraint of model (8). Also we must have the optimal path of investment z m = (t) satisfying at each moment of time t the optimality condition: c(z m )/ z m - p m (t) 0, at each t This implies for every positive level of investment the equivalence of marginal investment cost with the optimal shadow price, i.e., c(z m )/ z m = p m (t), for z m (t) > 0 (2)
0 In addition the adjoint variable p m (t) must satisfy the transversality condition rt lim e pm( t) = 0 (3) t T rt i.e., lim e pm( t) xm( t) = 0 t T 2 If the investment cost function c(z m ) is of a quadratic form, i.e., c(z m ) = (/2) α z m, α > 0, then the optimality conditions become * * m m z ( t) = p ( t) / α * * * m m m x& = ( p / α) δ x (4) * * m m p& = ( r + δ ) p µ * rt * * lim e pm( t) xm( t) = 0 t T with asterisks denoting optimal values. ote that the investment cost function includes potential losses from fluctuations in prices of capital and the cost of building capacity ahead of demand. * * The optimal trajectories { xm t pm t t T} ( ), ( );0 < < determined by the system (4) of necessary conditions have several interesting economic implications for efficiency. * * First of all, the steady state solutions ( xm pm) would be stable if the following conditions hold:, on the optimal trajectory defined by (4) * * pm / α < > δxm according as x m > < x m * and (r + δ) p m < > µ* according as p m > < p m *
since the differential equations above are linear. Also one could combine the two differential equations above to derive a single second order linear differential equation as follows: * * * m m m α&& x αrx& αδ( r + δ ) x + µ * = 0 (5) Its characteristic equation is u 2 - ru - δ(r + δ) = 0 which shows the two roots to be real and opposite in sign, i.e, u > 0, u 2 < 0. Thus the steady * * state pair ( xm pm), has the saddle point property. On assuming a fixed steady state value for µ*, the transient solution of equation (5) can be written as * * * * µ u t xm( t) = xm( 0) e ( r + ) 2 µ + ( r ) αδ δ αδ + δ * where x m ( 0 ) is the initial value of x * ( t) at t=0. ote that the constant term A in the solution m * * u t u t xm( t) = A e + A e 2 µ 2 + αδ( r + δ) has to be set equal to zero in order to satisfy the transversality conditions (3). Finally, if the observed path of capital expansion equals the optimal path, i.e., x m (t) = x * m ( t ) for every t, then the DEA model (8) would exhibit dynamic efficiency; otherwise any divergence of the two paths would generate inefficiency over time. In that case the conditional production function given an inoptimal stock of capital inputs would exhibit myopic inefficiency. 3. The stochastic implications The dynamic efficiency models above have one basic shortcoming in that it is assumed that the input prices are all deterministic and contain no errors. Since these are future prices they are
2 unlikely to be known for certain. Farrell [0] who is credited in being the first to originate the nonparametric method of determining a production and cost frontier never recommended using market prices of inputs. He raised two basic objections. One is that this efficiency measure will be seriously biased if the observed input prices are widely fluctuating, e.g., it may happen that a firm is not allocatively efficient, if its observed inputs are adjusted to the past or the expected future prices which differ from the current ones. A second objection is that the price or allocative efficiency measure is very sensitive to errors of measurement in estimating factor prices. However these objections are much less valid when we can incorporate the risk averse attitude of firms or DMUs in the objective functions of the DEA efficiency models (4) and (8) themselves. Reasons are two-fold. One is that the risk averse agent would allow an extra cost due to price fluctuations, thus preferring stable costs and profits over fluctuating ones. For instance in the field of agricultural production, where Farrell applied his nonparametric efficiency measurement the farmers have been observed to be highly risk averse to the uncertainty of price fluctuations and their crop substitution policies have reflected this risk aversion to a significant degree. Second, in case of long term investments in farm the farmers are known to prefer flexibility in choice between multiple crops and multiple inputs such as different varieties of fertilizers. Stochastic variations in prices can be modeled in the DEA framework in three different ways. One is through the chance-constrained interpretation of the inequality constraints, which assumes that the constraints are random and have a finite probability of being violated. Secondly, one could apply the two-stage programming approach called programming under recourse which decomposes the decision rule in two stages. In the first stage one uses an initial estimate of the random elements and computes on a conditional basis the first stage optimal
