Improvement of Technical Efficiency of Firm Groups

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Improvement of Technical Efficiency of Firm
Groups
Louisa Andriamasy∗
Caepem
Universit´e de Perpignan
Walter Briec†
Caepem
Universit´e de Perpignan
St´ephane Mussard‡
Chrome
Universit´e de Nˆımes
Abstract
Cooperation between firms can never improve the technical effi-
ciency of any coalition of firms. This standard result of the pro-
ductivity measurement literature is based on the directional distance
function computed on firm groups. Directional distance functions are
usually defined on the standard sum of input/output vectors. In this
paper, the aggregation of input/output vectors is generalized thanks
to an isomorphism in order to capture three results: the coopera-
tion improves technical efficiency ; the cooperation reduces technical
efficiency ; and finally the cooperation between firms yields no varia-
tion of technical efficiency, i.e., the distance function is quasi linear.
The improvement of technical efficiency is shown to be compatible
with semilattice technologies. In this case, the firms merge according
to their inputs only because constraints are imposed on outputs, and
conversely, they may merge according to the outputs they can produce
because some limitations are imposed on the use of inputs.
Keywords: Aggregation, Cooperative games, Distance functions, Produc-
tivity, Technical efficiency.
JEL Codes: D21, D24.
∗Caepem, IAE, Universit´e de Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860
Perpignan Cedex ; Email: andriamasy@univ-perp.fr
†Caepem, IAE, Universit´e de Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860
Perpignan Cedex ; Email: briec@univ-perp.fr
‡Chrome Universit´e de Nˆımes, Rue du Dr Georges Salan, 30000 Nˆımes, - e-mail:
mussard@lameta.univ-montp1.fr, Research fellow Lameta Universit´e de Montpellier,
Gr´
edi Universit´e de Sherbrooke, Liser Luxembourg.
1

1 Introduction
The cooperation between firms can never improve the technical efficiency of
any given coalition (industry). This impossibility has become a standard
result in the productivity measurement literature, see Briec, Dervaux and
Leleu (2003) or F¨are, Grosskopf and Zelenyuk (2008). The firm game is
the transferable utility game (TU-game) that exhibits this impossibility (see
Briec and Mussard, 2014), in other words, the core interior of the firm game
is empty. The result is based on the directional distance functions applied
to the technology of the industry, which relies on the standard sum of sets
(technologies of the firms), see e.g. F¨are, Grosskopf and Li (1992) and Li and
Ng (1995). Li and Ng (1995) point out that the standard sum of sets can yield
different results, in particular, it depends on the convexity of the technology
set. Li and Ng’s (1995) result follows the one due to Førsund and Hjalmarsson
(1979) in which inputs/outputs are simply averaged thanks to an arithmetic
mean. Li and Ng (1995) show that this particular aggregation provides a
bad representation of the technical efficiency of the industry. In other words,
the choice of the aggregation process, for aggregating input/outputs vectors
or equivalently aggregating technologies of firms, has a crucial impact on the
productivity measures. Since then, this question has been pending for a long
time, and from the best of our knowledge, no attempt has been made to
overcome this problem of aggregation, so that the standard sum is always
used in the efficiency literature to analyze the cooperation between firms.
In cooperative games, Lozano (2012) show that firm groups may take
benefit from cooperation when they share data about inputs and outputs.
The firms have also the possibility to merge in a so-called production games
(Lozano, 2013). In firm games, Briec and Mussard (2014) show that any
given firm coalition may always improve its allocative efficiency if the in-
put/output vectors are simply aggregated with the standard sum. However,
the impossibility outlines before is always met, i.e., the inefficiency of the
industry is always greater than the sum of the inefficiencies of each firm. In
other words, the technical bias that represents the difference between the
two aforementioned inefficiencies, is always positive. Coalitions of firms are
said to be sub-efficient because their cooperation increases the technical inef-
ficiency of the group. As a consequence, the core of the firm game is empty:
no firm can improve its technical efficiency by joining any given coalition.
In Data Envelopment Analysis (DEA), Post (2001) suggests a concave
transformation of the input/output vectors in order to limit the number
of observations to be non attainable, that is, input/output combinations
being outside the envelopment of the data that characterizes the production
technology. This point is crucial because decisions are taken on the basis of
samples. If the sample size is low and if an important quantity of data is
not exploitable by the current mathematical techniques, then it is difficult
2

