European Journal of Operational Research 241 (2015) 555–563

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European Journal of Operational Research

journal homepage: www.elsevier.com/locate/ejor

Interfaces with Other Disciplines

Primal and dual dynamic Luenberger productivity indicators

Alfons Oude Lansink∗, Spiro Stefanou, Teresa Serra

Wageningen University, Department of Business Economics, Hollandseweg 1, 6706 KN Wageningen, Netherlands

a r t i c l e i n f o

Article history:

Received 6 November 2013

Accepted 5 September 2014

Available online 30 September 2014

Keywords:

Directional distance function

Dynamics

Luenberger

Dairy sector

Productivity

a b s t r a c t

This paper develops primal and dual versions of the dynamic Luenberger productivity growth measures that

are based on the dynamic directional distance function and intertemporal cost minimization, respectively.

The empirical application focuses on panel data of Dutch dairy farms over the period 1995–2005. Primal

dynamic Luenberger productivity growth averages 1.5 percent annually in the period under investigation,

with technical change being the main driver of annual change. Dual dynamic Luenberger productivity growth

is −0.1 percent in the same period. Improvements in technical inefficiency and technical change are partly

counteracted by deteriorations of allocative inefficiency, with large dairy farms presenting a slightly higher

productivity growth than small dairy farms.

© 2014 Elsevier B.V. All rights reserved.

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. Introduction

The characterization and measurement of economic performance

n both theory and practice continues to claim considerable atten-

ion in the literature. The major attention of economic performance

enters on the measurement of efficiency and productivity growth.

he economics literature on efficiency has produced a wide range of

roductivity growth measures (see Balk, 2008 for a comprehensive

reatment).

The setting of the decision environment plays a crucial role in the

odeling framework and the characterization of results. The static

odels of production are based on the firm’s ability to adjust instan-

aneously and ignore the potential dynamic linkages of production

ecisions. The business policy relevance to distinguishing between

he contributions of variable and capital factors to inefficiency or pro-

uctivity growth is clear. For example, when variable factor use is

ot meeting its potential, remedies can include better monitoring of

esource use; when asset use is not meeting potential, remedies can

nclude training programs to enhance performance or even a review

f the organization of assets in the production process to take advan-

age of asset utilization. The weakness underlying the static theory of

roduction in explaining how some inputs are gradually adjusted has

ed to the development of the dynamic models of production where

urrent production decisions constrain or enhance future production

ossibilities.

Allowing for the presence of dynamic adjustment leads produc-

ivity growth measurement to include a scale and technical change

∗ Corresponding author. Tel.: +31612395145.

E-mail address: alfons.oudelansink@wur.nl (A. O. Lansink).

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ttp://dx.doi.org/10.1016/j.ejor.2014.09.027

377-2217/© 2014 Elsevier B.V. All rights reserved.

ffects (as in the static theory) in addition to capital stock adjustment

nd the impact of the changing shadow values on long-run equi-

ibrium capital stocks and investment (Luh & Stefanou, 1991). This

ecomposition can be further elaborated to account for efficiency

hange (Rungsuriyawiboon & Stefanou, 2008).

The characterization of dynamic efficiency can also build on the

djustment cost framework that implicitly measures inefficiency as

temporal concept as it accounts for the sluggish adjustment of

ome factors. In a nonparametric setting, Silva and Stefanou (2007)

evelop a myriad of efficiency measures associated with the dy-

amic generalization of the dual-based revealed preference approach

o production analysis found in Silva and Stefanou (2003). In a

arametric setting, Rungsuriyawiboon and Stefanou (2007) present

nd estimate the dynamic shadow price approach to dynamic cost

inimization.

An intriguing prospect is to incorporate the properties of the dy-

amic production technology presented in Silva and Stefanou (2003)

nto the directional distance function framework, which can ex-

loit the Luenberger productivity growth measurement. The direc-

ional distance function offers the powerful advantage of focusing

n changes in input and output bundles, inefficiency and the tech-

ology. Such a productivity measure based on the directional dis-

ance function has its origins in Chambers, Chung, and Färe (1996)

ho defined a Luenberger indicator of productivity growth in the

tatic context. A growing literature employing this approach has

merged more recently (see Balk, 2008; Boussemart, Briec, Kerstens, &

outineau, 2003; Briec & Kerstens, 2004; Chambers & Pope, 1996;

hambers et al., 1996; Färe & Grosskopf, 2005; Färe & Primont, 2003).

he dual approach to measuring Luenberger productivity growth in

he static context has been elaborated by e.g. Färe, Grosskopf, and

argaritis (2008), but has hardly been applied in the literature.

556 A. O. Lansink et al. / European Journal of Operational Research 241 (2015) 555–563

Fig. 1. Luenberger indicator of dynamic productivity growth.

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This paper develops primal and dual dynamic Luenberger produc-

tivity growth indicators that are based on the dynamic directional

distance function and the intertemporal cost function, respectively.

The adverbs ‘primal’ and ‘dual’ refer to the models that are underlying

the computation of the productivity indicators, i.e. the intertemporal

cost function used for computing the dual dynamic Luenberger pro-

ductivity growth indicator is dual to the primal distance function that

underlies the computation of the primal dynamic Luenberger produc-

tivity growth indicator. The primal Luenberger productivity growth

indicator is decomposed to identify the contributions of efficiency

growth and technical change, while the dual Luenberger productivity

growth indicator offers a further decomposition to identify the impact

of quasi-fixed factor disequilibrium and allocative efficiency change.

An illustration of these measures is applied to a panel of Dutch dairy

farms over 1995–2005.

The next section develops the primal and dual measures of dy-

namic productivity growth and its decomposition. This is followed

by the empirical application to the panel of Dutch dairy farms which

uses the results of a previously estimated dynamic directional dis-

tance function found in Serra, Oude Lansink, and Stefanou (2011) to

generate the primal and dual measures of productivity growth and

their respective decompositions. The final section offers concluding

comments.

