Comparisons of Economic Inefficiency Between Output and Input Measures of Technical Inefficiency Using the Fourier Flexible Cost Function

Journal of Productivity Analysis, 22, 123–142, 2004

# 2004 Kluwer Academic Publishers. Manufactured in The Netherlands.

Comparisons of Economic Inefficiency Between

Output and Input Measures of Technical Inefficiency

Using the Fourier Flexible Cost Function

TAI-HSIN HUANG

Department of Economics, Tamkang University, Taipei Hsien, Taiwan, R.O.C.

MEI-HUI WANG

Department of Banking and Finance, Tamkang University, Taipei Hsien, Taiwan, R.O.C.

Abstract

The current paper constructs a Fourier flexible cost function, which is commonlyknown to be a more general function form than the typical translog form, and canglobally approximate a true (but unknown) cost function. Both allocative andtechnical inefficiencies are considered using the Fourier function in the context of theparametric approach. The former is modeled using shadow input prices and thelatter is formulated either by adding an extra term of scale parameter (when theFarrell’s (1957) input technical inefficiency is assumed), or by correcting all the termsinvolving output quantities by a scale parameter (when the Farrell’s output technicalinefficiency is assumed). It is found that sample banks could save up to 23% of totalcosts, within the range of 3 and 69% uncovered by the previous works, in whichallocative inefficiency plays a more important role than technical inefficiency.Furthermore, the cost of misallocated labor input alone constitutes more than 80%of total allocative inefficiency. Financial deregulation starting from 1991 in Taiwanappears to have improved economic efficiency of the banking industry.

JEL Classification: C14, D21, G21

Keywords: technical inefficiency, allocative inefficiency, Fourier flexible function

1. Introduction

Economic inefficiency of a firm, also referred to as X-inefficiency, is composed oftechnical inefficiency ðTIÞ and allocative inefficiency ðAIÞ. Farrell’s (1957) radialmeasure of TI implies two distinct definitions as long as the production function failsto exhibit constant returns to scale. A firm is said to be technically efficient in outputif it is able to produce maximal output from a given set of inputs, and an inputtechnically efficient firm is capable of using minimal inputs to obtain a given set ofoutputs. Naturally, the two definitions lead to different specifications and functional

forms of the cost function and the corresponding share equations. An allocativelyefficient firm is one which is able to use inputs in optimal proportions, i.e., it employsfactors up to the point where the marginal rate of technical substitution between anytwo inputs equals the ratio of corresponding input prices.Kumbhakar (1997) claimed that the appropriate measure of TI is one of input-

saving, since output is treated as exogenous under a cost-minimizing framework.However, in lieu of prior knowledge of the firm’s production function and whatquantities (inputs or outputs) over which the managers have most control,1 it isnecessary to estimate both output and input measures of TI. This is because thesemeasures provide quite different information unrelated to the exogeneity of outputs.Moreover, as will be shown shortly, both measures are introduced into the costsystem through the same dual cost function, but are nevertheless different withrespect to inefficiency, which is assumed to be either output- or input-oriented.Over the past two decades, the translog cost function has been extensively applied

to studies of economies of scale and scope, mergers, as well as X-efficiency. However,this widely used functional form is frequently criticized as merely capable of locallyapproximating a true but unknown cost function, being derived from a second-orderTaylor expansion of an arbitrary function about a point. McAllister and McManus(1993) specifically avoided an application of this specification to banks of all sizes,because it would force large and small banks to lie on a symmetric U-shaped rayaverage cost curve. These authors, as well as Wheelock and Wilson (2001), notedthat the use of the translog function tends to produce biased measures of scaleeconomies for banks of various sizes.An important new development occurred when Gallant (1981, 1982) introduced

the Fourier flexible function form (hereafter the FF function), which consists of twomain components. The first component of the FF is a translog function with somemodifications, while the second component is a trigonometric Fourier series.2 Hence,the FF function is more general than the ordinary translog form alone and nests it aspecial case. It is important to note that these two components are not independentof each other. In addition to trigonometric functions, some well-known orthogonalpolynomials, representing the basis functions of a Fourier series, have indeed beenproposed in the literature. Subsection 2.1 will mention this.Gallant (1982) showed that the FF function is able to approximate the true

function as closely as desired in Sobolev norm. In addition, tests on parameterrestrictions, such as for constant returns, homothetic technology, or separability,have adequate nominal size, when the number of parameters of the FF function aremade dependent on the sample size. Specifically, in a large sample, the estimates ofelasticity of substitution from the FF function form have only a trivial bias, as hasbeen found by Elbadawi et al. (1983) and Chalfant and Gallant (1985). Empiricalstudies, such as McAllister and McManus (1993), Berger et al. (1997), Berger andMester (1997), and Mitchell and Onvural (1996), have all found that for bankingdata the FF function produces better fits than the translog form. Berger and Mester(1997) argued that a close fit of the data for the estimated efficient frontier is crucialin evaluating efficiency, because inefficiencies are assessed as deviations from thisfrontier.

124 HUANG AND WANG

Despite its superior properties, the FF function has not attracted much of theattention of empirical researchers. Notable exceptions to this have been McAllisterand McManus (1993), Mitchell and Onvural (1996), Ivaldi et al. (1996), Berger andDeYoung (1997), Berger et al. (1997), Berger and Mester (1997), DeYoung et al.(1998), Altunbas et al. (2000), and Huang and Wang (2001). This is possibly due tothe technical difficulties in constructing and estimating it. These articles abovenevertheless fail to incorporate AI into their statistical models, leading to potentialspecification error.3 Furthermore, most of these articles ignore the relation betweenthe two components of the FF function; excepting Mitchell and Onvural (1996),Ivaldi et al. (1996), and Huang and Wang (2001). In particular, the scaled variablesof outputs and input prices are typically used in the second component (the Fourierseries) only, whereas these variables should be used in both components (translogand Fourier series). For purposes of comparison, we tentatively refer to this modellacking inter-relatedness as the ‘‘independent model.’’ Being incomplete with respectto Gallant’s original formulation, the independent model may be unable to reachclose approximation in the Sobolev norm and may result in inconsistent parameterestimates.The current paper presents a complete econometric model utilizing a parametric

approach that avoids all of the aforementioned weaknesses. Specifically, the modeluses an artificial FF cost function, simultaneously involving both TI and AI basedon Gallant (1981, 1982), and uncovers new evidence on the X-efficiency of Taiwan’scommercial banks. In contrast to many recent studies, the association between thetranslog part and the Fourier series of an FF function is explicitly incorporated.Both output and input technical inefficiency measures are estimated in separate FFcost functions. In addition, for the purpose of comparison, translog cost frontiers arealso estimated.The remainder of the paper is as follows. Section 2 formulates a complete FF cost

frontier, which accommodates either the output or the input measure of TI and AI,while Section 3 briefly describes the data set and performs various estimations andcalculations. Section 4 concludes the paper.