to propose accurate indicators for taking decisions. We aim at extending
Post’s (2001) suggestion about data transformation in a cooperative game
framework, without taking recourse to DEA. The interesting feature relying
on the change of variables of input/output vectors is the derivation of a
flexible aggregation process, an aggregator from now on, that captures either
improvement or decline of technical efficiency of firm coalitions.
The aggregation of technologies is generalized thanks to an aggregator,
precisely a Φ-aggregator, inspired from Ben-Tal (1977) who studied the al-
gebraic properties of the aggregation process Φ underlying the generalized
mean introduced by Hardy, Littlewood and P´olya (1934) and characterized
by Acz´el (1966) and Eichorn (1979). The aggregation bias, i.e. the difference
between the inefficiency of the firm coalition and the sum of each firm’s inef-
ficiency, takes different forms with respect to the nature of the isomorphism
Φ. (i) The aggregation bias is null: there is no variation of technical efficiency
inherent to the cooperation. We retrieve the well-known result due to Briec
et al. (2003) and F¨are et al. (2008) as a special case: the directional distance
function is (quasi) linear. (ii) The bias is positive, so that the cooperation
between firms is impossible. (iii) The bias is negative: the aggregate ineffi-
ciency of any given coalition of firms decreases with cooperation. The core
of the firm game may be non void in this case. For that purpose, we begin
with a general (non specified) distance function and a general aggregation
process in order to understand the implications of a null bias when inputs
and outputs are transformed by a general isomorphism Φ. The conclusion
is clear, the distance function is quasi linear, i.e. there is no gain/loss for
a firm to join a coalition. Consequently, the employ of a Φ-aggregator is
robust for the measurement of group (technical) efficiency for any given dis-
tance functions. Also, postulating the homogeneity property of the distance
function – such as the well-known directional distance function, introduced
by Chambers, Chung and F¨are (1996, 1998) – the Φ-aggregator is found to
be the generalized mean.
On the basis of the generalized mean aggregator, two limit cases – in
the neighborhood of infinity – are introduced. The first one enables Kholi’s
(1983) technology to be characterized for a group of firms. It is a particular B-
convex technology, introduced by Briec and Horvath (2009). This coalitional
technology is shown to be consistent with the traditional assumptions of the
literature. It is a compact upper semilattice respecting a free disposal as-
sumption. The second one is a particular B−1-convex technology introduced
by Briec and Liang (2011). It is a compact lower semilattice also relevant
with a free disposal assumption. It is shown that those two aggregated tech-
nology sets enable the paradox of the positive technical bias to be solved.
Indeed, the negative bias is obtained by specifying two firm games, the in-
put fixed firm game and the output fixed firm game. The input fixed firm
game postulates that the cooperation between firms is related to the use of
3

inputs only, because some constrains of production are imposed on the indus-
trial sector in order to limit the number of outputs (pollution limitations).
On the contrary, the output fixed firm games is defined on the possibility
to improve the amount of outputs when the firms are limited by a given
amount of inputs (resource limitations). Those results are derived thanks to
the directional distance function applied on the aggregated data. The input
[output] firm game defined on upper semilattice technologies yields a nega-
tive [positive] bias. On the contrary, the output [input] firm game defined
on lower semilattice technologies yields a negative [positive] bias. Finally,
if the aggregation bias is supposed to be submodular, then the core of the
game is always non empty, i.e., the joint cooperation improves the technical
efficiency (negative bias).
The outline of the paper is as follows. Section 2 sets the notations. Section
3 defines the firm game and the directional distance function. Section 4
is devoted to the characterization of the aggregated technology when the
data are transformed by an isomorphism `a la Ben-Tal (1977). Then, the
results about the exact aggregation are exposed for the directional distance
function. Section 5 explores the negative bias supported by the directional
distance function. Section 6 introduces semilattice technologies, where it is
shown that coalitions of firms may increase their technical efficiency either by
putting in common their inputs or their outputs (input/output firm games).
Section 7 closes the article.
2 Setup
The set of firms (players) is K:= {1, . . . , |K|}, where |K| ≡ #{K}. The
subsets of the grand coalition Kare denoted by S. A transferable utility
game, i.e. a TU-game, is a pair (K, v), where vis defined as v: 2|K| →R+
such that v(∅) := 0, with R+the non-negative part of the real line and R++
its positive part (with Rn
+and Rn
++ its n-dimensional representation). The
set of all maps vis denoted Γ, such that v(S) provides the worth of coalition
S. A valued solution ϕ(v) is the pay-off vector of the TU-game (K, v) that
is a |K|-dimensional real vector that represents what the firms could take
benefit from cooperation. The valued solution of the TU-game is assumed
to satisfy the standard axioms.
Linearity:ϕ(α1v1+α2v2) = α1ϕ(v1) + α2ϕ(v2), for all maps v1, v2∈Γ and
α1, α2∈R.
Symmetry: for any given pay-off vector ϕ= (ϕ1, . . . , ϕk, . . . , ϕ|K|), then
ϕk(v) = ϕπ(k)(v) for all permutations, where a permutation is given by
v(π(S)) = v(S) for all S ⊆ K and v∈Γ.
Efficiency:Pk∈K ϕk(v) = v(K), for all v∈Γ.
4