2. The primal Luenberger indicator of dynamic productivity

growth

The primal Luenberger indicator of dynamic productivity growth

is defined through a dynamic directional distance function. Let

yt ∈ �M++ represent a vector of outputs at time t, xt ∈ �N+ de-

note a vector of variable inputs, Kt ∈ �F++ the capital stock vec-

tor, It ∈ �F+ the vector of gross investments and Lt ∈ �C++ a vector

of fixed inputs for which no investments are allowed. The produc-

tion input requirement set can be represented as Vt(yt : Kt, Lt) ={(xt, It) : (xt, It)can produce yt given Kt, Lt}. The input requirement

set is defined by Silva and Stefanou (2003) and assumed to have the

following properties: Vt(yt : Kt, Lt) is a closed and nonempty set, has

a lower bound, is positive monotonic in xt , negative monotonic in It ,

is a strictly convex set, output levels increase with the stock of capital

and quasi-fixed inputs and are freely disposable.

The input-oriented dynamic directional distance function �Dit(yt,

Kt, Lt, xt, It; gx, gI) can be defined as follows:

�Dit(yt, Kt, Lt, xt, It; gx, gI)

= max{β ∈ � : (xt − βgx, It + βgI) ∈ Vt(yt : Kt, Lt)},gx ∈ �N

++, gI ∈ �F++, (gx, gI) �= (0N, 0F) (1)

if(xt − βgx, It + βgI

) ∈ Vt(yt : Kt, Lt) for some β , �Dit(yt, Kt, Lt, xt, It;

gx, gI) = −∞, otherwise. The distance function is a measure of the

maximal translation of (xt, It) in the direction defined by the vector

(gx, gI), that keeps the translated input combination interior to the set

Vt(yt : Kt, Lt). Sinceβgx is subtracted from xt andβgI is added to It , the

directional distance function is defined by simultaneously contracting

variable inputs and expanding gross investments. As shown by Silva,

Oude Lansink, and Stefanou (2009), �Dit(yt, Kt, Lt, xt, It; gx, gI) ≥ 0 fully

characterizes the input requirement set Vt(yt : Kt, Lt), being thus an

alternative primal representation of the adjustment cost production

technology.

Extending the Luenberger indicator of productivity growth de-

fined by Chambers et al. (1996) to the dynamic setting by using the

dynamic directional distance function (assuming Variable Returns to

Scale) leads to:

LP(·) = 1

2

{[ �Dit+1(yt, Kt, Ltxt, It; gx, gI)

− �Dit+1(yt+1, Kt+1, Lt+1xt+1, It+1; gx, gI)

]

t

+[ �Di

t(yt, Kt, Ltxt, It; gx, gI)

− �Dit(yt+1, Kt+1, Lt+1xt+1, It+1; gx, gI)

]}(2)

his indicator provides the arithmetic average of productivity change

easured by the technology at time t + 1 [the first two terms in

q. (2)] and the productivity change measured by the technology at

ime t [the last two terms in Eq. (2)].

The Luenberger indicator of dynamic productivity growth is illus-

rated graphically in Fig. 1 (for ease of exposition, it is assumed that

utput is the same in both periods; the capital stock K differs across

eriods). The quantities of inputs and investments at time t and time

+ 1 are denoted as (xt, It) and (xt+1, It+1), respectively. The dynamic

irectional distance function measures the distance to the isoquants

t time t and time t + 1, which is denoted as �Dit

(yt, Kt, Lt, xt, It; gx, gI

).

he Luenberger indicator of dynamic productivity growth can be

ecomposed into the contributions of technical inefficiency change

�TEI) and technical change (�T):

P(·) = �T + �TEI (3)

he decomposition of productivity growth is obtained from Eq. (2)

y adding and subtracting the term [ �Dit+1(yt+1, Kt+1, Lt+1, xt+1, It+1;

x, gI)− �Dit(yt, Kt, Lt, xt, It; gx, gI)]. Technical change is computed as

he arithmetic average of the difference between the technology (rep-

esented by the frontier) at time t and time t + 1, evaluated using

uantities at time t [first two terms in Eq. (4)] and time t + 1 [last two

erms in Eq. (4)]:

T = 1

2

⎧⎪⎪⎪⎨⎪⎪⎪⎩

[ �Dit+1(yt, Kt, Lt, xt, It; gx, gI)

− �Dit(yt, Kt, Lt, xt, It; gx, gI)]

+ [ �Dit+1(yt+1, Kt+1, Lt+1, xt+1, It+1; gx, gI)

− �Dit(yt+1, Kt+1, Lt+1, xt+1, It+1; gx, gI)]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(4)

echnical change can be seen in Fig. 1 as the average distance be-

ween the two isoquants. This involves evaluating the isoquants using

uantities at time t, �Dit+1(yt, Kt, Lt, xt, It; gx, gI)− �Di

t(yt, Kt, Lt, xt, It;

x, gI), and quantities at time t + 1, �Dit+1(yt+1, Kt+1, Lt+1, xt+1, It+1;

x, gI)− �Dit(yt+1, Kt+1, Lt+1, xt+1, It+1; gx, gI). Dynamic technical inef-

ciency change is the difference between the value of the dynamic

irectional distance function at time t and time t + 1:

TEI = �Dit

(yt, Kt, Lt, xt, It; gx, gI

)− �Di

t+1

(yt+1, Kt+1, Lt+1, xt+1, It+1; gx, gI

)(5)

echnical inefficiency change is easily seen from Fig. 1 as the differ-

nce between the distance functions evaluated using quantities and

echnologies in period t and period t + 1.

A. O. Lansink et al. / European Journal of Operational Research 241 (2015) 555–563 557

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8

. The dual Luenberger indicator of dynamic productivity growth

It is assumed that firms are intertemporally cost minimizing and

hus they take their decisions in accordance with the following opti-

ization problem:

t(yt, Kt, Lt, wt, ct) = minx,I

∫ ∞

t

e−rt[w′txt + c′

tKt]dt

.t.

˙ t = It − δKt

�i(yt, Kt, Lt, xt, It; gx, gI

) ≥ 0

(6)

here wt ∈ �N++ is a variable input price vector, ct ∈ �F++ is a vector

f capital rental prices, δ is a diagonal matrix containing depreciation

ates and r is the discount rate. Within this framework, Kt is a vector

f initial capital stocks at a certain point in time. Capital is acquired

hrough gross investment, It which depreciates at a fixed proportional

ate, δ. Under our dynamic cost minimization framework, we assume

hat firms choose investment so as to minimize the present value of

he sum of future production costs over an infinite time horizon.