2. The Econometric Models

This section will first construct an FF cost function and then incorporate TI and AIinto the FF function. Estimation techniques are outlined.

2.1. The FF Cost Function

The procedure for constructing an FF cost function is briefly discussed here andreaders are asked to refer to Gallant (1981, 1982) for details. A cost-minimizing firmis assumed to have an optimal logarithmic cost function ln C�ðP0;Y 0Þ, where P is anN61 vector of input prices and Y is an M61 vector of output quantities. Each

COMPARISONS OF ECONOMIC INEFFICIENCY 125

element of these two exogenous vectors has to be rescaled by

li ¼ ln pi þ ln ai > 0; i ¼ 1; : : : ;N;

qj ¼ mjðln yj þ ln ajÞ > 0; j ¼ 1; : : : ;M;

where the location parameters ln ai and ln aj are chosen as

ln ai ¼ �minfln pig þ 10�5; i ¼ 1; : : : ;N;

and

ln aj ¼ �minfln yjg þ 10�5; j ¼ 1; : : : ;M;

as suggested by Gallant (1982), Chalfant and Gallant (1985), and Mitchell andOnvural (1996).By virtue of the above procedure, the minimum values of these scaled elements are

guaranteed to be slightly greater than zero. Parameter mj ð j ¼ 1; . . . ;MÞ is thescaling factor of output j defined similar to that in Gallant (1982). Let x ¼ ðl0; q0Þ0 bea ðN þMÞ61 vector, where vectors l and q have the same dimensions as vectors Pand Y, respectively. Gallant (1981, 1982) found that a logarithmic version of FFform, fKðx0 j yÞ, can approximate the unknown true cost function f ðx0Þ ¼ lnC� asclosely as desired in Sobolev norm, where y is a row of unknown parameterscharacterizing the function.Function fKðx0 j yÞ is specifically defined as:

fKðx0 j yÞ ¼ u0 þ b0xþ 1

2x0cx

þXEa¼1

ba þ 2XJj¼1

uja cosð jlk0axÞ � vja sinð jlk0axÞ� �( )

; ð1Þ

with c ¼ �PE

a¼1 bal2kak

0a, and a ðN þMÞ-vector of elementary multi-index ka,

whose properties are discussed in detail by Gallant (1982), and hence are overlookedhere to save space. The chosen vectors ka are also not shown, but available uponrequest. The common scaling factor, l, for input prices is defined analogously as inGallant (1982). The expression in the braces corresponds to a truncated Fourierseries having a total of E terms. The determination of the truncation parameter Ewill be discussed shortly.Let k0a ¼ ðr0a; r0bÞ and b0 ¼ ðb01; b02Þ, where ra and b01 are both N61 vectors, while rb

and b02 are both M61 vectors. The corresponding share equations can then beexpressed as

S ¼ qqlfKðx0 j yÞ

¼ b1 � lXEa¼1

balk0axþ 2

XJj¼1

j uja sinð jlk0axÞ þ vja cosð jlk0axÞ� �( )

ra: ð2Þ

126 HUANG AND WANG

Microeconomic theory requires that a firm’s cost function be homogeneous of firstdegree in input prices and that the cost function be symmetric. These two sets ofrestrictions will be imposed on the FF and translog cost functions during estimation.Equations (1) and (2) constitute a simultaneous equations system with 1þN þM þ Eð1þ 2JÞ unknown parameters.Since only a finite number of observations are available, the original infinite

Fourier series must be cut to, say, E terms, where E is forced to be less than half ofthe sample size. Chalfant and Gallant (1985), Eastwood and Gallant (1991), andMitchell and Onvural (1996) have suggested the number of parameters to beestimated, in the context of an FF cost function system, be equal to the number ofeffective sample observations raised to the two-thirds power. This can produce bias-minimizing and asymptotically normal estimates.4 Such a procedure can help tojustify the particular value of E chosen.5 By contrast, Eastwood (1991) has developeda novel rule—an upwards F test truncation rule—to select the truncation parameterE so that linear semi-nonparametric estimators are consistent and asymptoticallynormal. Both procedures will be applied later.It is important to note that there are several alternative choices of basis functions

in the Fourier expansion other than the sine and cosine terms. Well-known examplesinclude orthogonal Laguerre, Legendre, Hermite, and Jacobi polynomials, as havebeen pointed out by Hardle (1993), Efromovich (1999), and Wheelock and Wilson(2001). Under certain conditions (see, for example, Hardle, 1993), the consistency ofthe estimated regression function follows. Wheelock and Wilson (2001) specificallyclaimed that there is a lack of studies on the impact of selecting distinct polynomialson regression results. Barnett et al. (1991), instead, suggested Bayesian estimation ofthe asymptotically ideal model (AIM) specification generated from the Muntz–Szatzseries expansion. The AIM model has to be estimated semi-nonparametrically andpartial sums of the series expansion must be specified due to finite number ofobservations in empirical studies.

2.2. Economic Inefficiency

This article adopts a parametric approach to measure TI and AI, which is also usedby Atkinson and Cornwell (1993, 1994a, b), Kumbhakar (1996a, b, 1997), andHuang (1999, 2000). To the best of our knowledge, this is the first time this approachhas been generalized to an FF function form. The strength of the approach lies in itbeing independent of an arbitrary and restrictive functional form and distributionalassumptions on disturbances. In addition, this approach can model technicalefficiency using both output- and input-orientated measures, whereas the standarderror-components (stochastic frontier) method merely models non-cost minimizingbehavior with a function of input-specific disturbances that come from the inputdemand or share equations.The parametric approach assumes that firms’ decisions are based on shadow input

prices, rather than actual input prices, because of regulation or slow adjustment topast changes in input prices. The shadow-scaled input prices l�i ’s ði ¼ 1; : : : ;NÞ are


related to the actual counterparts by

l�i ¼ li þ ln hi; i ¼ 1; : : : ;N;

where hi’s are additional unknown non-negative parameters, representing the extentto which shadow and actual input prices differ. It can be seen that hi ¼ 1 indicatesfull allocative efficiency for the ith input.The (log) actual cost function and the corresponding share equations, when input

