Japanese banking inefficiency and shadow pricing

Mathematical and Computer Modelling 48 (2008) 1854–1867

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Mathematical and Computer Modelling

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Japanese banking inefficiency and shadow pricingI

Hirofumi Fukuyama a,∗, William L. Weber b,1

a Faculty of Commerce, Fukuoka University, 8-19-1 Nanakuma, Jonan-Ku, Fukuoka 814-0180, Japanb Department of Economics and Finance, Southeast Missouri State University, Cape Girardeau, MO 63701, United States

a r t i c l e i n f o

Article history:Received 14 November 2006Received in revised form 27 February 2008Accepted 14 March 2008

Keywords:DEADirectional DEA specificationParametric LP specificationShadow pricingJapanese bankingBad (undesirable) output

a b s t r a c t

We estimate Japanese banking inefficiency and the shadow price of problem loans bytreating problem loans as a jointly produced undesirable by-product of the loan productionprocess. Our method uses the directional output distance function of Färe et al. [R. Färe,S. Grosskopf, D.-W. Noh, W.L. Weber, Characteristics of a polluting technology: Theoryand practice, Journal of Econometrics 126 (2005) 469–492] and seeks the maximumexpansion of desirable outputs, such as loans and securities investments, and thesimultaneous contraction in undesirable outputs, such as problem loans. The directionaloutput distance function is estimated using data envelopment analysis and a parametriclinear programming method. While the two methods give similar estimates of inefficiency,the shadow price estimates for the two methods diverge.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

From 1980 to 1990 the Japanese money supply grew at an annual 9.1% rate. The increased money supply promptedJapanese banks to ease lending and resulted in a stock and real estate price bubble. In 1989, the Nikkei Index peaked at38,916. However, by the early 1990s, the stock price and real estate bubble had burst leaving Japanese banks with highlevels of non-performing loans on their balance sheets. As a percent of GDP, bank loans in Japan are more than doublethose of US banks making the Japanese economy highly susceptible to problems in the banking industry. From 1990 to 2000Japanese bank loan losses grew from less than 1% of bank assets to more than 2.5% of bank assets. (Weber and Devaney [1])Although nonperforming loan problems have recently eased, the balance sheets of both small banks and large banks are stillplagued by the inability of borrowers to pay interest and repay principal. Any attempt to measure the performance of banksin response to policy reforms or merger activities must account for problem loans. Drake and Hall [2] and Altunbas et al. [3]treat problem loans as an exogenous factor influencing bank technical efficiency or bank costs. Hughes and Mester [4] andMcAllister and McManus [5] also control for problem loans in their cost/scale efficiency estimates for U.S. banks. Fukuyamaand Weber [7,8] control for bank risk via a bank equity capital constraint in a DEA (Data Envelopment Analysis) model, sothat banks with similar risk profiles are compared with each other.

Following Berger and Humphrey’s [6] discussion concerning bank risk and non-performing loans on bank efficiency,Drake and Hall [2] treat problem loans as a fixed input in their DEA efficiency analysis. However, non-performing loans area by-product of the loan production process and do not occur until after a loan has been made. Therefore, non-performing

I An earlier version of this paper was presented at No. 17th seminar on OR applications (Osaka University, Suita-Shi, Osaka), October 14, 2006. Thisresearch is partially supported by the Grants-in-aid for scientific research, fundamental research (A) 18201030 and (B) 19310098, the Japan Society for thePromotion of Science.∗ Corresponding author. Tel.: +81 92 871 6631x4402; fax: +81 92 864 2938.

E-mail addresses: [email protected] (H. Fukuyama), [email protected] (W.L. Weber).1 Tel.: +1 573 651 2946; fax: +1 573 651 2947.

0895-7177/$ – see front matter © 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2008.03.004

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H. Fukuyama, W.L. Weber / Mathematical and Computer Modelling 48 (2008) 1854–1867 1855

loans are more accurately modeled as an undesirable or bad output, rather than as an input. Moreover, from a technicalmodeling stand point, if bad outputs are treated as inputs, the resulting output possibility sets are unbounded.

To measure efficiency we use the directional output distance function which was introduced by Chambers, Chung, andFäre [9] in production theory as an extension of Luenberger’s [10] benefit function. This function serves as an inefficiencymeasure as it seeks the maximum simultaneous expansion of good (desirable) outputs and contraction of bad outputs.Decision-making units are efficient if they cannot simultaneously expand good outputs and contract bad outputs, giveninputs. For the Japanese bank production process we explicitly model problem loans as the bad output that is a joint by-product of the loan production process. Park and Weber [11] measured Korean bank efficiency during 1992–1997 using thedirectional output distance function. We extend their work by estimating the shadow price of problem loans. Färe et al. [12]were the first to use the directional distance function to measure environmental efficiency and estimate the shadow price ofan undesirable polluting output (sulfur dioxide emissions). We use the method of Färe et al. [12] and estimate the shadowprice of problem loans via a parametric directional distance function. We also estimate the shadow price of problem loansby solving a dual DEA directional distance function.

Shadow pricing models have at least two uses. First, if all outputs have observable market prices, a shadow pricingmodel can be used to determine if the mix of outputs is consistent with revenue maximization. However, market prices forundesirable outputs are frequently unobserved. In these cases a second use of shadow pricing is to infer the unobserved pricefrom the market price of a desirable output and knowledge of the physical tradeoff between the desirable and undesirableoutput. We pursue this second case as we show that the shadow price of problem loans equals the decline in value of adesirable output needed to reduce the undesirable output by one unit. As such, the shadow price measures the opportunitycost of reducing the bad output by one unit.