The Hamilton–Jacobi–Bellman equation corresponding to the

ptimization program can be expressed as:

Wt

(yt, Kt, Lt, wt, ct

)= min

x,I

{w′

txt + c′tKt + WKt(yt, Kt, Lt, wt, ct)′(It − δKt)

+λ �Dit(yt, Kt, Lt, xt, It; gx, gI)

}(7)

here WKt(yt, Kt, Lt, wt, ct) is the firm-specific, shadow value of cap-

tal and the Lagrangian multiplier, λ, can be shown to be the cost

ndicator λ = WKt(yt, Kt, Lt, wt, ct)′gI − w′tgx (Silva et al., 2009). The

ynamic dual form of the Luenberger dynamic productivity growth

ndicator is formulated in terms of the differences between observed

osts and minimum costs as follows,

D(·) = 1

2

⎡⎣ (w′

txt+c′tKt+WKt+1,t(It−δKt))−rWt+1(yt ,Kt ,Lt ,wt ,ct)

wtgx−WK t+1,tgI

+ (w′txt+c′

tKt+WKt(It−δKt))−rWt(yt ,Kt ,Lt ,wt ,ct)wtgx−WKtgI

⎤⎦

− 1

2

⎡⎢⎣

(w′t+1xt+1+c′

t+1Kt+1+WKt+1(It+1−δKt+1))−rWt+1(yt+1,Kt+1,Lt+1,wt+1,ct+1)

wt+1gx−WK t+1gI

+ (w′t+1xt+1+c′

t+1Kt+1+WKt ,t+1(It+1−δKt+1))−rWt(yt+1,Kt+1,Lt+1,wt+1,ct+1)

wt+1gx−WKt ,t+1gI

⎤⎥⎦

(8)

his indicator computes the arithmetic mean of two components.

he first component consists of two ratios in which the second ra-

io measures the difference between observed shadow cost of pro-

uction at time t SCt = wtxt + ctKt + WKt(It − Kt), and the minimum

hadow cost measured by the optimal value function at time t using

he prices in time t [i.e., rWt(yt, Kt, Lt, wt, ct)]. The first ratio mea-

ures the difference between the observed and minimum shadow

osts using prices and quantities at time t and the frontier in t + 1.

he differences between the observed and minimum shadow costs

n the first and second ratios are scaled by the shadow value of the

irection vector, implying that the ratios are unit free. Note that the

hadow price of capital, WKt+1,t , in the first ratio is measured from

he cost frontier at time t + 1 and prices and quantities at time t;

.e., WKt+1,t = WKt+1(yt, Kt, Lt, wt, ct). The third and fourth ratios are

imilar to the first two ratios and measure the difference between

bserved and minimum shadow costs using prices and quantities at

ime t + 1. The shadow price of capital (WKt,t+1) in the fourth ratio is

easured from the cost frontier at time t and prices and quantities at

ime t + 1 (i.e. WKt,t+1 = WKt(yt+1, Kt+1, Lt+1, wt+1, ct+1)). Note that

D(·) is only defined in case the denominator in Eq. (8) is non-zero.

his condition is satisfied if at least one of the directional vectors gx

nd gI are non-zero.

As in the primal case, the dual dynamic Luenberger productivity

ndicator can be decomposed to identify the contributions of tech-

ical change (�TD) and technical efficiency change (�TEI). But now

hat this measure embodies an optimization objective (intertemporal

ost minimization), we can additionally address the contribution of

llocative inefficiency change (�AEI) and the change in the shadow

alue of capital (�SV):

D(·) = �TD + �TEI + �AEI + �SV (9)

hile the first three components of the right-hand-side of Eq. (9) have

irect analogs to the static case, the component �SV requires some

laboration. Once we allow for disequilibirum in quasi-fixed factor

se, it is clear from Eq. (7) that the notion of an internally generated

hadow price of capital plays the role of a price component for the

et investement infusions. In particular, changes in the captial stock

ead to shifts in the shadow price of capital, which must be addressed

n the productivity growth indicator.

Technical change is computed as the arithmetic mean of the nor-

alized distance between the optimal value frontiers, evaluated at

rices and quantities in period t and period t + 1, respectively, as

TD(·) = 1

2

[(rWt(yt, Kt, Lt, wt, ct))

wtgx − WKtgI− rWt+1(yt, Kt, Lt, wt, ct)

wtgx − WKt+1,tgI

]

+ 1

2

[rWt(yt+1, Kt+1, Lt+1, wt+1, ct+1)

wt+1gx − WKt ,t+1gI

− rWt+1(yt+1, Kt+1, Lt+1, wt+1, ct+1)

wt+1gx − WKt+1gI

](10)

The overall inefficiency change is given by:

LOEI(·)

=⎡⎣ (w′

txt+c′tKt+WKt(It−δKt))−rWt(yt ,Kt ,Lt ,wt ,ct)

wtgx−WKtgI

− (w′t+1xt+1+c′

t+1Kt+1+WKt+1(It+1−δKt+1))−rWt+1(yt+1,Kt+1,Lt+1,wt+1,ct+1)

wt+1gx−WKt+1gI

⎤⎦

(11)

Allocative inefficiency change can be identified as the difference

etween overall inefficiency change Eq. (11) and the primal estimate

f technical inefficiency change in Eq. (5):

AEI = �LOEI − �TEI (12)

The component to indicate a change over time is the shadow value

f capital, WK. The change in the capital stock is driving changes in

K. In the case of the dual form of the dynamic Luenberger produc-

ivity indicator, the last component is the change in shadow cost of

roduction, SCt , which is driven by the change in the shadow value of

apital, which yields

SV(·) = 1

2

[(w′

txt + c′tKt + WKt+1,t(It − δKt))

wtgx − WKt+1,tgI

− (w′txt + c′

tKt + WKt(It − δKt))

wtgx − WKtgI

]

+ 1

2

[(w′

t+1xt+1 + c′t+1Kt+1 + WKt+1(It+1 − δKt+1))

wt+1gx − WKt+1gI

− (w′t+1xt+1 + c′

t+1Kt+1 + WKt ,t+1(It+1 − δKt+1))

wt+1gx − WKt ,t+1gI

]

(13)

. Empirical application

Our empirical application focuses on a sample of specialized dairy

arms in the Netherlands. Farm-level data are obtained from the Eu-

opean Commission’s Farm Accountancy Data Network (FADN) and

over the period 1995–2005. To ensure that milk output is the main

arm output, we select those farms whose milk sales represent at least

0 percent of total farm income. The dataset is an unbalanced panel

558 A. O. Lansink et al. / European Journal of Operational Research 241 (2015) 555–563

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that contains 2614 observations on 639 farms that, on average, stay

in the sample during 4 years.