TI is assumed, are expressed as (see Atkinson and Cornwell, 1994a, for a detailedderivation):

ln C ¼ � ln Bþ fKðx�0 j yÞ þ lnXi

S�i h

�1i ; ð3Þ

and

Si ¼S�i h

�1iP

i S�i h

�1i

; i ¼ 1;L;N; ð4Þ

where lnC represents the (log) actual cost for a representative firm, x�0 ¼ ðl�0; q0Þ; fK 6ðx�0 j yÞ is the constructed FF shadow cost function, S�

i ¼ qfkðx�0 j yÞ=ql�i ; i ¼ 1; : : : ;N,is the shadow cost share of the ith input, and Si is its observed counterpart. The TImeasure B (0 < B � 1 by definition) reflects the degree to which the actual input costdeviates from its minimal counterpart given the outputs; hence, the term � lnBcaptures a possible cost reduction from improving technical efficiency.A higher value for the input-saving technical inefficiency coefficient B, implies a

more technically efficient firm. When B ¼ 1, the firm is said to be fully technicallyefficient. The fact that technical inefficiency B only appears in the cost equationimplies that B increases all of a firm’s input demands proportionately, leaving inputshares intact. As will be seen shortly, � lnB can be viewed as the firm-specificintercept term.When output TI is specified, the shadow cost system can be written as

lnC ¼ fKðx��0 j yÞ þ ln

Xi

S��i h�1

i ; ð5Þ

and

Si ¼S��i h�1

iPi S

��i h�1

i

; i ¼ 1; : : : ;N; ð6Þ

where all notations are the same as in equations (3) and (4), except thatx��0 ¼ ðl�0; q�0Þ; q�j ¼ qj � lnA, with 0 < A � 1 by definition, j ¼ 1; : : : ;M, andS��i ¼ qfkðx��0 j yÞ=ql�i ; i ¼ 1; : : : ;N. In contrast to the model of input TI, technical

inefficiency A appears not only in the cost equation, but also in the cost shares. Thisindicates that output TI exerts distinct influences on different input demands. Theterm � lnA will later be identified as a firm-specific parameter, representing the

128 HUANG AND WANG

extent to which a firm’s production deviates from its potential outputs given an inputmix.After adding classical random disturbances to these equations to capture random

shocks orthogonal to the right-hand-side elements (including the AI and TI terms),the parameters under consideration can be estimated through a non-linear iterativeseemingly unrelated regression technique. This technique can be proved to beequivalent to maximum likelihood when convergence is achieved. Clearly only N � 1of the N cost shares can be included during estimation, since singularity of thecovariance matrix of the stochastic disturbances would otherwise result.6

Technical inefficiencies A and B are assumed to vary across firms, but not overtime. The assumption of time-invariant technical inefficiency simplifies ourestimation procedure, while illustrating the superior fit of the FF model. Specifically,A and B are normalized to unity for the most efficient firm (say firm e) in the sample,and the relative technical efficiency of firm kð6¼ eÞ to firm e is defined asAk=Ae ¼ Ak ðBk=Be ¼ BkÞ. For a sample of F firms, only F � 1 firm-specificAk ðBkÞ can be estimated.Allocative inefficiency parameters hi’s are similarly assumed to be time-invariant,

but free to vary across firms. One of the N inputs (say the Jth) must be selected as thenumeraire, with hJ then normalized to unity for each firm in the sample. It followsthat the remaining N � 1 hi’s can then be estimated for all firms. For any firm, avalue of hi < 1 ðhi > 1Þ implies that the firm inappropriately uses more (less) of inputi relative to the chosen numeraire than that required by cost-minimization.

3. Data and Results

Panel data from 1981 to 2001 on 22 of Taiwan’s domestic banks are employed.Among them, 11 are large public banks—large relative to the remaining 11 privatebanks—in terms of total assets. Three output categories can be identified, i.e.,investment ðY1Þ, which is composed of government and corporate securities, short-term loans ðY2Þ, and long-term loans ðY3Þ. Moreover, deposits and borrowed moneyðX1Þ, labor ðX2Þ, and physical capital ðX3Þ, including dollar values of net premisesand fixed assets, are regarded as inputs, on the basis of the financial intermediationapproach.The input price of X1; p1, is defined as all interest payments divided by the dollar

value of X1. The price of X2; p2, is calculated as the employees’ compensation dividedby the number of full-time-equivalent employees, while the last input price, p3, isgenerated by dividing occupancy and fixed asset expenditures through X3. Thedefinitions used here for p1; p2, and p3 are exactly the same as those employed byHuang (1999, 2000), and are nearly identical to the ones exploited by Mitchell andOnvural (1996) and Hunter et al. (1990). Sample statistics of the relevant variablesare listed in Table 1.Since one bank in the sample started operations in 1982, there are a total of 461

observations in the sample. The two-thirds power of the effective sample points(1,383) is roughly 124. Due to the importance of the size effect on TI and AI and in


order to make convergence easier, the entire sample is classified into five groupsaccording to banks’ total assets. Each group is viewed as if it were an individual firm.In fact, only four firm-specific intercepts/parameters (one of them is normalized tounity) plus 10 shadow parameters need to be estimated, in addition to the originalparameters in cost function.7 Thus, optimum ka as defined above by the two-thirdsrule, is E ¼ 48.Eastwood’s (1991) upwards F test truncation rule is also utilized to help determine

the truncation parameter, k, on the Fourier series for our sample of n ¼ 461observations. Following his notation, we choose d as either 0.6 or 0.7, thus satisfyingthe requirement of d >

ffiffiffiffiffiffiffi0:2

p. Next, the rule r ¼ 0:2k, which defines how many

coefficients will be tested in each F-test, is selected so as to satisfy the necessaryasymptotic property.8 Finally, an increasing sequence fkig is defined, with initialvalue k0 ¼ k�n, upper bound k0n, and 0 < k�n < k0n � n=2 for all n � 1, aski ¼ minfki�1 þ rðki�1Þ; k0ng. When setting k0 ¼ 9, the recursive definition of ki hasvalues of 12, 14, 17, 20, 24, 29, 35, 42, etc.The critical values of the test, F�

i , are calculated according to the formulaminfd2ð2ki þ 1Þ=ð4riÞ; 2g. They are roughly 0.9 to 1.0 (1.20 to 1.26) given the chosend ¼ 0:6ð0:7Þ, the function r, and the sequence fkig. After performing a series of tests,we find that only the F-test statistic F29;n ¼ 0:97 is below unity and near the critical

Table 1. Sample statistics.