The purpose of this paper is four-fold. First, we present a directional DEA (data envelopment analysis) framework wherethe term “DEA” was coined by Charnes, Cooper, and Rhodes [13] and Banker, Charnes, and Cooper [14] as an extension ofFarrell’s [15] original efficiency analysis. Second, we present and estimate a parametric LP model (linear programming) usingthe loss minimizing procedure of Aigner and Chu [16]. In this step we estimate the quadratic directional output distancefunction introduced by Chambers [17]. Third, we present an empirical illustration of our two methods using data on Japanesebanks that operated in fiscal years 2002 through 2004. We choose this period because data on non-performing loans areavailable for most Japanese banks. Fourth, we compare our estimates of the shadow price for problem loans that are found forthe DEA and parametric LP specifications of the directional distance function. In this step we relax a monotonicity conditionfor the bad output in order to identify the region of the output possibility frontier where each bank operates. Our modelallows for the possibility of output congestion in that banks can operate along either an upward-sloping, flat, or downwardsloping portion of the output possibility frontier. While the deterministic quadratic directional distance function yields apoint estimate of the shadow price for a given bank, our DEA method provides a range of shadow price estimates for thatsame bank.

2. Characterizing the bank technology

2.1. The directional distance function and derivation of shadow prices

Let y ∈ R M+

, b ∈ RJ+, and x ∈ RN

+denote vectors of M good (desirable) outputs, J bad (undesirable) outputs, and N inputs.

The production process is modeled from Shephard’s [18,19] production theory with bad outputs. The underlying technologyis defined by the bank output possibility set:

P(x) =(y, b) : x ∈ R N

+can produce (y, b) ∈ R M

+× R

J+

, (1)

which is the set of all good and bad output vectors that are producible from a fixed level of input. In addition to the standardregularity assumptions, the bank technology (1) satisfies strong disposability of good outputs:

y ≥ y′ ∈ R M+

and (y, b) ∈ P(x) for x ∈ R N+⇒

(y′, b

)∈ P(x). (2)

Eq. (2) states that if the good output vector is decreased it is still producible for a given level of inputs and bad outputs. Wealso assume that inputs are strongly disposable:

x′ ≥ x ∈ R N+⇒ P(x′) ⊇ P(x). (3)

Strong disposability of inputs states that if inputs are increased from x to x′, the resulting bank output possibility set P(x′) isno smaller than the original output possibility set P(x).

While good outputs and inputs satisfy strong disposability, we assume the vector of bad outputs and good outputssatisfies only joint weak disposability:

(y, b) ∈ P(x) and 0 ≤ θ ≤ 1⇒ (θy, θb) ∈ P(x). (4)

Weak disposability implies that there is an opportunity cost of reducing bad outputs in that some good output must beforegone. Finally, good and bad outputs are null-joint if

(y, b) ∈ P(x) and b = 0⇒ y = 0. (5)

1856 H. Fukuyama, W.L. Weber / Mathematical and Computer Modelling 48 (2008) 1854–1867

The condition of null-jointness was introduced by Shephard and Färe [20] and has been discussed for environmentalproblems by Färe et al. [12].2 Good and bad outputs are null-joint if the firm cannot produce good outputs without theproduction of some bad outputs. That is, some bad outputs will be produced as byproducts when a firm produces goodoutputs.

So far we have used sets and properties of those sets to characterize the technology, but we wish to have a functionalrepresentation for estimating purposes. Relative to the output possibility set (1), define the directional output distancefunction as

EDyb(x, y, b; gy, gb) = max

[β : (y+ βgy, b− βgb) ∈ P(x)

], (6)

where gy = (gy1, gy2, . . . , g

yM) ∈ R M

+and gb = (gb1, g

b2, . . . , g

bJ ) ∈ R

J+ are scaling directions for good outputs and bad outputs.

The directional output distance function gives the simultaneous maximum expansion of good outputs and contraction ofbad outputs given inputs for the directional vectors, (gy, gb). When EDyb(x, y, b; gy, gb) = 0 a producer is technically efficientsince no additional good output and contraction in bad output is feasible, given inputs. When a producer is technicallyinefficient it has EDyb(x, y, b; gy, gb) > 0. Higher values of EDyb(x, y, b; gy, gb)indicate greater technical inefficiency. Variousdirectional vectors can be chosen. For instance, if gy = (1, . . . , 1) and gb = (1, 1, . . . , 1), the directional output distancefunction gives the maximum unit expansion in good outputs and unit contraction in bad outputs. If gy = (y1, y2, . . . , yM) andgb = (0, 0, . . . , 0), the directional output distance function gives the proportional expansion in good outputs, holding badoutputs constant. If gy = (0, . . . , 0) and gb = (b1, . . . , bJ), the directional output distance functions gives the proportionalcontraction in bad outputs, holding good outputs constant.

The directional output distance function provides a complete characterization of the technology, in the sense that

Complete Technological Characterization : (y, b) ∈ P(x)⇔ EDyb(x, y, b; gy, gb) ≥ 0. (7)

The directional output distance function also has the translation property

EDyb(x, y+ Ωgy, b− Ωgb; gy, gb) = EDyb(x, y, b; gy, gb)− Ω, Ω ∈ R (8)

which states that if a good output vector is expanded by Ωgy and a bad output vector is contracted by Ωgb then the valueof the resulting directional output distance function is equal to the difference between the value of the original directionaldistance function and Ω .