Let x∈Rn
+and y∈Rm
+be the input and output vectors, respectively.
The technology Tof the firms satisfies the following basic assumptions:
(T1): (0n,0m)∈T, (0n, y)∈T=⇒y= 0mi.e., no free lunch;
(T2): the set A(x) = {(u, y)∈T:u6x}of dominating observations is
bounded ∀x∈Rn
+,i.e., infinite outputs cannot be obtained from a
finite input vector;
(T3): Tis closed;
(T4): ∀z= (x, y)∈T, (x, −y)6(u, −v) =⇒(u, v)∈T,i.e., fewer outputs
can always be produced with more inputs, and inversely;
(T5): ∀β≥0, if (x, y)∈Tthen (β x, βy)∈T,i.e. the technology satisfies
constant returns to scale.
Given a production set one can define an input correspondence L:Rm
+−→
2Rn
+and an output correspondence P:Rn
+−→ 2Rm
+such that:
T=(x, y)∈Rn+m
+:x∈L(y)=(x, y)∈Rn+m
+:y∈P(x).(2.1)
The literature on technology aggregation (see e.g. Li and Ng, 1995) de-
fines the technology of the grand coalition as a standard sum of input and
output vectors. Let (xk, yk)∈Rn+m
+be the input-output vectors of firm k
whose technology is Tk. Following this specification, the technology of any
given coalition Sis the standard sum of the technologies Tkof each firm
k∈ S:
TS:= X
k∈S
Tk=n X
k∈S
xk,X
k∈S
yk: (xk, yk)∈Tk, k ∈ So.(2.2)
In the remainder of the paper, 0n[0m] is the n-dimensional [m-dimensional]
vector of zeros, 11nthe n-dimensional vector of ones, Nthe set of (strictly)
positive integers, and finally ≥(≤) denotes inequalities over scalars and >
(6) over vectors.
3 Directional Distance Functions and Firm
Game
The directional distance function introduced by Chambers, Chung and F¨are
(1996, 1998)1DT:Rn+m
+×Rn+m
+−→ R+involving a simultaneous input and
1See also Chambers and F¨are (1998) and Chambers (2002) for more details on direc-
tional distance functions.
5