Variables Mean Std. Dev.

COSTþ 20,787.33 22,072.59

yþ1 48,438.11 65,694.47

yþ2 80,167.25 86,264.30

yþ3 153,007.42 204,472.66

p1 0.0597 0.0222

p2 0.8033 0.3226

p3 0.5333 0.5417

S1 0.7029 0.0857

S2 0.1382 0.0517

S3 0.1589 0.0693

ln COST� 9.2233 1.4372

ln y�1 9.7825 1.7707

ln y�2 10.4496 1.5923

ln y�3 10.9786 1.6534

ln p�1 � 2.8740 0.3214

ln p�2 � 0.3042 0.4251

ln p�3 � 0.8153 0.5510

l1 0.8661 0.3214

l2 1.2569 0.4251

l3 2.1378 0.5510

q1 3.4671 0.7616

q2 3.3805 1.0920

q3 3.2051 1.0194

Notes: þMeasured in millions of New Taiwan’s Dollar. *ln denotes the natural logarithm. l1; l2 and l3 are

scaled log-input prices, while q1; q2, and q3 are scaled log-output quantities. Number of observations: 461.

130 HUANG AND WANG

values, and the remaining statistics are all greater than 1.57 for ki � 24. It isconservatively suggested that the truncation point be 35, which is smaller than thechoice made by the simpler two-thirds rule and is far less than n=2.Parameters of both the FF and translog efficiency-adjusted shadow cost

functions for both output- and input-oriented models are estimated and reportedin Table 2. However, the coefficient estimates of the Fourier series are not shown,but available upon request, and the hypothesis that these coefficients are joint zero(corresponding to the translog form) is statistically rejected using the Wald test.One is led to conclude that the FF cost function is indeed more relevant than thetranslog form in approximating a firm’s cost function. Therefore, inferences of X-efficiency based upon the FF function may be more reliable and reasonable thanthe translog form.9

Table 3 shows the estimates of the firm-specific TI measures of A and B, as well asAI measures of h. All these measures tend to increase with bank size, up to group 3,and then, decline as bank size grows. Except for the best-practice group and groups 1and 4 of the FF functions, TI parameter estimates for the remaining groups are allsignificantly different from, and moderately less, than unity, implying that TIappears to be small, but pervasive in the banking sector. Banks in groups 1, 3, and 4reach their production frontiers, as their estimated TI parameters are eithernormalized to be unity or statistically indistinguishable from unity. All the TIparameter estimates of A’s and B’s for the five groups are quite close to one anotherfor both the FF and translog functions.10 Both estimates of A and B from the FFfunctions exceed their translog counterparts, implying that the translog form tendsto underestimate these parameters, and in turn, to overstate the increase in costs dueto TI (see below).AI parameter estimates of labor ðX2Þ are greater than unity; irrespective of which

functional form and which measure of TI are applied. This implies that the samplebanks incline towards under-utilization of labor, relative to the arbitrarily chosennumeraire (deposits and borrowed money, X1). AI parameter estimates of capitalðX3Þ are all less than unity for the FF functions and these estimates are found to beinsignificantly different from zero for output-oriented model, implying that capitalare apt to be over-utilized by the sample banks. Results from translog functions aremixed. AI estimates of capital for the first two groups fall short of unity, while theremaining groups exceed unity. Viewed from this angle, the FF form appears to bemore robust than the translog form.One can infer that if X2 were selected as the numeraire, then all AI parameters of

X1 and X3 would be estimated to be less than unity. Banks have to decrease the use ofX1 and X3 to eliminate their AI and lower total costs. By reducing deposits andborrowed money, the capital adequacy ratio (CAR) of banking firms will increase,thus maintaining their solvency. In fact, the Banking Law of Taiwan was modified in1989 so as to force banks to hold a CAR of at least 8%.If AI parameters are assumed to be the same among banks for the same input,

then the industry-level input-specific AI parameters can be estimated with andwithout imposing technical efficiency. Table 3 shows the degree to which theimposition of technical efficiency these input-specific AI parameter estimates for the


Table 2. Estimates of the Fourier cost function and the translog cost function—input measure.