To evaluate the tradeoff between good and bad outputs we link the physical representation of the technology asrepresented by the directional output distance function with the maximal revenue function used by Färe et al. [12]. Therevenue function takes the form

R(x, p, q) = maxy,b

[py− qb : EDyb

(x, y, b; gy, gb

)≥ 0

], p ∈ R M

+, q ∈ RJ, (9)

where p is the vector of good output prices and q is the vector of bad output prices. Note that py = p1y1 + · · · + pMyMand qb = q1b1 + · · · + qJbJ . It is usually assumed that the vector of bad output prices is positive so that the production ofbad outputs reduces revenue. We allow the frontier of P(x) to have either a positive or negative slope by not imposing anysign restrictions on bad output prices, q. Assuming that EDyb(x, y, b; gy, gb) is differentiable in (y, b), the shadow price of theundesirable output is found by solving (9) and using the envelope theorem. The shadow pricing formulas take the form:

pm = −∂EDyb

(x, y, b; gy, gb

)∂ym

×

(pgy + qgb

)∀m

qj =∂EDyb

(x, y, b; gy, gb

)∂bj

×

(pgy + qgb

)∀j,

(10)

where we assume pgy + qgb 6= 0. Taking the ratio of qj to pm and rearranging, the shadow prices of bad outputs equals

qj = −

∂EDyb(x,y,b;gy,g)∂bj

∂EDyb(x,y,b;gy,gb)∂ym

× pm ∀j,∀m. (11)

In the following two subsections, we show how to estimate the directional output distance function and the shadow priceof bad outputs using DEA and using a parametric method that estimates a quadratic form of the directional output distancefunction.

2 While we impose null-jointness in the DEA estimation, it is not imposed in the deterministic estimation process.


2.2. Directional DEA specification

In this subsection we present a directional DEA specification and its primal and dual directional LP programs. Anobservation for decision-making unit k is represented as (xk, yk, bk) = (x1k, . . . , xNk, y1k, . . . , yMk, b1k, . . . , bJk) where thereare i = 1, . . . , K decision-making units. We assume that observed inputs and outputs are positive.3 Relative to the data set(xk, yk, bk) : i = 1, 2, . . . , K, we construct the reference technology as the DEA output possibility set:

P(x) =

(y, b) :

K∑i=1

xniλi ≤ xn,∀n;K∑

i=1ymiλi ≥ ym,∀m;

K∑i=1

bjiλi = bj,∀j;K∑

i=1λi ≤ 1,λi ≥ 0,∀i

, (12)

where λi (i = 1, . . . , K) are intensity variables. The restriction∑K

i=1 λi ≤ 1 models non-increasing returns to scale.4 Relativeto P(x), the primal-DEA directional output distance function for bank k is computed as

EDyb(xk, yk, bk; gy, gb) = max

λ,β

[β :

K∑i=1

xniλi ≤ xnk,∀n;K∑

i=1ymiλi ≥ ymk + βg

ym,∀m;

K∑i=1

bjiλi = bjk − βgbj ,∀j;

K∑i=1λi ≤ 1, λi ≥ 0,∀i;β free

]. (13)

The corresponding dual directional output distance function is computed as

ν(xk, yk, bk; gy, gb)xk − µ(xk, yk, bk; g

y, gb)yk − σ(xk, yk, bk; gy, gb)bk + ω(xk, yk, bk; g

y, gb)

= minυ,µ,σ,ω

[νxk − µyk − σbk + ω :

N∑n=1νnxni −

M∑m=1

µmymi −

J∑j=1σjbji + ω ≥ 0,∀i;

M∑m=1

µmgym −

J∑j=

σjgbj = 1; µm ≥ 0,∀m; νn ≥ 0,∀n;ω ≥ 0,σ free

]. (14)

We refer to the formulation consisting of (13) and (14) as the directional DEA specification. Because of the translationproperty (8) there are shadow price vectors that satisfy

µ(xk, yk + Ωgy, bk − Ωgb; gy, gb) = µ(xk, yk, bk; gy, gb), Ω ∈ R and

σ(xk, yk + Ωgy, bk − Ωgb; gy, gb) = σ(xk, yk, bk; gy, gb), Ω ∈ R .

(15)

For the results of (15) we assume that the technology does not change when the vector (yk, bk) changes in the direction of(gy, gb

). The Appendix gives a proof of (15).

If bad outputs are ignored and the directional vector for desirable outputs is chosen as gy = (gy1, . . . , gyM) = (y1, . . . , yM)

the directional output distance function is related to the standard Farrell measure of efficiency as

EDyb(xk, yk, 0; yk, 0) = Fy(xk, yk)− 1, (16)

where

Fy(xk, yk) = maxλ,ϕ

[ϕ :

K∑i=1


i=1ymiλi ≥ ϕymk,∀m; ∀j;

K∑i=1λi ≤ 1, λi ≥ 0,∀i

](17)

is the output-oriented Farrell [15] specification (often called a BCC model after Banker, Charnes, and Cooper [14]). Includingbad outputs in the technology and choosing directional vectors gy = (gy1, . . . , g

yM) = (y1, . . . , yM) and gb = (gb1, g

b2 . . . , gbJ ) =

(−b1,−b2, . . . ,−bJ), the Farrell measure of efficiency is related to the directional distance function as:

EDyb(xk, yk, bk; yk,−bk) = Fy(xk, yk, bk)− 1, (18)

where

Fy(xk, yk, bk) = maxλ,ϕ

[ϕ :

K∑i=1


i=1ymiλi ≥ ϕymk,∀m;

K∑i=1

bjiλi = ϕbjk,∀j;K∑

i=1λi ≤ 1, λi ≥ 0,∀i

]. (19)

The Shephard output distance function equals the reciprocal of (19). Such a function was estimated by Coggins andSwinton [21] in their study of electric utilities. A problem with (19) is it gives the maximum proportional expansion inboth good and bad outputs whereas policy-makers and managers are usually interested in maximizing the good output andminimizing the bad output.

3 This assumption along with joint-weak disposability will lead to the implementation of the null-joint property. See Färe et al. [12].4 This restriction was suggested by one of the referees in order to impose jointly weak disposability of good and bad outputs, since no bank in our data

set produces zero amounts of good and bad outputs.