We distinguish one output, two variable inputs, two quasi-fixed

inputs and two fixed inputs to keep the vector of estimated parame-

ters to a manageable size. Output, y, is defined as a farm’s total output

and includes milk, livestock and livestock products, crops and crop

products and other output. The two variable inputs are variable costs

other than feed, x1, and feed expenses, x2. Variable x1 is an aggre-

gate input that includes veterinary expenses, energy, contract work,

crop-specific costs and other variable input costs. Breeding livestock

is considered as a quasi-fixed input, K1. Machinery and buildings, also

defined as quasi-fixed inputs, are aggregated into K2. Variables y, x1,

x2, K1 and K2 are measured at constant 1995 prices. Total utilized

agricultural area, L1, measured in hectares, and total labor input, L2

measured in annual working units (AWU), are assumed to be fixed

inputs. Labor was assumed to be a fixed input because approximately

95 percent of the labor input was coming from the farm family in the

sample period.

Since output and input prices are unavailable from FADN, country-

level price indices are taken from Eurostat’s New Cronos Dataset. Net-

puts measured in monetary values are defined as implicit quantity in-

dices by computing the ratio of value to its corresponding Tornqvist

price index. Depreciation rates considered for buildings, machinery

and breeding livestock are 3 percent, 10 percent and 25 percent, re-

spectively. The interest rate (r) is defined as the average, over the

period 1995–2005, of the annual interest rate for 10 years’ maturity

government bonds (Eurostat) and is equal to 4.97 percent. Following

Epstein and Denny (1983), Pietola and Myers (2000) and others, we

assume that the current price of a quasi-fixed input can be derived

as the discounted sum of the future rents on the depreciated asset.

Based on this assumption, the rental cost price of capital is measured

as ci = (r + δi)zi, where zi is the quasi-fixed asset price (defined as a

Tornqvist price index).

Table 1 provides descriptive statistics for the variables used in the

analysis. With quantity and price indexes used to construct the data,

milk production accounts for 90 percent of the value of production,

and variable expenses (of which 40 percent are feed expenses) are

43 percent relative to the value of production. The observed long-

run cost represents almost 70 percent of total value of output. While

breeding livestock gross investments are substantial, I1, net invest-

ments, K̇1, represent only 0.25 percent of K1, which is due to the milk

quota system regulating EU’s dairy sector and limiting this sector’s

growth. It should be noted though that the dairy quota system in the

Netherlands allows farms to continue to grow by buying or leasing

additional quota. Although the milk quota limits the possibilities for

Table 1

Descriptive statistics for the variables used in the analysis

Variable Description

Y Total output (index)

C Observed long-run cost (index)

K1 Breeding livestock (index)

K2 Buildings and machinery (index)

L1 Land (hectares)

L2 Labor (AWU)

x1 Variable inputs, except feed (index)

x2 Feed (index)

I1 Gross investments in breeding livestock (in

I2 Gross investments in machinery and buildin

K̇1 Net investments in breeding livestock (inde

K̇2 Net investments in machinery and building

p Output price (index)

w1 Variable inputs’ price (excluding feed) (inde

w2 Feed price (index)

c1 Breeding livestock rental price (index)

c2 Machinery and buildings rental price (index

Number of observations: 2614.

rowth of the dairy herd, it does not prevent modernization of dairy

oldings that, on average, have net investments in machinery and

uildings of almost 7 percent per year.

The estimation of the dynamic directional distance function and

ynamic cost function can be done parametrically as in this study or

onparametrically. The nonparametric approach more easily allows

or a further decomposition of productivity into the contributions of

cale and congestion (see Epure, Kerstens, & Prior, 2011). In addi-

ion, the nonparametric approach allows for firm-specific measure-

ents of technical change, while the parametric approach requires

ssuming equal technical change across the farms in the sample or

roups of farms within the sample as in latent class models. However,

he nonparametric approach can be subject to computational issues

hen there is not wide variation in the benchmark technology which

s largely determined by those firms identifying the boundary; i.e.,

(y(t)|K(t)) such that �D(yi, Ki, xi, Ii; gx, gI) ≥ 0. With the computa-

ional problem being conditioned on input–output bundles and the

ata set for the dynamic factors exhibiting limited variation over this

eriod, the dual based method can lead to limited variation in the

hadow values.

The empirical application builds on the parametric estimation of

he dynamic directional distance function presented in Serra et al.

2011), using the Dutch dairy farming data set described above. Quan-

ification of the dynamic directional distance and optimal value func-

ions was achieved by econometric estimation. Following Chambers

2002) and Färe, Grosskopf, Noh, and Weber (2005), the quadratic

unction was used as a parametric specification for the directional

istance function. Dynamic cost inefficiency is obtained by estimating

quadratic specification of the optimal value function. The empirical

odel and the results of the estimation are presented in Appendix A

nd are more elaborately discussed in Serra et al. (2011). The study

ielded a dynamic directional distance function that is increasing in

ariable, quasi-fixed and fixed inputs and decreasing in output and

nvestment demand, and a dynamic cost frontier that is increasing in

rices of variable and quasi-fixed inputs and decreases with capital

tock.

. Results

Table 2 presents the computation of the primal dynamic Luen-

erger productivity growth and its decomposition into the contri-

utions of technical change (�T) and technical inefficiency change

�TEI). Productivity grows on average with 1.5 percent per year, al-

hough the annual changes are fluctuating between −5.3 percent and

2.7 percent. The average productivity growth of 1.5 percent indicates

.