output variation in the direction of a pre-assigned vector g= (gi, go)∈Rn+m
+
is defined as:
DT(x, y;g) = sup
δδ∈R: (x−δgi, y +δgo)∈T.(3.1)
In the sequel, the directional distance is such that DT(x, y;g)≥0, i.e., the
cases of infeasibilities for which (x, y)/∈Tare not reported. For a group
of |K| firms with technology Tk, the technical aggregation bias is defined as
follows (see Briec, Dervaux and Leleu, 2003):
AB(K;g) := DT X
k∈K
(xk, yk); g!−X
k∈K
DTk(xk, yk;g).(3.2)
It provides the loss of technical efficiency due to the cooperation between the
firms of group K. The aggregation bias may be nil, in this case the exact
aggregation condition is:
DT X
k∈K
(xk, yk); g!=X
k∈K
DTk(xk, yk;g).(3.3)
Under the assumptions (T1)-(T4), the exact aggregation is possible when-
ever:
(i) the technologies Tkare identical and the input set is one-dimensional;
(ii) the firms use the same technique and (T5)-(T6) hold.
Actually, those results (see Briec et al., 2003) are merely dependent on the
structure of the aggregation process, i.e. the standard sum used to describe
the aggregated technology of the group of firms. Indeed, the standard sum
is used in a cooperative game, the so-called firm game, in order to aggregate
inputs and outputs.
Definition 3.1 A firm game is a collection {K, v(S) : S ⊆ K} such that:
(x,y) : 2K→Rn+m
+∀S ∈ 2K,S 6=∅,where (x,y) (S) := Xk∈S "xk, yk∈TS,
with v: 2K→Rn+m
+→R+, v(S) := DTS◦(x,y) (S),
and (x,y) (∅) := 0, v (∅) := 0 by convention.
The game v(S) provides the value of the directional distance function DTS(·)
related to any given firm coalition Swith technology TS:
v(S) = DTS X
k∈S "xk, yk;g!,for all S ⊆ K.(3.4)
On this basis, the technical aggregation bias is non negative. As suggested
by Post (2001), it seems that the way the inputs/outputs are aggregated has
serious implications on the efficiency measures. Precisely, the measurement
of technical efficiency and its bias depend on particular aggregators.
6

4 Technology Aggregators: characterization
The literature on firm groups outlines an impossibility according to the di-
rectional distance function: the collaboration between firms cannot improve
the technical efficiency of the group. This impossibility was proven indepen-
dently by Briec, Dervaux and Leleu (2003) and F¨are, Grosskopf and Zelenyuk
(2008). Their result may be rewritten by choosing a general distance function
denoted fS:Rn+m
+×Rn+m
+−→ R+. The game vf(S) is the distance function
for any given coalition S, such that vf(S)≡fSfor all S ⊆ K. Following this
notation, setting fthe directional distance function, their result is:
vf(K)−X
k∈K
vf({k}) = AB(K;g)≥0,(4.1)
where the distance fgauges technical efficiency (the distance between one
point and the frontier of the technology). The distance of the group is, for
any given type of technology, always greater than the sum of the individual
distances. The same conclusion holds for all possible coalitions S ⊆ K if fis
also chosen to be the directional distance function (see Briec and Mussard,
2014):
vf(S)−X
k∈S
vf({k}) = AB(S;g)≥0.(4.2)
This result reports a sub-efficiency related to the cooperation between firms.
It is inherent to the standard additive form of the aggregation of vectors.
However, following Ben-Tal (1977), Eichorn (1979) or Blackorby and Russell
(1999), there exist many other forms of aggregation. For instance, Blackorby
et al. (1981) characterize an isomorphism widely employed in welfare eco-
nomics, specially for welfare, inequality and poverty indices. In the following
lines, the proposed isomorphism is directly inspired from Ben-Tal (1977) and
Blackorby et al. (1981) who studied the algebraic properties of the aggregator
underlying the generalized sum (mean).
4.1 Generalized Sum
Let dbe a positive integer and let Φ : X−→ Rdbe a bijective map, where
Xis an arbitrary set. From Ben-Tal (1977) we consider on Xthe algebraic
operators Φ
+ and Φ
.defined for all x, y ∈Xand for all α∈Rby:
xΦ
+y= Φ−1 Φ(x) + Φ(y)and αΦ
·x= Φ−1 α·Φ(x).(4.3)
The Φ-sum, denoted
Φ
P, of (x1,··· , xd)∈Rnis defined by2
Φ
X
i∈[d]
xi= Φ−1X
i∈[d]
Φ(xi).(4.4)
2For ease of exposition, for all d∈N, [d] := {1, . . . , d}.
7