Fourier Cost Function Translog Cost Function

Parameter Estimate Standard Errors Parameter Estimate Standard Errors

Intercept � 12.998*** 4.4471 Intercept 0.8709*** 0.2918

l1 0.3745*** 0.0275 ln p1 0.4326*** 0.0213

l2 0.5990*** 0.0283 ln p2 0.5686*** 0.0257

q1 4.4637*** 1.4408 ln y1 � 0.0860 0.0878

q2 4.1936*** 1.3562 ln y2 0.0922 0.0806

q3 � 2.2732 1.6369 ln y3 0.8383*** 0.0809

l21 � 0.1044*** 0.0088 ln p16 ln p1 � 0.1601*** 0.0088

l22 � 0.0972*** 0.0093 ln p26 ln p2 � 0.1727*** 0.0077

l23 � 0.0058** 0.0026 ln p36 ln p3 � 0.0248*** 0.0054

q21 5.1367*** 1.7982 ln y16 ln y1 0.0229 0.0196

q22 4.8326*** 1.6206 ln y26 ln y2 0.2176*** 0.0193

q23 5.5643*** 1.5990 ln y36 ln y3 0.1985*** 0.0203

l16l2 � 0.0979*** 0.0092 ln p16 ln p2 � 0.1540*** 0.0088

l16l3 � 0.0065** 0.0030 ln p16 ln p3 � 0.0061** 0.0030

l16q1 0.0055 0.0080 ln p16 ln y1 0.0085** 0.0040

l16q2 � 0.0161*** 0.0042 ln p16 ln y2 0.0081** 0.0036

l16q3 � 0.0733*** 0.0050 ln p16 ln y3 0.0450*** 0.0043

l26l3 0.0007 0.0006 ln p26 ln p3 � 0.0187*** 0.0038

l26q1 � 0.0055 0.0080 ln p26 ln y1 0.0008 0.0048

l26q2 0.0161*** 0.0042 ln p26 ln y2 � 0.0129*** 0.0042

l26q3 0.0733*** 0.0050 ln p26 ln y3 � 0.0579*** 0.0042

q16q2 � 1.4591 1.0288 ln y16 ln y2 0.0276* 0.0166

q16q3 � 3.1274*** 1.1584 ln y16 ln y3 � 0.0307* 0.0176

q26q3 � 2.5410*** 0.7592 ln y26 ln y3 � 0.2024*** 0.0138

Output Measure

Log-likelihood 2,123.53 Log-likelihood 1,900.18

Intercept � 14.956*** 4.3164 Intercept 0.8625*** 0.3154

l1 0.3793*** 0.0264 ln p1 0.3882*** 0.0215

l2 0.6078*** 0.0259 ln p2 0.6190*** 0.0257

q1 4.8041*** 1.3408 ln y1 � 0.1351 0.0911

q2 4.8860*** 1.3714 ln y2 0.1252 0.0844

q3 � 2.5544* 1.5489 ln y3 0.8371*** 0.0842

l21 � 0.1090*** 0.0076 ln p16 ln p1 � 0.1633*** 0.0069

l22 � 0.1057*** 0.0082 ln p26 ln p2 � 0.1748*** 0.0060

l23 � 0.0024 0.0025 ln p36 ln p3 � 0.0249*** 0.0054

q21 5.3280*** 1.7151 ln y16 ln y1 0.0199 0.0197

q22 5.0681*** 1.5435 ln y26 ln y2 0.2149*** 0.0197

q23 5.8740*** 1.5539 ln y36 ln y3 0.2014*** 0.0203

l16l2 � 0.1062*** 0.0080 ln p16 ln p2 � 0.1566*** 0.0071

l16l3 � 0.0028 0.0030 ln p16 ln p3 � 0.0067** 0.0031

l16q1 0.0029 0.0080 ln p16 ln y1 0.0083** 0.0040

l16q2 � 0.0147*** 0.0042 ln p16 ln y2 0.0102*** 0.0037

l16q3 � 0.0741*** 0.0049 ln p16 ln y3 0.0478*** 0.0041

l26l3 0.0005 0.0005 ln p26 ln p3 � 0.0182*** 0.0037

l26q1 � 0.0029 0.0080 ln p26 ln y1 � 0.0009 0.0048

l26q2 0.0147*** 0.0042 ln p26 ln y2 � 0.0151*** 0.0042

l26q3 0.0741*** 0.0049 ln p26 ln y3 � 0.0605*** 0.0041

q16q2 � 1.4543 1.0015 ln y16 ln y2 0.0316* 0.0168

q16q3 � 3.3300*** 1.1246 ln y16 ln y3 � 0.0273 0.0176

q26q3 � 2.6421*** 0.7269 ln y26 ln y3 � 0.2067*** 0.0141

Log-likelihood 2,124.46 Log-likelihood 1,905.89

Notes: ***Significant at the 1% level. **Significant at the 5% level. *Significant at the 10% level.

132 HUANG AND WANG

Table 3. Estimates of efficiency parameters—input measure.

Fourier Cost Function

Allocative Efficiency

Number of Firms

Groups Total Assetsþ Technical Efficiency Labor X2 Capital X3 (Sample Size)

I 12,000–50,000 0.9528*** (0.0317) 4.0491*** (0.4735) 0.0610** (0.0287) 4 (84)

II 50,001–90,000 0.9617*** (0.0200) 5.8833*** (0.7627) 0.3518** (0.1736) 5 (104)

III 90,001–220,000 1 6.7482*** (0.9043) 0.4129** (0.2041) 4 (84)

IV 220,001–460,000 0.9932*** (0.0262) 2.2224*** (0.3390) 0.1483** (0.0730) 4 (84)

V 460,001–800,000 0.8836*** (0.0272) 2.2680*** (0.3246) 0.2008** (0.0997) 5 (105)

Input-specific (with TI) 4.3055*** (0.5419) 0.3015** (0.1389) 22 (461)

Input-specific (without TI) 4.3283*** (0.5041) 0.2682* (0.1406) 22 (461)

Estimates of efficiency parameters—output measure



Number of Firms


I 12,000–50,000 0.9586*** (0.0246) 4.7487*** (0.5614) 0.0250 (0.0264) 4 (84)

II 50,001–90,000 0.9717*** (0.0129) 6.7260*** (0.8035) 0.1467 (0.1592) 5 (104)

III 90,001–220,000 1 8.4165*** (1.0230) 0.1867 (0.2017) 4 (84)

IV 220,001–460,000 0.9918*** (0.0177) 2.9282*** (0.4225) 0.0634 (0.0693) 4 (84)

V 460,001–800,000 0.8851*** (0.0259) 1.9216*** (0.2541) 0.0482 (0.0521) 5 (105)

Input-specific (with TI) 4.5950*** (0.5591) 0.2918** (0.1412) 22 (461)

Input-specific (without TI) 4.3284*** (0.5041) 0.2682* (0.1406) 22 (461)

Estimates of efficiency parameters—input measure.

Translog Cost Function


Number of Firms


I 12,000 – 50,000 0.7607*** (0.0318) 8.7921*** (1.5959) 0.6381*** (0.1727) 4 (84)

II 50,001 – 90,000 0.8061*** (0.0252) 8.7391*** (1.4390) 0.9161*** (0.2402) 5 (104)

III 90,001 – 220,000 0.9086*** (0.0259) 19.325*** (2.0680) 1.6029*** (0.4269) 4 (84)

IV 220,001 – 460,000 1 14.208*** (2.8506) 1.3572*** (0.3673) 4 (84)

V 460,001 – 800,000 0.7428*** (0.0251) 6.7872*** (1.4935) 1.2412*** (0.3134) 5 (105)

Input-specific (with TI) 1.2309*** (0.2501) 0.0825 (0.0563) 22 (461)

Input-specific (without TI) 1.4563*** (0.2666) 0.0173 (0.0595) 22 (461)

Estimates of efficiency parameters—output measure



Number of Firms


I 12,000 – 50,000 0.7173*** (0.0327) 7.2677*** (1.3869) 0.5699*** (0.1487) 4 (84)

II 50,001 – 90,000 0.7716*** (0.0244) 7.5742*** (1.2060) 0.8266*** (0.2102) 5 (104)

III 90,001 – 220,000 0.8998*** (0.0238) 18.5213*** (1.7695) 1.5184*** (0.4005) 4 (84)