2.3. Parametric linear programming (LP) specification

The translog function has been widely used in producer theory to estimate production functions, cost functions, andShephard distance functions. Because it is multiplicative, the parameters of the translog function can be restricted to satisfyvarious homogeneity properties. However, the translog function is not well suited for additive measures of inefficiencyderived from the directional distance function, nor can the parameters of the translog function be restricted to satisfy thetranslation property. As suggested by Chambers [17] a suitable candidate for the directional distance function is a quadraticfunctional form. The quadratic function serves as a second order approximation to the true but unknown production relationand its parameters can be restricted to satisfy the translation property.

To estimate the directional output distance function we choose a common directional vector for each bank so that thedirectional vector does not have to be parameterized as part of the quadratic function. In the empirical section we investigatea technology with M = 2 good outputs and J = 1 bad output. We choose directional vectors gy = (1, 1) and gb = (1). Otherdirectional vectors can be chosen but they must be common for all observations. The quadratic directional output distancefunction takes the form:

EDyb

(xk, yk, bk; g

y, gb)= α+

N∑n=1αnxnk +

M∑m=1

βmymk +

J∑j=1γjbjk

+12

N∑n=1

N∑n′=1

αnn′xnkxn′k +12

M∑m=1

M∑m′=1

βmm′ymkym′k +12

J∑j=1

J∑j′=1γjj′bjkbj′k

+

N∑n=1

M∑m=1

δnmxnkymk +

N∑n=1

J∑j=1ηnjxnkbjk +

M∑m=1

J∑j=1ρmjymkbjk. (20)

We impose symmetry conditions on the second order terms in (20) so that

i. αnn′ = αn′n, n 6= n′

ii. βmm′ = βm′m, m 6= m′

iii. γjj′ = γj′ j, j 6= j′.

(21)

In addition, the translation property (8) imposes restrictions:

i.M∑

m=1βmg

ym −

J∑j=1γjg

bj = −1

ii.M∑

m′=1βmm′g

′

m −

J∑j=1ρjmg

bj = 0 ∀m

iii.J∑

j′=1γjj′g

bj′ −

M∑m=1

ρjmgym = 0 ∀j

iv.M∑

m=1δnmg

ym −

J∑j=1ηnjg

bj = 0 ∀n.

(22)

Färe et al. [12] employ the quadratic form (20) in order to analyze the effects of environmental regulation on US electricutilities. Two additional restrictions due to the complete characterization property (7) and monotonicity (2) are imposed.These restrictions take the form:

i. EDyb

(xk, yk, bk; g

y, gb)≥ 0 ∀k

ii. ∂EDyb

(xk, yk, bk; g

y, gb)/∂ymk ≤ 0 ∀m,∀k.

(23)

To estimate the parameters of (20), we use Aigner and Chu’s [16] deterministic linear programming procedure where theparameters are estimated by minimizing the sum of the deviations of each bank’s observation (xk, yk, bk) from the frontier.Thus, we solve

minK∑

k=1

EDyb

(xk, yk, bk; g

y, gb)− 0

(24)

subject to the translation property (22), the symmetry restrictions (21), and restrictions requiring feasibility andmonotonicity given by (23). The objective function in (24) minimizes the sum of deviations of the directional output distancefunctions from the efficient output frontier based on the directions, gy = (1, 1) and gb = (1).

Although Färe et al. [12] impose monotonicity restrictions on the bad outputs of the form, ∂EDyb(·)

∂bj≥ 0, we do not, so as

to allow the output frontier, P(x), to be positively or negatively sloped. We think that not imposing these restrictions willallow a better comparison of shadow prices derived from the deterministic method described above and the DEA methoddescribed in the next section.


3. Shadow pricing of bad outputs

3.1. Directional DEA specification

One way of estimating shadow prices using DEA is to estimate the dual form of the directional output distance functiongiven by (14) and then use the estimates of u,σ, and p to obtain an estimate of the bad output price, q. In our model weassume that Japanese banks produce two desirable outputs (loans and other interest bearing assets) and a single bad output(non-performing loans). We use the price of output 2, p2, to determine the shadow price of the bad output. EvaluatingEq. (11) for the dual directional output distance function (14) we obtain:

q1 = −p2σ1

µ2, µ2 6= 0. (25)

However, it is clear that supporting hyper-planes are not necessarily unique for an efficient observation and in fact there isthe possibility of multiple supporting hyper-planes in the case of the directional DEA specification. As an alternative to (25),another shadow pricing method would be to obtain upper and lower bounds on the shadow price of the bad output bysolving a series of fractional programming problems. We obtain bounds on the shadow price of the single bad output fromthe market price of output 2. The upper bound for the bad output shadow price is the solution to the following fractionalprogramming problem:

qmax1 = max

υ,µ,σ1,ω

[−σ1p2k

µ2:

3∑n=1νnxnk −

2∑m=1

µmymk − σ1b1k + ω = D∗;2∑

m=1µmg

ym − σ1g

b1 = 1;

3∑n=1νnxnk −

2∑m=1

µmymk − σ1b1k + ω ≥ 0,∀k;µm ≥ 0,m = 1, 2;µ2 > 0; νn ≥ 0, n = 1, 2, 3;ω ≥ 0,σ1free]

,

(26)where D∗ = EDyb(xk, yk, bk; gy, gb) is the solution to (13) and (14). The dual variable for problem loans, σ1, and theconstant, ω, can take either negative or positive values, whereas the dual variables (v, µ) associated with inputs andoutputs are restricted to be nonnegative and thus satisfy monotonicity. Note that if y∗k = yk + EDyb(xk, yk, bk; gy, gb)gy andb∗1k = b1k − EDyb(xk, yk, bk; gy, gb)g

b1, the restriction

∑3n=1 νnxnk −

∑2m=1 µmy∗mk − σ1b

∗

1k + ω = 0 in (26) can be replaced by∑3n=1 νnxnk −

∑2m=1 µmymk − σ1b1k + ω = D∗. This result is due to (15).