The subset Xendowed with these algebraic operators has some proper-
ties very similar to those of a vector space. Indeed, let Ebe an arbitrary
nonempty set and let φ:E−→ Rbe an isomorphism. One can define over
Ethe operations defined ∀λ, µ ∈Eby:
λφ
+µ=φ−1(φ(λ) + φ(µ)) and λφ
.µ=φ−1(φ(λ).φ(µ)) .(4.5)
From Ben-Tal (1977) the set φ(R) endowed with the algebraic operators
φ
+ and φ
.is a scalar field. A vector space can then be constructed as the
Cartesian product of an isomorphic transformation of the scalar field R, that
is Ed, in the case where the bijective map Φ is defined for all u∈Rdand all
x∈X=Edby:
Φ(x) = (φ(x1), . . . , φ(xd)) and Φ−1(u) = φ−1(u1), . . . , φ−1(ud).(4.6)
It follows that E=φ−1(R) is endowed with a total order defined by:
λφ
≤µ⇐⇒ φ(λ)≤φ(µ).(4.7)
Obviously (Ed,φ
+,φ
.) is a vector space where the algebraic operators φ
+ and φ
·
are those defined above. It is then clear that if B={v1, . . . , vd}is a basis
of Rdthen Bφ:= Φ−1(B) = {Φ−1(v1), . . . , Φ−1(vd)}is a basis of the vector
space (Ed,φ
+,φ
.).
4.2 Characterization of the aggregator
We first begin our investigations with a general (non specified) multidimen-
sional aggregator. It is a map that transforms the data, i.e., a function
whose images are monotonic transformations of inputs and outputs. Let Φ
be a general aggregator (isomorphism) such that Φ−1:Rd
+−→ Ed
+with
d∈N:Φ
X
k∈S
zk:= Φ−1!X
k∈S
Φ(zk)".(4.8)
In the following, we say that Φ is a canonical φ-isomorphism if there exists
a real valued bijective map φ:E+−→ R+such that for all z∈Rd
+:
Φ(z) := (φ(z1), . . . , φ(zd)) .(4.9)
By definition, the Φ-sum of the production technologies is:
Φ
X
k∈S
Tk=n
Φ
X
k∈S
xk,
Φ
X
k∈S
yk: (xk, yk)∈Tk, k ∈ So.(4.10)
In this respect, the technology of the coalition Sis defined as follows.
8

Definition 4.1 –Coalitional Technology (CT) – The aggregated tech-
nologies TS
Φ, for all S ⊆ K such that |S| ≥ 1, are defined as follows:
TS
Φ:=
Φ
X
k∈S
Tk.
Note that whenever |S| = 1, TS
Φ=Tk. The distance function fSof coali-
tion S ⊆ K yields the distance between one point and the frontier of the
technology TS
Φ:
vf(S)≡fS Φ
X
k∈S
xk,
Φ
X
k∈S
yk!.(4.11)
The distance function exists because some firms merge in order to improve
their technical efficiency, the so-called firm game.
Definition 4.2 A firm game defined on the algebraic operators Φ
+and Φ
·is
a collection {K, vf(S) : S ⊆ K,Φ}such that the coalitional technology is
defined as follows:
(x,y) : 2|K| →En+m
+for all S ∈ 2|K|,such that |S| ≥ 1, where
(x,y) (S) := Φ
P
k∈S
xk,
Φ
P
k∈S
yk∈TS
Φ,with
fS: 2|K| →En+m
+→R+, vf(S) := fS◦(x,y),and
(x,y) (∅) := 0n+m, vf(∅) = fS◦(x,y) (∅) = 0 by convention.
The technical aggregation bias of any given coalition is then modeled thanks
to the firm game. A negative [positive] bias is a sub[super]additive game
defined on the basis of the general aggregation process Φ. A negative [pos-
itive] bias represents an improvement [decline] of technical efficiency due to
the cooperation between firms. The technical aggregation bias (Φ-bias for
short) of a canonical isomorphism Φ defined with respect to a real valued
isomorphism φis defined as, for all S ⊆ K such that |S| ≥ 2:
ABΦ(S) := fS Φ
X
k∈S
xk,
Φ
X
k∈S
yk!−
φ
X
k∈S
fk(xk, yk).(4.12)
Definition 4.3 –Φ-sub[super]additivity (SUBΦ[SUPΦ]) –Let R,S ⊆ K
such that R∩S =∅. Let Φ : En+m
+−→ Rn+m
+be a canonical φ-isomorphism.
A firm game K, vf(S) : S ⊆ K,Φis defined to be sub[super]additive if:
ABΦ(S ∪ R)≤[≥] 0 ⇐⇒ vf(S ∪ R)≤[≥]vf(S)φ
+vf(R).
The technical aggregation bias is equivalently rewritten as, for all S ⊆ K
such that |S| ≥ 2:
ABΦ(S) = vf(S)−
φ
X
k∈S
vf({k}).(4.13)
9