Mean Standard deviation

199,665.76 115,708.47

137,006.94 75,100.78

68,747.85 39,215.14

204,077.17 141,387.32

44.73 24.18

1.71 0.64

52,075.09 28,278.93

34,513.88 21,574.47

dex) 17,358.42 13,565.17

gs (index) 24,754.31 53,066.53

x) 171.46 7,115.17

s (index) 13,851.36 49,641.54

0.99 0.04

x) 1.16 0.11

0.99 0.04

0.27 0.02

) 0.12 0.01

A. O. Lansink et al. / European Journal of Operational Research 241 (2015) 555–563 559

Table 2

Primal dynamic Luenberger productivity growth

(LP(·)) and its decomposition in technical change (�T)

and technical inefficiency change (�TEI).

Year LP(·) �T �TEI

1996 − 0.012 0.012 − 0.024

1997 0.127 0.012 0.115

1998 − 0.053 0.012 − 0.065

1999 0.044 0.012 0.031

2000 0.001 0.012 − 0.011

2001 0.009 0.012 − 0.004

2002 − 0.010 0.012 − 0.022

2003 0.034 0.012 0.022

2004 0.002 0.012 − 0.011

2005 0.015 0.012 0.002

Mean 0.015 0.012 0.002

Small farms 0.011 0.012 − 0.001

Large farms 0.018 0.012 0.006

KS test 0.099∗∗ 0.000 0.099∗∗

∗∗ Statistical significance at the 5 percent level.

t

f

i

t

p

b

m

o

m

r

A

T

t

H

d

i

f

t

o

i

A

l

t

c

d

c

1

s

T

p

t

e

P

v

o

c

s

t

c

p

m

a

Fig. 2. Evolution of productivity (LP), technical change (�T) and technical inefficiency

change (�TEI) from 1996 (1) till 2005 (10).

Table 3

Dual Luenberger dynamic productivity growth (LD(·)) and its decomposition in

technical change (�TD), shadow value change (�SV), technical inefficiency change

(�TEI) and allocative inefficiency change (�AEI).

Year LD(·) �TD �SV �TEI �AEI

1996 0.010 0.004 0.000 − 0.024 0.030

1997 0.050 0.004 0.000 0.115 − 0.068

1998 − 0.065 0.004 0.000 − 0.065 − 0.004

1999 0.058 0.005 0.000 0.031 0.022

2000 − 0.020 0.005 − 0.000 − 0.011 − 0.014

2001 0.015 0.005 − 0.000 − 0.004 0.014

2002 − 0.037 0.005 0.000 − 0.022 − 0.020

2003 0.014 0.005 0.000 0.022 − 0.013

2004 0.025 0.005 − 0.000 − 0.011 0.031

2005 − 0.043 0.005 − 0.000 0.002 − 0.050

Mean − 0.001 0.005 0.000 0.002 − 0.008

Small farms − 0.001 0.005 0.000 − 0.001 − 0.004

Large farms − 0.001 0.005 0.000 0.006 − 0.011

KS test 0.156∗∗ 0.098∗∗ 0.175∗∗ 0.099∗∗ 0.160∗∗

∗∗ Statistical significance at the 5 percent level.

n

a

p

t

o

t

d

i

a

hat every year during the sample period 1995–2006, Dutch dairy

armers produced 1.5 percent more output from the same quantity of

nputs.

Technical change is 1.2 percent per year and is the major contribu-

or, on average, to improvement of productivity. Hence, technical im-

rovements allowed Dutch dairy farmers to increase their production

y 1.2 percent per year during the sample period. Technical improve-

ents could have come from improvements of the genetic potential

f the dairy cows, improvements in feeding and improvements in the

ilking technology such as the increasing adoption of the milking

obot (André, Berentsen, Engel, de Koning, & Oude Lansink, 2010;

ndré, Berentsen, van Duinkerken, Engel, & Oude Lansink, 2010).

echnical inefficiency increases on average to make a positive con-

ribution to productivity growth of, on average, 0.2 percent per year.

owever, the fluctuation in technical inefficiency is large and is the

river of the year to year changes in productivity. Productivity growth

s slightly larger for large dairy farms (1.8 percent) than small1 dairy

arms (1.1 percent), a difference that is attributable to the higher con-

ribution of technical inefficiency change on large dairy farms. This

utcome suggests that large dairy farms better succeeded in improv-

ng the use of the current production technology than small farms.

ccording to the Kolmogorov–Smirnov (KS) test, differences between

arge and small farm indicators are significant, with the exception of

he technical change indicator. The annual contributions of technical

hange and technical inefficiency change to productivity growth are

isplayed in Fig. 2. The figure clearly shows that technical inefficiency

hange (�TEI) is the driver of changes in productivity in the period

996–2005.

Moving to the dual dynamic Luenberger productivity growth mea-

ure can help clarify the movement of technical inefficiency change.

able 3 presents the computation of the dual dynamic Luenberger

roductivity growth and its decomposition into the contributions of

echnical change (�TD), shadow value change (�SV), technical in-

fficiency change (�TEI) and allocative inefficiency change (�AEI).

roductivity declines on average with 0.1 percent per year, though

arying between −6.5 percent and 5.8 percent. Hence, this outcome

f the dual model suggests that Dutch dairy farmers produced 0.1 per-

ent less output from the same quantity of inputs per year during the

ample period 1995–2006. Both the average size and range of produc-

ivity growth are smaller than in the primal model (Table 2). Technical

hange is around 0.5 percent per year and is also smaller than in the

rimal model, but still the major contributor, on average to improve-

ent of productivity. The change in the shadow value of capital has

1 A farm was classified as large or small, depending on whether its production was

bove or below the median.

F

(

f

o impact on productivity growth. Technical inefficiency decreases on

verage by 0.2 percent and delivers the second largest contribution to

roductivity growth. The fluctuation in technical inefficiency is coun-

eracted partly by reverse changes in allocative inefficiency. However,

n average, allocative inefficiency change (�AEI) has a negative con-

ribution to productivity growth. The annual contributions of the dual

ynamic Luenberger productivity growth components are displayed

n Fig. 3 which indicates that technical inefficiency change (�TEI)

nd allocative inefficiency change are the main drivers of changes in

ig. 3. Evolution of productivity (LD), technical change (�TED), shadow value change

�SV), technical inefficiency change (�TEI) and allocative inefficiency change (�AEI)

rom 1996 (1) till 2005 (10).