We restrict our model to the case where µ2 > 0. Applying the Charnes–Cooper [22] transformation procedure, weconvert the fractional programming problem (26) into an LP problem.5 Setting t = 1/µ2; νn = tνn (n = 1, 2, 3); µm =

tµm (m = 1, 2); σ1 = tσ1,ω = tω, λ = λt, our estimate of the upper bound6 is:

qmax1 = max

ν,µ,σ1,ω,t

[−σ1p2k :

3∑n=1

νnxnk −2∑

m=1µmymk − σ1b1k + ω = D∗t;

M∑m=1

µmgym − σ1g

b1 = t;

3∑n=1

νnxnk −2∑

m=1µmymk − σ1b1k + ω ≥ 0, k = 1, . . . , K; t > 0; µ2 = 1;

µm ≥ 0,m = 1, 2; νn ≥ 0, n = 1, 2, 3;ω ≥ 0, σ1 free]

. (27)

Tone [23] employs a similar transformation procedure to estimate a slack-based measure of technical efficiency via linearprograms.

For the computation of the lower bound,7 we use

qmin1 = min

υ,µ,σ1,ω,t

[−σ1p2k :

3∑n=1

νnxnk −2∑

m=1µmymk − σ1b1k + ω = D∗t;

M∑m=1

µmgym − σ1g

b1 = t;

3∑n=1

νnxnk −2∑

m=1µmymk − σ1b1k + ω ≥ 0, k = 1, . . . , K; t > 0; µ2 = 1;

µm ≥ 0,m = 1, 2; νn ≥ 0, n = 1, 2, 3;ω ≥ 0, σ1 free]

. (28)

Infeasible solutions to either (27) or (28) will occur if µ2 = 0. Cases such as this occur when the output possibility set isvertical and would imply an infinitely negative shadow price.

5 If we use∑N

n=1 νnxnk −∑M

m=1 µmymk −∑J

j=1 σjbjk + ω = D∗ rather than∑N

n=1 νnxnk −∑M

m=1 µmy∗mk −∑J

j=1 σjb∗jk + ω = 0 in (27) and (28), the

estimation equation will be slightly different from (26).6 For the estimation of (26), we also impose the restriction that the objective is less than some large number. We choose 100,000 for that number and

assume that if the upper bound is 100,000, then the solution is infeasible.7 For the estimation of (28), we also impose two restrictions: (i) the objective is less than or equal to the upper bound found in (27), and (ii) the objective

is greater than some small number. We choose – 100,000 for that number and assume that if the lower bound is – 100,000, then the solution is infeasible.


3.2. Parametric LP specification

For the quadratic directional output distance function given by (20), the shadow price of the bad output can bedetermined by applying (11). In our model we have N = 3 inputs, M = 2 good outputs, and J = 1 bad output. We usethe market price of output 2 for bank k, p2k, to estimate the shadow price of the bad output. This shadow price estimate isfound as

q1k = −γ1 + γ11b1k + η11x1k + η21x2k + η31x3k + ρ11y1k + ρ21y2k

β2 + β12y1k + β22y2k + δ12x1k + δ22x2k + δ32x3k + ρ21b1k× p2k. (29)

In a study of the Canadian pulp and paper industry, Hailu and Veeman [24] estimate a translog distance function by imposingmonotonicity restrictions with respect to both undesirable and desirable outputs. They also restrict the shadow price ofthe undesirable output to be nonnegative as do Färe et al. [12]. Coggins and Swinton [21] also estimate a translog outputdistance function with monotonicity restrictions. Using the Coggins and Swinton framework and adopting the bad outputspecification of Park and Weber [11], Chaffai and Lassoued [25] estimate shadow prices of nonperforming loans in Tunisianbanking industry and find that the shadow prices are negatively correlated with bank risk.

4. Data and estimation results

Our data are from Nikkei’s Financial Quest for the fiscal years 2002 through 2004, corresponding to ending balance sheetsin the first quarter of 2003 and the first quarter of 2005. We have complete data on 126 banks in 2002, 122 banks in 2003,and 118 banks in 2004. We pool the data for the three years, thereby assuming a common technology for the banks duringthe three year period.

Following much of the literature on bank efficiency we regard a bank as a financial firm that transforms three variableinputs of labor (x1), physical capital (x2), and raised funds (x3) into two asset-based good outputs, loans (y1) and otherinterest bearing assets (y2), and a single bad output of non-performing loans (b1). Our input/output specification is consistentwith the asset or intermediation approach which is due to Sealey and Lindley’s [26] perspective on bank production. Thisview is also related to the user cost approach of Hancock [27]. In the asset approach deposits and other liabilities are viewedas inputs. Although there is an unsettled debate about whether deposits should be treated as an input or an output, weemploy the asset approach by following previous Japanese bank production studies such as Kasuya [28] and Fukuyama [29,30]. Glass, McKillop, and Morikawa [31] conclude that the choice between output classifications is of minor importance inthe study of cost economies in Japanese banking. Berger and Humphrey [32] provide a comprehensive discussion on thetreatment of deposits.

The bad output of non-performing loans is defined as the sum of problem loans, which are part of the total loans. Problemloans are computed by adding the balance of loans to bankrupt borrowers and the balance of non-accrual delinquent loans.A third type of problem loan is the balance of loans past due for three months or more. Data on this type of problem loanare available for only a small subset of banks in our sample and so we confine our definition of non-performing loans to thefirst two categories of problem loans.