IV 220,001 – 460,000 1 14.5073*** (2.5278) 1.3395*** (0.3520) 4 (84)


two FF and the two translog functions. It is clear that such affection seems to beignorable, especially for the FF function.As mentioned in Section 1, the independent models for the FF cost frontiers are

estimated and their parameter estimates are presented in the Appendix. As can beseen, not only are the parameter estimates of the translog parts in the independentmodels substantially different from their counterparts in Table 2, but the estimates ofTI and AI parameters are also far apart from one another. In particular, all the AIestimates now fall short of unity in the input-oriented model, while the opposite istrue for the output-oriented model (except for the capital input in the first twogroups), whose TI measures are all much lower than their counterparts in Table 3.This appears to confirm the statement made in Section 1 that the employment ofunscaled variables to the translog part of an FF cost function may give rise to biasedparameter estimates.Firm managers are more likely to be concerned about the impact of TI and AI on

costs rather than simply knowing of the presence of TI and AI. Table 4 translates TIand AI into average potential cost reductions for both the output and input models.Figures in the table are computed as one minus the ratio of fitted costs of achievingeither technical efficiency ðA ¼ B ¼ 1Þ, allocative efficiency ðh ¼ 1Þ, or both, to thefitted costs without assuming the appropriate firm efficiency. As expected, translogevidence on economic efficiency suggests that the overall potential cost savings aremuch higher than those of the FF functions. This may arise from the fact that the FFfunction is capable of fitting the data more closely due to its ability to globallyapproximate the underlying function.The presence of economic inefficiency raises a bank’s cost roughly 26% for the

input and output models, based on the translog cost functions. The mean value of26% implies average efficiency of 0.74. However, the corresponding numbers areabout 18 and 23% when the FF cost function is assumed, implying averageefficiencies of 0.82 and 0.77. After reviewing the results of 130 financial institutionefficiency studies, Berger and Humphrey (1997) found that the annual averageefficiency ranges from 0.31 to 0.97 of actual costs, with a mean of 0.79. Our translogand the FF evidence fall into this range.It is evident that X-inefficiency comes largely from AI rather than TI, if the FF

function is assumed, while the opposite is true, if the translog function is applied.The finding that TI plays a small role contradicts previous studies, such as Berger

Table 3. Continued.


Number of Firms

Groups Total Assets{ Technical Efficiency Labor X2 Capital X3 (Sample Size)

V 460,001 – 800,000 0.7134*** (0.0296) 4.3359*** (0.9390) 1.0837*** (0.2609) 5 (105)

Input-specific (with TI) 1.1815*** (0.2478) 0.0631 (0.0540) 22 (461)

Input-specific (without TI) 1.4563*** (0.2666) 0.0173 (0.0595) 22 (461)

Notes: Numbers in parentheses are standard errors. {Measured in millions of New Taiwan’s Dollar.

**Significant at the 5% level. ***Significant at the 1% level.

134 HUANG AND WANG

Table 4. Cost of inefficiency—input measure (%).


Technical

Inefficiency Allocative Inefficiency (%)

Economic

Inefficiency Number of Firms

Groups Total Assets{ (%) X2 X3 X2 þ X3 (%) (Sample Size)

I 12,000 – 50,000 4.72 16.54 5.13 20.85 24.59 4 (84)

II 50,001 – 90,000 3.83 17.01 1.37 18.14 21.28 5 (104)

III 90,001 – 220,000 0 13.65 1.01 14.53 14.53 4 (84)

IV 220,001 – 460,000 0.68 5.41 4.39 9.54 10.16 4 (84)

V 460,001 – 800,000 11.64 3.56 2.58 6.05 16.99 5 (105)

Full sample 4.50 11.13 2.82 13.66 17.65 22 (461)

1981–1991 6.62 14.18 2.81 16.65 22.25 22 (241)

1992–2001 2.27 10.68 4.85 14.97 16.92 22 (220)

Private banks 3.11 15.92 2.56 18.08 20.59 11 (231)

Public banks 5.90 6.32 3.07 9.21 14.69 11 (230)

Cost of inefficiency—output measure (%).


Technical


Economic



I 12,000 – 50,000 4.90 20.45 5.10 24.55 28.21 4 (84)

II 50,001 – 90,000 3.85 20.75 2.75 22.92 25.91 5 (104)

III 90,001– 220,000 0 17.96 2.17 19.75 19.75 4 (84)

IV 220,001– 460,000 1.22 8.96 5.66 14.06 15.11 4 (84)

V 460,001– 800,000 15.78 3.79 5.52 9.11 23.45 5 (105)

Full sample 5.58 14.17 4.23 17.88 22.68 22 (461)

1981–1991 7.59 16.00 3.54 19.07 25.50 22 (241)

1992–2001 3.17 12.90 6.10 18.23 20.90 22 (220)

Private banks 3.25 19.82 3.29 22.47 24.95 11 (231)

Public banks 7.92 8.51 5.18 13.27 20.40 11 (230)

Cost of inefficiency—intput measure (%)


Technical


Economic



I 12,000 – 50,000 23.94 14.08 0.70 14.68 35.10 4 (84)

II 50,001 – 90,000 19.39 12.69 0.03 12.72 29.64 5 (104)

III 90,001 – 220,000 9.14 15.05 0.83 15.75 23.46 4 (84)

IV 220,001 – 460,000 0 12.35 0.40 12.70 12.70 4 (84)

V 460,001 – 800,000 25.72 5.04 0.19 5.22 29.60 5 (105)

Full sample 16.26 11.57 0.40 11.92 26.41 22 (461)


and Humphrey (1991), Kumbhakar (1991), Berger et al. (1993), and Huang (1999,2000), all of whom did not employ the FF function. Specifically, Huang (1999,2000), who applied multiproduct translog shadow profit functions to examine X-efficiency with respect to Taiwan’s banking sector, showed that the average profitlosses due to TI outweigh the average profit reduction due to AI. It seems thatdifferent choices of functional forms, behavioral assumptions (cost minimizationor profit maximization), industries (banking or non-banking sectors), andcountries may result in distinct conclusions on the relative importance betweenAI and TI.The cost of misused labor alone constitutes more than 80% of total AI and

decreases as bank size grows, implying that larger banks tend to be more allocativelyefficient. On the contrary, there is no apparent trend in the cost of technical

Table 4. Continued.