560 A. O. Lansink et al. / European Journal of Operational Research 241 (2015) 555–563

Fig. 4. Evolution of the milk price index from 1996 (1) till 2005 (10).

T

c

0

i

a

a

m

p

a

v

p

a

n

h

s

l

n

b

t

t

n

6

b

d

T

t

1

fi

o

D

s

f

t

b

h

c

l

t

the dual Luenberger dynamic productivity growth indicator in the

period 1996–2005. Dual dynamic Luenberger productivity growth is

almost equal for small and large dairy farms. Allocative inefficiency

change makes a relatively large negative (−1 percent) contribution to

productivity growth of large dairy farms, suggesting that large dairy

farms have more problems in adjusting inputs to long run optimal

levels than small dairy farms. According to the KS test, differences

across large and small dairy farms’ dual productivity indicators are

significant.

Being a first effort to generate primal and dual measures of dy-

namic Luenberger productivity growth, our results are not directly

comparable with productivity growth measures generated in previ-

ous research as they are based on static models. Brümmer, Glauben,

and Thijssen (2002) measured and decomposed productivity growth

of Dutch dairy farms over the period 1991–1994, finding that techni-

cal change contributed 0.5 percent to productivity growth, similar to

the contribution generated by the dual dynamic productivity growth

measure in our study. Also, they find that technical efficiency change

had a 0.6 percent contribution to productivity growth, a value that is

close to our finding of 0.3 percent. In their static model, allocative ef-

ficiency change had a much more positive contribution (1.7 percent)

to productivity growth than our dynamic model (−0.8 percent). This

divergence between our results and Brümmer et al (2002) suggests

that farmers have more problems in finding an efficient allocation of

inputs and outputs in the long-run than in the short-run. It should be

kept in mind though that the period under investigation in our study

does not coincide with that of the Brümmer et al.’s study.

Fluctuations in milk prices over the period of analysis (see Fig. 4),

may explain the difficulties of producers to allocate resources effi-

ciently from a technical and economic point of view in the long-run.

Fluctuations in productivity growth, mainly driven by technical and

allocative inefficiency changes, are negatively correlated with milk

price fluctuations.2 This may suggest that farmers are conservative

(pessimistic) regarding price expectations and they devise produc-

tion structures that are optimal in low price frameworks. As a result,

during years of bad prices, behavior is more efficient than in good

price years. To the extent that this hypothesis is correct, progressive

decoupling of EU policies, reducing price supports, may lower farmer

price expectations, which may exacerbate inefficiencies during high

price years.

Also, Oude Lansink and Zhu (2009) analyzed productivity growth

of Dutch dairy farms in the period 1995–2004 using a static model

of production. Their results suggest a higher technical change than

predicted by our dynamic primal and dual model (3.6 percent).

2 The correlation coefficient is −0.4. Please note that the milk price index is used in

the computation of the output quantity index.

p

n

t

he contribution of technical efficiency change and allocative effi-

iency change were different from our model, i.e. −0.5 percent and

.7 percent, respectively.

Several studies have now measured the composition of productiv-

ty growth using the sample of Dutch dairy farms. The dynamic primal

nd dual approaches account for the presence of costs of adjustment

ssociated with changes in quasi-fixed factors of production in the

easurement of productivity growth. Moreover, the dynamic dual ap-

roach provides a richer decomposition of productivity than the static

pproach by measuring the contribution of the change in the shadow

alue of capital. The results of this study show that the dynamic ap-

roaches yield different results than the static approaches that were

pplied to the same data set. The contribution of technical change is

otably lower, and the contribution of technical efficiency change is

igher in the dynamic approach rather than the static approach. This

uggests that adjustment costs associated with investments trans-

ate into a lower contribution of technical efficiency change, when

ot accounted for properly, whereas the benefits of investments, i.e.

etter technology are overstated through a higher contribution of

echnical change. The results of the dual dynamic model indicate that

he contribution of shadow value change to productivity growth was

egligible in the study sample.

. Conclusions

This paper develops primal and dual measures of dynamic Luen-

erger productivity growth that are based on the dynamic directional

istance function and intertemporal cost minimization, respectively.

he empirical illustration focuses on a panel of Dutch dairy farms over

he period 1995–2005.

Average primal dynamic Luenberger productivity growth is

.5 percent in the period under investigation, with technical inef-

ciency change being the main driver of change. Productivity growth

f large farms is higher (1.8 percent) than of small farms (1.1 percent).

ual dynamic Luenberger productivity growth is −0.1 percent in the

ame period. Productivity growth of large dairy farms and small dairy

arms are almost equal (−0.1 percent). In the period under investiga-

ion, improvements in technical inefficiency are partly counteracted

y deteriorations of allocative inefficiency. Particularly, large farms

ave a negative (−1 percent ) contribution of allocative inefficiency

hange to productivity growth, suggesting that finding an optimal al-

ocation of inputs in the long-run is more problematic for large rather

han small dairy farms.

This study has demonstrated the value of the dynamic Luenberger

roductivity indicators in that it allows for identifying the compo-

ents of productivity growth. The dual dynamic Luenberger produc-

ivity growth indicator allows for a richer decomposition than the

A. O. Lansink et al. / European Journal of Operational Research 241 (2015) 555–563 561

p

e

m

i

c

g

i

R

i

p

f

o

t

t

A

w

u

t

i

f

t

s

t

h

D

P

p

∑

∑

∑

∑

S

a

d

m

c

0

w

I

−

F

q

s

f

F

n

c

C

w

i

t

s

d

r

δW

W

w

a

t

3 This variable represents gross investment in breeding livestock. Parameter esti-

rimal indicator as it also identifies contributions from allocative in-

fficiency change and change in the shadow value of capital.