Several researchers incorporate problem loans and risk into their studies of Japanese bank inefficiency. Altunbus et al. [3]investigate the impact of loan quality and risk on Japanese banks’ costs for the years 1993–1996. Accounting for loan quality,they find that more banks operate in the range of diseconomies of scale than when loan quality is ignored. Drake and Hall [2]measure technical and scale efficiency for 149 Japanese banks in 1997. They find that the increased efficiency gains fromsmall banks realizing greater scale economies and the gains to large banks from reducing scale diseconomies is reduced whenloan loss provisions are included as part of the production technology. Liu and Tone [33] use credit costs against possible loanlosses as an input in their analysis of bank efficiency. After controlling for environmental effects in the Japanese economy,they find that Japanese banks appear to be “learning by doing” as efficiency levels increased during the period 1997–2001.Drake, Hall, and Simper [34] also assert the necessity of treating loan loss provisions as an input or cost in their DEA analysis.Fukuyama and Weber [35] control for risk by including equity capital as a fixed input. Using 1994 data, Hori [36] controlsfor risk by subtracting non-performing loans from total loans and finds that the average regional bank can reduce costs byabout 20% by realizing greater technical efficiency. These studies all share the view that risk or problem loans are importantfactors affecting efficiency measurement.

Japanese banks have two main activities: the production of loans and the production of other business activities. Thetraditional business activity of Japanese banks is making loans financed by raised funds. Total loans on the balance sheetinclude both performing loans and non-performing loans. We estimate our models using two specifications of loans. Inmodel 1, we use total loans as y1. In model 2, we restrict the loan output to only performing loans which equals thedifference between total loans and non-performing loans. In a study of Korean bank efficiency, Park and Weber [11] considera specification of loan outputs similar to our model 2. The non-traditional output of other business activities, y2, includessecurities investments, trading assets, and foreign exchange transactions.

To calculate the shadow price of non-performing loans we use the observed price of good output 2 (p2) and knowledgeof the physical production tradeoff. The price of other interest bearing assets is computed as the ratio of earned income onother assets to the asset value of y2. As shown in (11), the shadow price of the bad output can be computed from the observedprice of any of the good outputs. However, since some loans yield interest for only a few months out of the year, the ratio of


Table 1Descriptive statistics for the pooled sample of 366 observations, 2002–2004

Variablea Mean Std. dev. Minimum Maximum

Assets 6.231 15.401 0.196 104.103Equity 0.248 0.499 0.006 3.954y1 (model 1)b 3.533 7.805 0.140 56.446y1 (model 2)c 3.417 7.622 0.129 54.995y2 1.581 4.086 0.030 29.417x1 2.185 2.987 0.245 19.496x2 0.065 0.137 0.003 1.295x3 5.439 13.192 0.182 87.948b1 0.116 0.201 0.004 1.690p2 0.012 0.004 0.003 0.044

a Units except for x1 are in trillions of yen. Employees (x1) are in thousands of persons. All financial data are deflated by the Japanese GDP deflator (baseyear= 2000). The pooled sample consists of 126, 122 and 118 banks in fiscal years 2002, 2003, and 2004.

b In model 1, y1 = total loans.c In model 2, y1 = total loans− nonperforming loans.

interest income on loans to total loans is probably not an accurate measure of the market price of loans. Therefore, we usep2 to derive the shadow price of problem loans.

The labor input equals the number of full-time employees at the end of each year. Physical capital equals the sum of thebook value of premises and equipment, suspense payments for unfinished construction, and surety deposits and intangibles.The raised funds input equals the sum of deposits, CDs, call money, bills sold, borrowed money, foreign exchange, and amiscellaneous item on the liability side of the balance sheet. The outputs and inputs are measured in trillions of yen exceptfor labor (x1) which is measured in thousands of employees. We deflate all financial data by the Japanese GDP deflator (baseyear= 2000) in order to allow for comparison across the years.

Table 1 provides descriptive statistics for the outputs, inputs, total assets, equity capital, and the price of output 2 for thepooled sample. The mean asset value is 1.8 times greater than the mean value of total loans. Bad loans average three percentof total loans. Equity capital is twice as large as bad loans.

We follow Färe et al. [12] and estimate the model using directional vectors gy = (1, 1) and gb = 1. This choice ofdirections means that the estimate of the directional distance function will give the potential simultaneous unit expansionin the loans and other interest bearing assets and contraction in non-performing loans if the bank were to become efficient.When a common directional vector is used to estimate the directional distance function, Färe and Grosskopf [37] show thateach bank’s inefficiency can be aggregated to an industry indicator of inefficiency when there is no allocative inefficiency.In addition, a common directional vector for all banks means that the directional vector does not have to parameterizedas part of the quadratic form of the directional distance function given by (20). Table 2 provides the parameter estimatesfor the quadratic form of the directional distance function which are used to estimate bank inefficiency and the shadowprice of non-performing loans. We tested whether the quadratic form of the directional output distance function satisfiednull-jointness using the method described in Färe et al. [12]. Null-jointness implies that if no bad output is produced thenno good output can be produced. Therefore, for a given level of inputs, null jointness holds if EDyb(x, y, 0; 1, 1) < 0 for y > 0.That is, a positive level of desirable output and zero amounts of the bad output are infeasible given the technology. Wesubstituted each bank’s observed input vector, observed loans and other assets, and a zero quantity for non-performingloans and evaluated the quadratic directional output distance function using the parameters reported in Table 2. For thissimulation we found that 320 banks (model 1) and 324 banks (model 2) had EDyb(x, y, 0; 1, 1) < 0 and thus satisfied nulljointness. This result indicates that non-performing loans are an undesirable by-product of the loan production process,much like pollution is an undesirable by-product in manufacturing.

Table 3 provides the estimates of bank inefficiency and the number of banks that define the frontier for the parametricdirectional distance function and for the distance function estimated by DEA. We also provide estimates of the Farrelloutput measure of efficiency (17). We examined the Spearman and Kendall rank correlation coefficients between each ofthe inefficiency measures estimated by DEA and the parametric approach. Although not reported, there was a significantpositive correlation between each inefficiency measure for the two estimation methods and for the two models.