Cost of inefficiency—intput measure (%)


Technical


Economic



1981–1991 19.77 11.05 0.38 11.38 28.99 22 (241)

1992–1997 15.91 12.57 0.41 12.92 26.95 22 (220)

Private banks 18.25 13.78 0.48 14.18 29.88 11(231)

Public banks 14.26 9.35 0.33 9.64 22.93 11(230)

Cost of inefficiency—output measure (%)


Technical


Economic



I 12,000 – 50,000 25.61 11.74 1.08 12.69 35.06 4 (84)

II 50,001 – 90,000 21.46 11.06 0.14 11.19 30.40 5 (104)

III 90,001 – 220,000 9.56 14.58 0.63 15.12 23.24 4 (84)

IV 220,001 – 460,000 0 12.19 0.36 12.50 12.50 4 (84)

V 460,001 – 800,000 28.19 0.76 0.03 0.78 28.76 5 (105)

Full sample 17.71 9.68 0.42 10.05 26.31 22 (461)

1981–1991 18.82 9.69 0.38 10.03 27.21 22 (241)

1992–1997 15.14 11.04 0.45 11.43 25.17 22 (220)

Private banks 19.83 12.18 0.61 12.71 30.11 11(231)

Public banks 15.58 7.18 0.23 7.38 22.50 11(230)

Note: {Measured in millions of New Taiwan’s Dollar.

136 HUANG AND WANG

inefficiency as bank size grows. To effectively reduce total cost, sample banks shoulddevote greater attention to optimizing the input mix, especially the labor usage. Sucha substantial allocative distortion is possibly due to the government’s regulationimposed on the banking industry through the first 11 years of the sample period. Onemay ask if financial deregulation in Taiwan since 1991 has enhanced banks’X-efficiency.Table 4 shows that technical efficiency indeed improves substantially during the

second sub-sample period (1992–2001), based on the FF cost functions. Thepotential percent cost reduction due to technical efficiency varies from 6.62 to2.27% and from 7.59 to 3.17%, respectively. By contrast, the translog function formdetects merely a slight improvement. Allocative efficiency is also found to haveimproved moderately during the second half of the sample period. Our results arecongruent with Leightner and Lovell (1998) and Gilbert and Wilson (1998), whoconstructed Malmquist indices to investigate the effect of financial liberalization(privatization) on productivity growth of Thai and Korean banks. Table 4 alsosummarizes the potential cost-reduction estimates for private and public banks. It isfound that public banks uniformly outperform private ones in terms of thepotential percent of cost-savings. These cost savings are mainly due to theachievement of allocative efficiency for the FF function. As for translog function,these cost savings are due to the achievement of technical, allocative, and overallefficiencies.To further support the above assertions, we ran three least squares regressions,

using the (log) values of possible cost reduction resulting from TI, AI, and EI ð¼TIþ AIÞ as the dependent variables. The (log) total assets and its square(representing bank size), deregulation dummy (d81), and ownership dummy (public)were chosen to be the explanatory variables. Regression results from the FFfunctions are shown in Table 5. To a large extent, all parameter estimates listed inthe table are in compliance with the foregoing, apart from variable d81 in equationlnðAIÞ of both input and output measures. This inconsistency appears to be quitereasonable, since after controlling for bank size and ownership, the sample banksmay take a longer time to adjust their input mix to achieve allocative efficiency.Along this line, when fixing bank size and time dummy, public banks are moreallocatively and economically efficient than private banks, while private banks aremore effective in production technique.

4. Summary and Conclusions

To shed light on the nature and extent of X-inefficiency in Taiwan’s banking sector,the current paper has estimated the output as well as input models using the moreflexible cost function, the FF function. This function is known as being able toglobally approximate the unknown true cost function, while the translog cost


frontier is merely a second-order and local approximation. Relative to best-practicefirms, banks operate inefficiently as they use around 18–23% more resources toproduce an equal amount of outputs, implied by the two FF cost frontiers. However,total potential cost savings are about 26%, implied by the two translog costfunctions, which is higher than that obtained by the FF functions. This is due to thefact that the FF function form has the ability to fit the data closer than the translogform by construction. In addition, the introduction of flexibility in the FF functionform can substantially minimize the confounding of specification error withinefficiency. Although the FF form appears to be more flexible and informativethan the translog form, its construction is by no means simple; hence, its popularityis restricted.The conclusions that can be drawn from the estimated FF shadow cost frontiers

are: (1) production inefficiency plays a smaller role than a misallocation of resourcesand raises an average bank’s cost about 4.50 and 5.58% for the two models, whileallocative distortions generate 13.66 and 17.88% higher costs than an allocativelyefficient bank; (2) a greater increase in total costs results from a relative lack of laborutilization than from relatively excessive capital usage to the numeraire; (3) financialderegulation appears to have a greater affect on sample banks’ allocative efficiencythan on their technical efficiency; and (4) inference from the output model is insubstantial agreement with that from the input model.

Table 5. Least squares regression results.

Input Measure Output Measure

lnðTIÞ lnðAIÞ lnðEIÞ lnðTIÞ lnðAIÞ lnðEIÞ

Intercept 0.5651***

(0.0691)

0.2294**

(0.1021)

0.7945***

(0.1186)

0.6078***

(0.1066)

0.0945

(0.1236)

0.7023***

(0.1482)

Asset � 0.1047***

(0.0127)

0.0285

(0.0187)

� 0.0759***

(0.0217)

� 0.1187***

(0.0195)

0.0633***

(0.0226)

� 0.0554**

(0.0271)

Asset6Asset 0.0051***

(0.0006)

� 0.0030***

(0.0009)

0.0021**

(0.0010)

0.0061**

(0.0009)

� 0.0046***

(0.0010)

0.0015

(0.0012)

d81 � 0.0181***

(0.0048)

0.0369***

(0.0071)

0.0188**

(0.0083)

� 0.0408***

(0.0075)

0.0493***

(0.0086)

0.0085

(0.0104)

Public 0.0068

(0.0053)

� 0.0285***

(0.0078)

� 0.0217**

(0.0091)

0.0108

(0.0082)

� 0.0330***

(0.0094)

� 0.0222**

(0.0113)

Adjusted R2 0.245 0.556 0.353 0.485 0.248 0.175

Notes:

d81 ¼ 1 if year � 1992;

¼ 0 otherwise:

Public ¼ 1 if Public bank;¼ 0 otherwise:

138 HUANG AND WANG

Appendix

Parameter estimates of the independent model.