While this study focuses on the dynamic microeconomic decision

aking implications for efficiency and productivity, there are several

nteresting policy-related issues that can be gleaned from the empiri-

al analysis. Technical change is a principal contributor to productivity

rowth followed by technical efficiency change. With technological

nnovations in this sector being driven externally, publicly supported

&D activities and policies that can facilitate private sector R&D be-

ng translated into marketed innovations are potential productivity-

romoting actions. Knowledge translating and outreach activities to

armers can certainly promote efficiency improvements. Further, rec-

gnizing that large farms can benefit from allocative efficiency gains,

argeted knowledge in translating and outreach activities tailored to

heir scope of activities can be fruitful.

ppendix A. Empirical model and results of estimation

The quantification of the dynamic directional distance function

as achieved by econometric estimation. The quadratic function was

sed as a parametric specification for the directional distance func-

ion as it offers the advantage that it can be easily restricted to sat-

sfy the translation property. The quadratic specification is a flexible

unctional form that is twice differentiable in all its arguments. Set-

ing gxi = 1, i = 1, . . . , N, gIj = 1, j = 1, . . . , F, M = 1 (i.e., we assume a

ingle-output firm) and including a time trend (t), the distance func-

ion for the firm h can be expressed as follows, where time indicators

ave been ignored for simplicity:

�ih(y, K, L, x, I, t; 1, 1) = a0 + ayy +

C∑n=1

aLnLn +F∑

j=1

aIjIj +N∑

i=1

axixi

+F∑

j=1

aKjKj + 1

2ayyy2 + 1

2

C∑n=1

C∑n′=1

aLnLn′ LnLn′ + 1

2

F∑j=1

F∑j′=1

aIjIj′ IjIj′

+ 1

2

N∑i=1

N∑i′=1

axixi′ xixi′ + 1

2

F∑j=1

F∑j′=1

aKjKj′ KjKj′ +C∑

n=1

ayLnyLn

+F∑

j=1

ayIjyIj +N∑

i=1

ayxiyxi +F∑

j=1

ayKjyKj +C∑

n=1

F∑j=1

aLnIjLnIj

+C∑

n=1

N∑i=1

aLnxiLnxi +C∑

n=1

F∑j=1

aLnkjLnKj+F∑

j=1

N∑i=1

aIjxiIjxi

+F∑

j=1

F∑j′=1

aIjKj′ IjKj′+F∑

j=1

N∑i=1

aKjxiKjxi + att (A.1)

arameter restrictions that need to be imposed for the translation

roperty to hold are:

F

j=1

aIj −N∑

i=1

axi = −1;

F∑j=1

F∑j′=1

aIjKj′−F∑

j=1

N∑i=1

aKjxi = 0;

F

j=1

ayIj −N∑

i=1

ayxi = 0; −N∑

i′=1

axixi′ +F∑

j=1

aIjxi = 0, i = 1, . . . , N;

F

j′=1

aIjIj′ −N∑

i=1

aIjxi = 0, j = 1, . . . , F; and

C

n=1

F∑j=1

aLnIj −C∑

n=1

N∑i=1

aLnxi = 0.

ymmetry restrictions are also imposed: aLnLn′ = aLn′Ln, aIjIj′ = aIj′Ij,

xixi′ = axi′xi, and aKjKj′ = aKj′Kj.

m

Following Kumbhakar and Lovell (2000) and Färe et al. (2005), the

ynamic quadratic directional input distance function can be esti-

ated using stochastic estimation techniques. The stochastic specifi-

ation of the distance takes the following form:

= �Dih(y, K, L, x, I, t; 1, 1)+ εh (A.2)

here εh = vh − uh, vh ∼ N(0, σ 2v ) is white noise and uh ∼ N+(0, σ 2

u ).n order to estimate Eq. (A.2), the translation property is used:

αh = �Dih(y, K, L, x − αh, I + αh, t; 1, 1)+ εh (A.3)

unction �Dih(y, K, L, x − αh, I + αh, t; 1, 1) corresponds to the

uadratic form in Eq. (A.1), with αh added to gross investments and

ubtracted from variable input quantities. By choosing αh specific

or each firm, variation on the left hand side of Eq. (A.3) is obtained.

ollowing Färe et al. (2005), αh is set equal to I1, which is the

ormalizing input in determining technical efficiency.3

Dynamic cost inefficiency is obtained by estimating the following

ost frontier model, (including a time trend):

h = rW

(y, K, L,

w2

w1,

c

w1, t

)− Wk

(y, K, L,

w2

w1,

c

w1, t

)′K̇

− Wt

(y, K, L,

w2

w1,

c

w1, t

)+ ξh (A.4)

here Ch = (w′x + c′K)/w1 is the observed long-run cost normal-

zed by the variable input price w1, W(y, K, L,w2w1

, cw1

, t) is the op-

imum cost where all input prices have been normalized with re-

pect to w1, Wk(y, K, L,w2w1

, cw1

, t)and Wt(y, K, L,w2w1

, cw1

, t)are its first

erivatives with respect to K and t respectively. The composite er-

or component is specified as ξh = γh + δh, where γh ∼ N(0, σ 2γ ), and

h ∼ N+(0, σ 2δ). By normalizing all input prices with respect to w1,

(·) is specified as:

(y, K, L,

w2

w1,

c

w1, t

)= b0 + byy + bw2

w2

w1+

F∑j=1

bcj

cj

w1

+F∑

j=1

bkjKj+C∑

n=1

bLnLn + 1

2byyy2 + 1

2bw2w2

(w2

w1

)2

+ 1

2

F∑j=1

F∑j′=1

bcjcj′cj

w1

cj′

w1+ 1

2

F∑j=1

F∑j′=1

bkjkj′ KjKj′

+ 1

2

C∑n=1

C∑n′=1

bLnLn′ LnLn′ + byw2y

w2

w1+

F∑j=1

bycjycj

w1

+F∑

j=1

bykjyKj+C∑

n=1

byLnyLn+F∑

j=1

bw2cjw2

w1

cj

w1+

F∑j=1

bw2kj

w2

w1Kj

+C∑

n=1

bw2Lnw2

w1Ln+

F∑j=1

F∑j′=1

bkjcj′ Kj

cj′

w1+

F∑j=1

C∑n=1

bcjLn

cj

w1Ln

+F∑

j=1

C∑n=1

bkjLnKjLn + btt (A.5)

ith the symmetry restrictions bcjcj′ = bcj′cj, bkjkj′ = bkj′kj, bLnLn′ = bLn′Ln,

nd bkjcj′ = bkj′cj imposed.