For the parametric specification of EDyb

(xk, yk, b

;

k1, 1)

, the sum of the inefficiency values for the pooled banks is 22.4 inmodel 1 and 23.3 in model 2. For the directional DEA specification, the summed inefficiency values are 22.3 (= 0.061×366)and 23.8 (= 0.065× 366) in models 1 and 2. The summed levels of inefficiency are similar for both estimation methods andgiven the significant positive correlation between estimated inefficiency, the two methods seem to give consistent estimatesof inefficiency.

Table 4 reports the means and the range of the shadow price estimates for the two models and the two estimationmethods where feasible solutions are found for the LP problems. The shadow price estimates indicate a relatively highcost of reducing bad loans. For instance, using the parametric LP estimates from model 2, the shadow price of bad loans is4.43. This price indicates that a 1 million yen reduction in bad loans reduces bank revenues by 4.43 million yen due to thereduction in desirable loans. To reduce bad loans by 1 million yen, the estimates range from a price of about 0.023 million yento 52.3 million yen. The partial derivative ∂EDyb (·) /∂y2 for the parametric LP specification was zero for three banks in model 1


Table 2Parametric LP specification estimates (n = 366, model 1 and 2)

Parameter Variable Coefficient (model 1) Coefficient (model 2)∑EDYB(x, y, b; 1, 1) Objective value 22.402 23.267

α Constant 0.034 0.030α1 x1 −0.110 −0.086α2 x2 1.672 1.039α3 x3 0.035 0.028β1 y1 −0.040 −0.034β2 y2 −0.003 0.000γ1 b1 0.958 0.966α11 x1 × x1 0.122 0.084α21 x2 × x1 −1.270 −0.654α31 x3 × x1 −0.028 −0.026α22 x2 × x2 −5.297 −7.569α32 x3 × x2 0.475 0.345α33 x3 × x3 0.003 0.006β11 y1 × y1 −0.017 −0.009β12 y1 × y2 0.007 0.006ρ11 y1 × b1 −0.009 −0.003β21 y2 × y1 0.007 0.006β22 y2 × y2 0.003 0.003ρ21 y2 × b1 0.011 0.009γ11 b1 × b1 0.001 0.005δ11 x1 × y1 0.017 0.021δ12 x1 × y2 −0.001 −0.002η11 x1 × b1 0.016 0.018δ21 x2 × y1 −0.102 −0.133δ22 x2 × y2 −0.204 −0.098η21 x2 × b1 −0.307 −0.231δ31 x3 × y1 0.003 −0.001δ32 x3 × y2 −0.005 −0.004η31 x3 × b1 −0.001 −0.005

Table 3Inefficiency estimates for the parametric and DEA distance functions

Variable K Estimation method Mean Std. dev. # of frontier firms Min. Max.

Model 1 (y1 = total loans, y2 = other interest bearing assets, b1 = non-performing loans)

EDyb(x, y, b; 1, 1) 366 Parametric 0.061 0.081 17 0 0.850EDyb(x, y, b; 1, 1) 366 DEA 0.061 0.068 27 0.000 0.731Fy(x, y) 366 DEA 1.161 0.100 20 1.000 1.401

Model 2 (y1 = total loans – non-performing loans, y2 = other interest bearing assets, b1 = non-performing loans)

EDyb(x, y, b; 1, 1) 366 Parametric 0.064 0.075 15 0 0.696EDyb(x, y, b; 1, 1) 366 DEA 0.065 0.073 25 0.000 0.780Fy(x, y) 366 DEA 1.195 0.102 19 1.000 1.491

and for four banks in model 2, which implies shadow prices of negative infinity for those banks. For model 2, the DEAestimates of the mean shadow price of non-performing loans is 3.91. When shadow prices are estimated via the dual forms(27) and (28), only 49 banks in model 1 and 38 banks in model 2 have µ2 6= 0 and a finite shadow price for the bad output.

The quadratic directional output distance function provides a second-order approximation to the true, but unknowndirectional output distance function, while the directional DEA specification given by (13) or (14) provides only a first orderapproximation. Since the estimation methods are nonparametric and the distributions of efficiency and shadow prices arenon-normal, we do not report standard errors. Instead, we graph the kernel distribution functions of shadow prices for thetwo methods and two models of bank production. The distributions functions given in Figs. 1 and 2 are smoothed histogramsof shadow prices based on a standard normal kernel density. (Pagan and Ullah [38]) We test whether the two distributionsare the same using the T-test proposed by Li [39]. The T-test has an asymptotic normal distribution and is used to testwhether two kernel distributions of a variable x, call them f (x) and g(x), are equal. As seen in the figures, the DEA firstorder approximation of shadow prices has a much tighter distribution than the shadow prices estimated from the secondorder approximation of the quadratic form. The T-test of Li [39] allows us to reject the null hypothesis of equal distributionfunctions of shadow prices. See Figs. 1 and 2.

For each bank, is the shadow price estimate from the quadratic directional distance function within the range of shadowprice estimates found by solving the DEA models (27) and (28)? Table 5 reports the range of answers to this question. Only49 banks had feasible solutions to the DEA problems solving the upper and lower bounds for the shadow price of problemloans in model 1 and in model 2. In model 1, three out of the 49 banks with feasible solutions to the DEA problems had


Table 4Shadow price estimates

Model K mean Std. dev. Min. Max.

Parametric LP estimates

q1 = −p2 ·∂EDyb(·)/∂b1∂EDyb(·)/∂y2

1 363 1.93 2.36 0.014 40.05

q1 = −p2 ·∂EDyb(·)/∂b1∂EDyb(·)/∂y2

2 362 4.43 5.16 0.023 52.3

DEA estimatesa ,b

q1 = −p2 ·σ1µ2

1 49 −0.50 2.90 −19.17 1.23q1 = −p2 ·

σ1µ2

2 49 3.91 10.43 −1.90 44.83

a For the DEA model, ∂EDyb (·) /∂b = −σ1(·), ∂EDyb (·) /∂y2 = −µ2(·).b For the DEA estimates we exclude banks for which qmax

1 = +∞ and qmin1 = −∞. However, if qmax

1 or qmin1 is finite, we use the finite value as the DEA

estimate. If qmax1 ≥ 0 ≥ qmin

1 , we use q1 = 0.