3 decision. Then at the second stage, the random variables are directly observed and based on the discrepancy of the prior estimate and the observed value a new decision vector is selected so as to minimize the expected penalty cost due to the discrepancy. Kall and Wallace [] have recently applied this approach to dynamic intertemporal models. Thirdly, one can incorporate price fluctuations in the risk averse behavior of DMUs and use a transformed objective function which minimizes an expected loss function. This approach is due to von eumann s utility theory under risk and this is widely applied in portfolio models in finance, e.g., the mean variance approach. We have decided to follow the third approach because of two reasons. One is that the first two approaches generally involve nonlinear programming problems in larger dimensions; these are usually more nonlinear than the quadratic, even when the random variables are normally distributed. Secondly, the two-stage programming under recourse which involves expected costs of recourse is more suitable for a discrete time framework. Here we are considering a more continuous time formulation for convenience, since we are exploiting Eulertype variational calculus to characterize the optimal path. Furthermore, the quadratic adjustment cots assumed in our formulation may be considered a close substitute of the expected recourse costs. Consider first the loss function model (4). As in the mean variance theory of investment portfolios, the objective function of the DEA model (4) may be replaced by a linear combination of mean and variance of input costs as follows: [ ] t Min L = ρ q ( t) x( t) + ( / 2) θ{ x ( t) V( t) x( t) + d ( t) Wd( t) + z ( t) Hz( t) } t=
4 where q( t), V(t) are the mean vectors and variance-covariance matrices of the input prices q i (t). Assuming the input prices to be statistically independent the optimal linear decision rules (5) would now be altered as * i i i * (α i + θv ii ) x i (t) = * wi x i (t-) + * ρwi x i (t+) + hi x$ ( t) q ( t) + µ ( t) (6) This shows that if the variance component v ii, or its weight θ or the mean input price q t i ( ) rises, then the optimal input usage declines. The opposite effect occurs when the desired level x$ i ( t) or * the marginal productivity µ i ( t ) of input rises. Secondly, if wi is zero but the input prices are not statistically independent, then the optimal decision rule can be specified as: x*(t) = (H + θv) - [µ*(t) + H x$( t )] (7) This implies increased (decreased) substitution in favor of (against) inputs which are relatively cheaper (costlier) in mean prices (price variances). ext consider the dynamic model (8) and assume the input price vector q(t) to be random with mean q and variance-covariance matrix V. The objective function now becomes c = T rt e ( q x( t) + ( / 2) θ x t Vx t ) dt + rqmxm t + xm t Vmx m t 0 ( ) ( ) ( ) ( / 2) θ 2 ( ) ( ) where θ, θ 2 are the nonnegative weights on the variances of input prices. Assuming an interior solution the optimal input demand functions x*(t), x * m ( t ) may now be written as x*(t) = (θ V) - (µ*(t) - $q ) (8) * * m 2 m m m x ( t) = ( θ V ) ( µ ( t) rq ) Clearly an increased risk aversion measured by the weights θ, θ 2 leads to a decline in the optimal input usage. Increased price variance has a similar effect. ormally the capital goods
5 prices are more fluctuating than the current input prices and hence its impact may be more severe on the use of capital inputs. ote that in the dynamic efficiency case we need the estimates on the mean input prices q and their variances. When panel data are available, these estimates may be made in the first stage from the cross-section samples and then in the second stage the optimal input demand functions and hence the optimal cost functions may be obtained from the DEA models. 