We first examine the existence of an aggregator when the Φ-bias is null.
The first result shows that the transformation of the data, generated by any
given aggregator Φ, yields a linear distance function defined up to the map
φ−1. This is a generalization of the untransformed case in which the distance
function is linear, see Briec et al. (2003) and F¨are et al. (2008).
Proposition 4.1 Let Φ : En+m
+−→ Rn+m
+be a canonical φ-isomorphism.
Under (T1)-(T4) and (CT), for all firm games {K, vf(S) : S ⊆ K,Φ}, the
following implication holds:
ABΦ(S) = 0,∀S ⊆ K=⇒vf(S) = φ−1 Xk∈S c·Φ(zk) + Xk∈S ck.
Proof:
Let ABΦ(S) = 0, such that Φ : En+m
+−→ Rn+m
+with its one-dimensional
representation φ:R+−→ E+. For all S ⊆ K such that |S| ≥ 2:
ABΦ(S) = 0 ⇐⇒ fS!Φ
X
k∈S
(xk, yk)"=
φ
X
k∈S fk(xk, yk).
Thus,
fS#Φ−1!X
k∈S
Φ(xk, yk)"$=φ−1!X
k∈S
φfk(xk, yk)".
Let us denote the vector zk:= (xk, yk)∈Rn+m
+such that uk:= Φ(zk), thus:
φ!fS#Φ−1!X
k∈S
uk"$"=X
k∈S
φfk(zk).
Set φ◦fS=: φSand φ◦fk=: φk, for all k∈ {1, . . . , |S|}. As zk= Φ−1(uk),
we get:
φS#Φ−1!X
k∈S
uk"$=X
k∈S
φkΦ−1(uk).
We recognize the well-known Pexider’s equation of solution (see Acz´el, 1966,
p.141):
φS◦Φ−1!X
k∈S
uk"=c·!X
k∈S
uk"+X
k∈S
ck;
φk◦Φ−1(uk) = c·uk+ck,
where the vector c∈En+m
+and the constants ck∈E+are set to be non-
negative in order to get well-defined distance functions (being non-negative).
The solution can be rewritten in a general setting as:
φ◦fk(zk) = c·Φ(zk) + ck,∀k∈ {1, . . . , |S|},
10

and,
φ◦fS Φ
X
k∈S
zk!=c·X
k∈S
Φ(zk) + X
k∈S
ck,∀S ⊆ K.
Thus,
vf(S) = fS Φ
X
k∈S
zk!=φ−1 c·X
k∈S
Φ"zk+X
k∈S
ck!,∀S ⊆ K.
The result proves that the transformation of the data thanks to the Φ-
aggregator enables the standard interpretation to be retrieved as a particular
case, that is, the distance function is linear (up to the map φ−1), as shown
by Briec et al. (2003) and F¨are et al. (2008) with the standard sum of sets.
Now, we impose more structure to the distance function in order to char-
acterize the Φ-aggregator. The homogeneity of degree one is known to be
well suited for the measure of technical efficiency, see Chambers, Chung and
F¨are (1996, 1998)3:fk(λxk, λyk) = λfk(xk, yk) for all λ≥0. We show that
if the technical Φ-bias is null, then Φ is found to be quasi-linear.4
Proposition 4.2 Under the assumptions (T1)-(T4) and (CT), for all firm
games {K, vf(S) : S ⊆ K,Φ}such that fSis homogeneous of degree one, if
Φ : En+m
+−→ Rn+mis a φ-canonical isomorphism, then the following are
equivalent:
(i) ABΦ(S) = 0.
(ii) vf(S) = "Pk∈S c·(zk)τ1
τ,for some τ6= 0.
Proof:
[(i) =⇒(ii)]. From Proposition 4.1, when the Φ-bias is null, setting c:=
(b1, . . . , bn+m), we get:
vf(S) = φ−1 X
k∈S
c·Φ(zk) + X
k∈S
ck!=φ−1 X
k∈S
n+m
X
ℓ=1
bℓφ(zk
ℓ) + X
k∈S
ck!.
If the distance function is homogeneous of degree one, then the function
φ−1"Pk∈S Pn+m
ℓ=1 bℓφ(λzk
ℓ) + Pk∈S ckinherits the homogeneity property of
vf(of degree one). Hence, for λ≥0, the following relation holds,
λφ−1 X
k∈S
n+m
X
ℓ=1
bℓφ(zk
ℓ) + X
k∈S
ck!=φ−1 X
k∈S
n+m
X
ℓ=1
bℓφ(λzk
ℓ) + X
k∈S
ck!,
3See also Chambers and F¨are (1998) and Chambers (2002).
4The homogeneity of degree one of the distance function is sometimes associated with
the constant returns to scale hypothesis (T5). This is the case for instance for the direc-
tional distance function.
11