Input Model Output Model

Parameter Estimate Standard Errors Estimate Standard Errors

Intercept � 29.8240** 11.9904 12.9579** 5.3017

ln p1 0.6857*** 0.0640 0.0341 0.0222

ln p2 0.3088*** 0.0651 1.0192*** 0.0254

ln y1 � 4.6483* 2.7913 1.5409** 0.6373

ln y2 7.6776** 3.2380 4.6022*** 1.3005

ln y3 6.2909*** 2.2313 � 6.2854*** 1.2578

ln p16 ln p1 � 0.0391*** 0.0092 � 0.1275*** 0.0035

ln p26 ln p2 � 0.0271*** 0.0070 � 0.1350*** 0.0047

ln p36 ln p3 � 0.0023** 0.0010 � 0.0262*** 0.0040

ln y16 ln y1 � 1.2154** 0.4941 0.1411** 0.0572

ln y26 ln y2 � 2.0920* 1.1326 � 1.6388*** 0.3485

ln y36 ln y3 � 2.9985*** 0.9057 0.0409 0.3425

ln p16 ln p2 � 0.0320*** 0.0076 � 0.1181*** 0.0040

ln p16 ln p3 � 0.0071** 0.0030 � 0.0094*** 0.0034

ln p16 ln y1 � 0.0026 0.0021 0.0085** 0.0037

ln p16 ln y2 0.0062*** 0.0021 0.0189*** 0.0032

ln p16 ln y3 0.0201*** 0.0050 0.0492*** 0.0036

ln p26 ln p3 0.0049** 0.0020 � 0.0168*** 0.0033

ln p26 ln y1 0.0028 0.0021 0.0008 0.0044

ln p26 ln y2 � 0.0063*** 0.0021 � 0.0277*** 0.0036

ln p26 ln y3 � 0.0200*** 0.0050 � 0.0646*** 0.0039

ln y16 ln y2 0.1049 0.4525 0.2208*** 0.0639

ln y16 ln y3 1.4237*** 0.4583 � 0.3643*** 0.0981

ln y26 ln y3 1.1994** 0.4776 0.8953*** 0.2788

Log-likelihood 2,130.75 2,182.30

Parameter estimates of the independent model (continue 1).

Input Model


Groups Total Assets{ Technical

Efficiency

Labor X2 Capital X3 Number of Firms

(Sample Size)

I 12,000 – 50,000 0.8918***

(0.0320)

0.5086***

(0.1491)

0.0136**

(0.0060)

4 (84)

II 50,001 – 90,000 0.8896***

(0.0232)

0.5044***

(0.1449)

0.0218**

(0.0098)

5 (104)

III 90,001 – 220,000 0.9488***

(0.0224)

0.4600***

(0.1311)

0.0255**

(0.0115)

4 (84)

IV 220,001 – 460,000 1 0.4493***

(0.1294)

0.0268**

(0.0124)

4 (84)

V 460,001 – 800,000 0.8861***

(0.0186)

0.4044***

(0.1202)

0.0262**

(0.0120)

5 (105)


Acknowledgments

The first author would like to thank the National Science Council, Executive Yuan,Taiwan, Republic of China, for providing financial support (NSC89-2415-H-032-003). We are grateful for comments from Prof. Paul Wilson and an anonymousreferee.

Notes

1. Researchers are often encouraged to employ the input measure of TI when firms have particular

orders to fill, since input quantities are likely to be the main decision variables. By contrast, an output

measure of TI would appear more suitable if firms are restricted to a fixed quantity of resources.

2. The Fourier series can be equivalently expressed as complex-valued exponential representations. See

Gallant (1982).

3. Atkinson and Cornwell (1994a) suggested that technical and allocative inefficiencies are highly

correlated in the airline industry, and the bias from assuming technical or allocative inefficiency alone

is significant.

4. The number of effective sample observations is defined to be equal to the number of observations of

the sample multiplied by the number of equations to be simultaneously estimated.

5. In almost all applications in this area, J is set to unity.

6. This is because the N cost shares must sum to unity, and these shares are linearly dependent.

7. The authors attempted to alternatively estimate the statistical models without classification, which

implied a further 51 extra TI and AI parameters must be added to the models. Even though the same

number of Fourier series may be deleted to maintain 124 unknown parameters, the likelihood

functions consistently failed to converge against many sets of initial values. This was due partly to the

fact that those inefficiency parameters entered the models highly non-linearly, and partly because of

the small number of observations available.

Parameter estimates of the independent model (continue 2).

Output Model


Groups Total Assets{Technical

Efficiency Labor X2 Capital X3

Number of Firms

(Sample Size)

I 12,000 – 50,000 0.2513***

(0.0100)

6.3940***

(0.5488)

0.7441***

(0.1045)

4 (84)

II 50,001 – 90,000 0.1009***

(0.0055)

1.8130***

(0.1767)

0.6739***

(0.0919)

5 (104)

III 90,001 – 220,000 0.7460***

(0.0347)

21.2767***

(1.8225)

5.1310***

(0.6341)

4 (84)

IV 220,001 – 460,000 1 8.6820***

(0.9906)

1.9527***

(0.2602)

4 (84)

V 460,001 – 800,000 0.3159***

(0.0185)

2.2559***

(0.3390)

1.5907***

(0.2542)

5 (105)

Notes: The parameter estimates of the Fourier series are not shown to save space. {Measured in millions of

New Taiwan’s Dollar. ***Significant at the 1% level. **: Significant at the 5% level. *Significant at the 10%

level.

140 HUANG AND WANG

8. Eastwood (1991) suggested the rule r ¼ 0:1k and argued that its performance is good enough in

experiments, even though the resulting values of r are small. See Eastwood and Gallant (1987).

9. In fact, we have estimated four nested models of the FF cost frontier for the input-oriented

specification, with the models differing in their number of trigonometric terms. The estimated models

are E ¼ 0; 9; 18, and 35. Using the Wald test, we do not accept the null hypothesis of either E ¼ 9 or

E ¼ 18 against the alternative hypothesis of E ¼ 35.

10. In a single output model, the finding that Ai ¼ ð< or >ÞBi for a firm is consistent with constant

(increasing or decreasing) returns to scale.

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Documents

Comparisons of Economic Inefficiency Between Output and Input Measures of Technical Inefficiency Using the Fourier Flexible Cost Function