4. An empirical application To illustrate the application of some of our efficiency models we have utilized the time series data set for international airlines previously studied by Schefczyk [3]. We have used the period 988-94 for 4 airlines, each having three inputs (x,x 2,x 3 ) and two outputs (y,y 2 ) all measured i logarithmic units. The input output data set exhibits widespread fluctuations for the airlines industry due to various regulatory controls and cost uncertainties. One main reason for cost uncertainties is the relative fixity of the capacity-related cost elements, e.g., acquisition of aircraft, development of route systems etc. have a multiperiod character. This is why capital cost is considered an important long run factor in airlines operations. Three inputs and two outputs, each in logarithmic units are used as follows: x = available ton kilometer which reflects aircraft capacity, x 2 = total operating cost net of depreciation and amortization costs and x 3 = total nonflight assets defined as total assets minus flight equipment at cost net of depreciation and amortization. The two outputs are: y = revenue passenger kilometer and y 2 = nonpassenger revenue at current prices. Three sets of computations are reported here, all based on 7 selected airlines which are representative of the whole group. The first set (Tables and 2) shows the changes in three types
6 of efficiency over time, e.g., technical efficiency, allocative efficiency and the scale efficiency. Here scale efficiency (SE) is measured by the sum (SE=β +β 2 +β 3 ) of the coefficients of the log linear production function y = β 0 + β x + β 2 x 2 + β 3 x 3 This is different from the DEA measure of scale efficiency in terms of the sign of the intercept term β 0. Tables and 2 show three interesting results. First of all, only one out of 7 airlines exhibits 00% technical and allocative efficiency for each year over the period 988-94, i.e., Singapore airlines. For the overall sample of 4 airlines this pattern persists. Second, the scale efficiency changes over time also varies significantly over time. This is mainly due to the changes in the capacity input, thus implying that the dynamics of efficiency variations is very important i the airlines industry. The second set of computation in Table 3 reports the moment estimates of the distribution of optimal and actual outputs. One interesting feature of this calculation is that the optimal output distribution is more stable than the actual output distribution, e.g., the mean optimal output is higher and the variance lower. This suggests that the filtering of observed input output data in terms of the optimal values tends to impart a degree of robustness to the DEA estimates. The third set of computation reported in Table 4 uses the two subsets: the efficient and the inefficient units of the DEA model in each year to estimate a log linear production function over the whole period (988-94) by the method of ordinary least squares. Two striking features come out. One is that the capacity input (x ) is the most dominant and significant input affecting the scale economies in the airlines industry. The other two inputs are either insignificant or with wrong signs. Secondly the inefficient units are significantly different from the efficiency units
7 when tested by the standard Chow test. ote however that the least squares estimates have an upward bias when compared with the DEA estimates. It would be interesting to estimate the impact of price variations of capital inputs (x ) on the DEA efficiency. However the requisite data are not available for the whole period. Hence we could not estimate the optimal decision rules (7) which incorporate the impact of price variance. 5. Conclusions The static versions of DEA models usually compute the relative efficiency of a DMU in a cluster, when there are no price data for inputs and outputs. In many situations however the unit price data are available through cost data, but such data usually involve variations over time due to rigidity of some capacity-based inputs. Given these input prices or their expected changes over time, one could compute an efficient level of various inputs and hence the optimal costs for any DMU. Thus the overall efficiency with its two components: the technical and allocative efficiency may be directly computed by a DEA model. Thus the DEA model yields in this case an optimal value of the control variable, e.g., optimal level of input demand. When the input is capital or capacity-based resource, this would yield an optimal time profile of input demands. This would characterize a dynamic cost and production frontier. When the input prices vary due to stochastic factors, the input price fluctuations may be incorporated into the risk averse behavior of DMUs and thus a risk averse efficiency frontier may be specified. Here these various extensions of the DEA model are developed both analytically and empirically. The empirical applications to international airlines data show that changes in efficiency over time 988-94 have been very significant in the air transport market.