if and only if Pk∈S ck= 0 and φ−1is homogeneous of degree τ6= 0. Then,
φ−1(λt) = tτφ−1(λ) =: tτκ1,
and,
φ−1(λt) = λτφ−1(t) =: κ2φ−1(t).
Thus,
φ−1(t) = tτκ1
κ2
=: κtτ, κ26= 0, τ 6= 0,
with κ≥0 in order to get a non-negative distance function. If the distance
function vf(S) is supposed to be homogeneous of degree one, then the dis-
tance function is a mean of order τ6= 0 and the aggregator φis a power
function.
[(ii) =⇒(i)]. Since Φ(x) = (xτ
1, . . . , xτ
n+m) for τ6= 0, we deduce from the
functional form of vf(S) that for all k∈ S and κ≥0:
vf({k}) = κc·zk,
and the implication follows.
In sum, the transformation of the data, thanks to the isomorphisms Φ,
enables the usual aggregation bias to be linked with the quasi-linearity of
some homogeneous distance functions fS. A similar result based on the
directional distance function was formerly found by Briec et al. (2003) in
the case where the data are not transformed, i.e., Φ would be reduced to the
identity map in our framework. In this case, the core of the firm game would
be represented by one point. Indeed, the core of the firm game is:
CΦ:= (ϕ∈E|K|
+:X
k∈S
ϕk≤vf(S),∀S ⊂ K)\(X
k∈K
ϕk=vf(K)).(4.14)
Proposition 4.3 Under the assumptions (T1)-(T4),(CT) and (SUBΦ), for
all firm games {K, vf(S) : S ⊆ K,Φ}such that Φ = IdEd
+, the following are
equivalent:
(i) ABΦ(S) = 0.
(ii) CΦ={vf({1}), . . . , vf({|K|})},◦
CΦ=∅.
Proof:
[(i) =⇒(ii)]: Let ϕk:= vf({k}). Note that ABΦ(S) = 0 implies that
Pk∈S ϕk=v(S) for all S ⊆ K. The vector ϕ:= (ϕ1, . . . , ϕ|K|) is an imputa-
tion since it respects ϕk≤vf({k}) and Pk∈K ϕk=v(K).As a consequence,
since ϕk=vf({k}), then ϕis the core. The core CΦis then reduced to one
point and so ◦
CΦ=∅.
[(ii) =⇒(i)]: If the pay-off point {vf({1}), . . . , vf({|K|})}is the core, then
by definition, Eq.(4.14), we get that Pk∈S ϕk≤vf(S), for all S ⊂ K. Since
12