Fig. 1. Kernel density estimates of shadow prices for model 1 deterministic vs. DEA estimates Qi Li’s T-statistic= 8.02.

Fig. 2. Kernel density estimates of shadow prices for model 2 deterministic vs. DEA estimates Qi Li’s T-statistic= 12.68.

∂EDyb (·) /∂y2 = 0 for the parametric LP specification. For the remaining 46 banks, only 14 banks had a shadow price estimatedfrom the parametric quadratic distance function within the DEA lower and upper bound range. For the 49 banks with feasiblesolutions for the DEA upper and lower bound shadow price problems, five banks had ∂EDyb (·) /∂y2 in the parametric LPspecification. For the remaining 44 banks, only 12 banks had a shadow price estimated from the quadratic form within theDEA lower and upper bound range.


Fig. 3. Regions of shadow bad output prices: model 1.

We summarize the range of estimates for the shadow price of non-performing loans for the two models in Figs. 3 and 4.When the directional output distance function is estimated as a quadratic form using the parametric LP method, more than98% of banks are shown to operate on the positively sloped portion of the production possibilities frontier for models 1 and2. For the DEA estimates, when a feasible solution is given, 45% (22 out of 49) of banks in model 1 and 59% (29 out of 49)in model 2 operate on the positively sloped portion of the production possibilities frontier. Another 41% (model 1) and 39%(model 2) operate on the horizontal portion of the production possibilities frontier. These results suggest non-performingloans for banks are like polluting by-products of manufacturing firms or electric utilities. Thus, a meaningful analysis of theperformance of Japanese banks can be had only when estimates of inefficiency account for this undesirable output.

5. Conclusions

In this paper we examined the extent of Japanese banking inefficiency by modeling non-performing loans as a jointlyproduced undesirable by-product of the loan production process. Our estimates of bank inefficiency were derived from thedirectional output distance function which was estimated using two methods. In our first method, we specified a quadraticform for the directional output distance function and estimated it by minimizing the deviation of each bank’s observation


Fig. 4. Regions of shadow bad output prices: model 2.

of inputs and outputs from the frontier. In the second method, we estimated the directional output distance function usingDEA. The inefficiency estimates for the two methods had a significant positive rank correlation and gave consistent estimatesof bank inefficiency. We also estimated the shadow price of non-performing loans from the two methods. The estimates ofthe quadratic directional distance function gave a point estimate of the shadow price of non-performing loans for each bank.For the DEA technology we solved two fractional programming problems and obtained a lower and upper bound estimateof the shadow price of non-performing loans. The empirical evidence indicates that when feasible solutions to the DEAproblems exist, a clear majority of banks had point estimates for the shadow price of problem loans outside the range foundby solving the fractional programming problems. Although both methods gave similar estimates of bank inefficiency, thequadratic form allowed us to estimate the shadow price of non-performing loans for 99% of the banks in our data set, whilethe DEA method gave us shadow prices for only 13% of the banks. Our results suggest that non-performing loans are anundesirable output for Japanese banks, much like pollution is an undesirable by-product of electric utilities. A consequenceof our findings is that researchers examining the efficiency of Japanese banks should control for non-performing loans as anundesirable by-product of the loan production process.


Table 5Comparison of the shadow price of non-performing loans for the two estimation methods

Are bad output shadow prices from the parametric LP specification within ranges of upper and lower bounds from the directional DEA specification?

Resulta Frequency (%) Cumulative frequency (%)

Model 1

No 32 (8.74%) 32 (8.74%)µ2 = 0 317 (86.61%) 349 (95.36%)∂EDyb(·)/∂y2 = 0 3 (0.82%) 352 (96.17%)Yes 14 (3.83%) 366 (100%)

Model 2

No 32 (6.01%) 32 (8.74%)µ2 = 0 317 (89.34%) 349 (95.35%)∂EDyb(·)/∂y2 = 0 5 (1.37%) 354 (96.72%)Yes 12 (3.28%) 366 (100%)

a “No”, means the parametric LP estimate of the shadow price is not in the range of shadow price estimates from the DEA method. If µ2 = 0, the DEAmethod is infeasible. If ∂EDyb(·)/∂y2 = 0, the parametric LP method yields a shadow price estimate of positive infinity. If “Yes”, the parametric LP estimateof the shadow price is within the range of shadow price estimates from the DEA method.

Appendix. Proof of the Eq. (15)

Proof. We have µ · gy − σ · gb = 1 for all nonnegative directions(gy, gb

)6= 0. Suppose gy1 > 0 but gym = 0 (∀m =

2, . . . ,M) and gbj = 0 (∀j = 1, . . . , J). Thus we have µ1 · gy1 = 1 and µ1(x, y + Ωgy, b − Ωgb; gy, gb)gy1 = 1 or

µ1(x, y+ Ωgy, b− Ωgb; gy, gb) = 1/gy1. Letting Ω = 0, we have µ1(x, y, b; gy, gb) = 1/gy1, implying that

µ1(x, y+ Ωgy, b− Ωgb; gy, gb) = µ1(x, y, b; gy, gb).

Similarly, we have

µm(x, y+ Ωgy, b− Ωgb; gy, gb) = µm(x, y, b; gy, gb) ∀m = 2, . . . ,M

σj(x, y+ Ωgy, b− Ωgb; gy, gb) = σj(x, y, b; gy, gb) ∀j = 1, . . . , J.

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Japanese banking inefficiency and shadow pricing