Results of the estimation of the directional distance function and

he cost function are presented in Tables A.1 and A.2.

ates changed very little with the choice of αh however.

562 A. O. Lansink et al. / European Journal of Operational Research 241 (2015) 555–563

Table A.1

Directional distance function parameter estimates.

Parameter Estimate Standard error Parameter Estimate Standard error

a0 − 4.85E−02∗∗ 2.39E−02 aI1x2 1.18E−01∗∗ 3.95E−02

ay − 1.02E+00∗∗ 5.71E−02 aI1I2 4.50E−03 4.79E−03

aL1 3.70E−01∗∗ 4.80E−02 aK1K1 − 2.36E−01∗ 1.25E−01

aL2 − 1.31E−01∗∗ 4.84E−02 aK1K2 1.69E−02 5.13E−02

ax1 3.87E−01∗∗ 3.33E−02 aK2K2 − 3.85E−03 2.25E−02

ax2 5.49E−01∗∗ 3.67E−02 ayI2 − 1.35E−02 1.19E−02

aI2 − 1.72E−02∗∗ 8.63E−03 ayx1 1.03E−01∗ 6.19E−02

ak1 − 6.28E−02 6.90E−02 ayx2 − 2.18E−01∗∗ 7.85E−02

ak2 4.96E−02∗ 3.01E−02 ayK1 2.28E−02 7.90E−02

ayy 5.48E−01∗∗ 1.55E−01 ayK2 − 6.53E−03 6.60E−02

aL1L1 1.01E−01∗ 5.48E−02 aI1K2 − 2.28E−02 2.45E−02

aL1L2 − 1.56E−01∗∗ 4.76E−02 aI2K1 2.61E−02 3.02E−02

aL2L2 − 4.83E−02 5.77E−02 aI2K2 − 1.84E−02 3.03E−02

ax1x1 − 3.50E−01∗∗ 4.45E−02 aK1x2 − 4.73E−02 7.43E−02

ax2x2 − 1.13E−01∗ 6.24E−02 aK2x1 − 8.60E−02∗ 4.51E−02

ax1x2 2.23E−01∗∗ 3.92E−02 aK2x2 2.08E−02 3.93E−02

aI2x1 1.27E−02∗ 5.94E−03 aK1x1 1.94E−01∗∗ 7.49E−02

ayL1 1.40E−01∗ 7.97E−02 aL2x2 − 2.92E−02 5.85E−02

ayL2 − 1.99E−01∗∗ 7.65E−02 aL1K1 − 1.54E−01∗∗ 7.85E−02

aL1I2 − 2.63E−01∗∗ 2.35E−02 aL2K1 2.60E−01∗∗ 8.17E−02

aL2I1 − 1.37E−01∗∗ 3.55E−02 aL1K2 1.34E−02 3.53E−02

aL2I2 2.65E−01∗∗ 2.37E−02 aL2K2 1.93E−02 3.68E−02

aL1x1 − 2.94E−01∗∗ 4.78E−02 at 7.32E−02∗∗ 6.44E−03

aL1x2 − 1.03E−01∗∗ 4.95E−02 σε 1.97E−01∗∗ 8.15E−03

aL2x1 4.24E−01∗∗ 5.30E−02 λε 1.53E+00∗∗ 2.12E−01

∗ Statistical significance at the 10 percent level.∗∗ Statistical significance at the 5 percent level.

Table A.2

Cost function parameter estimates.

Parameter Estimate Standard error Parameter Estimate Standard error

b0 5.20E+00 1.56E+01 byc2 − 5.52E+00 4.14E+00

by 5.70E+00 4.42E+00 byL1 − 2.25E+00 1.44E+00

bw2 − 5.40E+01∗∗ 1.29E+01 byL2 6.74E−01 1.26E+00

bc1 − 2.70E+00 8.90E+00 bw2c1 − 8.22E+01∗∗ 2.54E+01

bc2 2.35E+01∗∗ 9.18E+00 bw2c2 2.24E+00 7.30E+00

bL1 3.33E+00 4.29E+00 bw2L1 − 1.05E+01∗∗ 3.82E+00

bL2 − 6.66E+00∗ 3.77E+00 bw2L1 5.11E+00 3.38E+00

byy 1.69E+00 1.97E+00 bc1L1 3.63E+00 3.46E+00

bw2w2 1.29E+02∗∗ 3.20E+01 bc2L1 4.21E+00 3.67E+00

bc1c1 8.38E+01∗∗ 2.69E+01 bc1L2 − 7.49E−01 2.25E+00

bc1c2 5.18E+00 6.20E+00 bc2L2 1.71E−01 3.00E+00

bc2c2 − 2.84E+01∗∗ 8.52E+00 bk1 − 1.98E−03 2.49E−03

bL1L1 8.86E−01 1.14E+00 bk2 7.43E−03 2.52E−02

bL1L2 1.09E+00 9.22E−01 bk1k1 − 8.81E−04 8.77E−04

bL2L2 8.08E−01 1.14E+00 bk1k2 − 4.72E−04 5.24E−04

byw2 2.01E+01∗∗ 4.14E+00 bk2k2 − 8.85E−03∗∗ 2.21E−03

byc1 − 3.89E+00 3.86E+00 byk1 1.70E−03 1.54E−03

byk2 1.46E−02∗∗ 5.50E−03 bk2L1 − 1.88E−03 4.51E−03

bw2k1 1.83E−03 2.01E−03 bk2L2 8.99E−03 6.33E−03

bw2k2 2.52E−02 1.68E−02 bt − 5.66E−01 7.19E−01

bk1c1 − 7.18E−04 1.47E−03 σξ 1.92E−01∗∗ 1.06E−02

bk2c1 4.93E−04 1.67E−03 λξ 1.83E+00∗∗ 3.39E−01

bk2c2 − 5.15E−02∗∗ 2.09E−02

bk1L1 − 3.47E−04 5.53E−04

bk1L2 3.69E−04 5.28E−04

∗ Statistical significance at the 10 percent level.∗∗ Statistical significance at the 5 percent level.

B

B

B

C

C

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