8
9 Table. Efficiency changes over time (988-94) Airlines Technical Efficiency TE k = θ* Allocative Efficiency AE k Cathay Pacific 988 990 992 994 988 990 992 994.0 0.920 0.89 0.90 0.94 0.95 0.89 0.94 Singapore.0.0.0.0.0.0.0.0 Quantas 0.97 0.920 0.905 0.899 0.85 0.87 0.9 0.92 British Airways American Airlines 0.907 0.92 0.902 0.894 0.8 0.85 0.87 0.89 0.896 0.92 0.989 0.920 0.9 0.93 0.95 0.98 KLM 0.734 0.702 0.70 0.705 0.65 0.66 0.69 0.7 Japan Airlines 0.724 0.702 0.698 0.724 0.7 0.75 0.8 0.70
20 Table 2. Estimated trends in scale efficiency SE = β + β 2 + β 3 over time (988-94) SE i (t) = a i + b i SE i (t-) Airlines $a i $ bi R 2 Cathay Pacific 0.02 0.92** 0.90 Singapore 0.3 0.96** 0.97 Quantas 0.093 0.96* 0.8 British Airways 0.0 0.89* 0.82 American Airlines 0.932 0.94* 0.9 KLM 0.805 0.84* 0.74 Japan Airlines 0.80 0.82* 0.76 ote. One and two asterisks denote significant t-statistics at 5 and % respectively.
2 Table 3. Moment estimates of the distribution of optimal and actual outputs (988-94) Optimal DEA output Actual output 988-90 99-94 988-90 99-94 Mean ' ( µ ) 9.93 9.942 9.32 9.02 Variance (µ 2 ) 0.248 0.22 0.46 0.532 Coefficient of variation (CV) 0.050 0.046 0.073 0.080 Skewness coefficient (γ ).350 0.28 0.365.423 Kurtosis coefficient (γ 2 ) 0.928 0.94 0.954 0.989 otes:. γ = µ 3 / µ 2 3 2 /, γ 2 = (µ 4 / µ 2 2 ) - 3 µ ' = mean, µ 2 = variance, µ 3, µ 4 are third and fourth order moments around the mean. 2. The second and fourth order moments have been corrected for grouping by applying Sheppard s corrections. 3. Here the estimates of composite output are based on a weighted average of the passenger and nonpassenger revenue with weights 0.8 and 0.2 respectively.
22 Table 4. Least squares estimates of the log linear production function (988-94) Efficient units Inefficient units Intercept x x 2 x 3 R 2 45. 72.. 0. 79 ( t = 054. ) ( 354. ) ( 6. ) ( 59. ) 0.543 0. 92 8. 0. 8 ( 0. 3) ( 2. 45) ( 72. ) ( 62. ) 0.490 otes:. Here the DEA models are used for ranking the efficient and inefficient units. 2. Chow test for testing the difference between the two sets of coefficients yields a significant statistic at 5% level.
23 References [] Charnes, A., Cooper, W.W., Lewin, A.Y. and Seiford, L.M., (Eds.) 994. Data Envelopment Analysis: Theory, Methodology and Applications. Kluwer Academic Publishers, Boston. [2] Sengupta, J.K., 995. Dynamics of Data Envelopment Analysis: Theory of Systems Efficiency. Kluwer Academic Publishers, Boston. [3] Fare, R., and Grosskopf, S., 996. Intertemporal Production Frontiers. Kluwer Academic Publishers, Boston. [4] Sueyoshi, T., 992. Measuring technical, allocative and overall efficiencies using DEA. J. Operational Research Society, 43:4-55. [5] Holt, C.C., Modigliani, F., Muth, J., and Simon, H.A., 960. Planning Production Inventory and Workforce. Prentice Hall, Englewood Cliffs,.J. [6] Sengupta, J.K., 996. Dynamic aspects of data envelopment analysis, Economic otes, 25:43-64. [7] Treadway, A.B., 970. Adjustment costs and variable inputs in the theory of the firm. J. Economic Theory, 2:329-347. [8] Kennan, J., 979. The estimation of partial adjustment models with rational expectations. Econometrica, 47:44-456. [9] Sengupta, J.K., 994. Measuring dynamic efficiency under risk aversion. European J. Operational Research, 74:6-69. [0] Farrell, M.J., 957. The measurement of productive efficiency. Journal of Royal Statistical Society, Series A., 20:253-290. [] Kall, P., and Wallace, S.W., 994. Stochastic Programming. John Wiley, ew York.
24 [2] Sengupta, J.K., 992. On the price and structural efficiency in Farrell s model. Bulletin of Economic Research, 44:28-300. [3] Schefczyk, M., 993. Operational performance of airlines: an extension of traditional measurement paradigms. Strategic Management Journal, 4:30-37.