vf({k}) = ϕk, then Pk∈S vf({k})≤vf(S), for all S ⊂ K. By subadditivity,
we get Pk∈S vf({k})≥vf(S), and so AB(S) = 0 for all S ⊆ K.
Finally, we have seen that the use of an aggregator Φ being a power
function, in order to deal with heterogeneous firms, allows well-defined ho-
mogeneous distance functions fSto be derived when the aggregation bias
ABΦis null. As a consequence, it is of interest to test whether the aggrega-
tor is compatible with the celebrated directional distance function to gauge
technical efficiency.
5 Biases of the Directional Distance Function
We show in this section, with numerical examples, that the aggregator is
enough flexible to capture different technical biases (positive, negative and
null) with the directional distance function. Before, we specify the power
function characterized in the previous section, and we subsequently define the
resulting aggregator (the generalized mean) and the aggregated directional
distance function.
5.1 Power Functions
For all α∈]0,+∞[, let φα:R−→ Rbe the map defined by:
φα(λ) = λαif λ≥0
−|λ|αif λ≤0.(5.1)
For all α6= 0, the reciprocal map is φ−1
α:= φ1
α. It is first quite straightforward
to state that: (i)φαis defined over R+; (ii)φαis continuous over R+;
(iii)φαis bijective over R+. Throughout the section, for any vector x=
(x1, . . . , xd)∈Rd
+we use the following notations:
Φα(x) = φα(x1), . . . , φα(xd)= x1α, . . . , xdα=xα.(5.2)
It is then natural to introduce the following algebraic operation over Rn
+:
xα
+y= Φ−1
α Φα(x) + Φα(y)and λα
·x= Φ−1
α(φα(λ)Φα(x)) .(5.3)
In this case (φα(R),+,·) is a scalar field since φα(R) = R.
Let us focus on the case α∈]− ∞,0[. The map x7→ xαis not defined at
point x= 0. Thus, it is not possible to construct a bijective endomorphism
on R. However, it is possible to construct an operation preserving at least
associativity. For all α∈]− ∞,0[ we consider the function φαdefined by:
φα(λ) = 


λαif λ > 0
−(|λ|)αif λ < 0
+∞if λ= 0.
(5.4)
13

In such a case M:= φα(R) = R\{0}∪{+∞}. Moreover, let us construct the
application Φα:Rd−→ Md, defined by Φα(x1, . . . , xd) = (φα(x1), . . . , φα(xd)).
For all α < 0, let us consider the algebraic operators α
+ and α
·defined by:
xα
+y= Φα−1(Φα(x) + Φα(y)) and λα
·x= Φα−1(φα(λ).Φα(x)) .(5.5)
In such a case (R,α
+,α
·) is not a scalar field because there is not a neutral
element. Notice that (R,α
+,α
·) admits 0 as an absorbing element. It is easy
to check that for all λ∈R, 0 α
+λ= 0. This comes from the fact that for
all µ∈M,µ+∞=∞ ∈ M. Thus Rd,α
+,α
·is not a Φα-vector space.
However, the addition α
+ is well defined over Rdand it is trivial to check
that associativity holds. For the purpose of the paper, the fact that Mdoes
not contain a neutral element is not a problem since we consider operations
defined on Rd
+. If α= 0, we denote φ0:R+−→ Rthe map defined by:
φ0(λ) = ln(λ) if λ > 0
−∞ if λ= 0.(5.6)
The reciprocal map is:
φ0−1(λ) = exp(λ) if λ∈R
0 if λ=−∞.(5.7)
It is then possible to construct an algebraic operator summing the elements
of Rn
+. 1 is a neutral element of (R,0
+,0
·) and ∞is an absorbing element.
5.2 The generalized mean
When the distance function is homogeneous of degree one, the aggregator Φ
is the generalized mean investigated by Ben-Tal (1977), Eichorn (1979) and
Blackorby et al. (1981). Then, the aggregation process found in the previous
section is defined as follows. Let φα:R−→ Mbe the injective map defined
for all t > 0 by φα(t) = tα. For all (t1, . . . , tℓ)∈Rℓ
+, an one-dimensional
aggregator is given by:
φα
X
k∈[ℓ]
tk=

Pk∈[ℓ](tk)α1
α∀α > 0
Qk∈[ℓ]tkα= 0.
(5.8)
If α < 0, then:
φα
X
k∈[ℓ]
tk=

Pk∈[ℓ](tk)α1
αif minktk>0
0 if minktk= 0.
(5.9)
14

Note that the well-known arithmetic mean is obtained when α= 1. On the
other hand, using L’hospital rule yields the geometric mean:
lim
α−→0
φα
X
k∈[ℓ]
tk=Y
k∈[ℓ]
tk.(5.10)
In order to aggregate technologies (input/output vectors), we take re-
course to a multidimensional map, the so-called Φα-aggregator.
Definition 5.1 –Φα-Aggregation –Let Φα:Rd−→ Mdbe an injective
map defined by:
Φα(z1, . . . , zd) = (φα(z1), . . . , φα(zd)).
For all collections Z={zk:k∈ S} ∈ Rd
+, a Φα-aggregator is given by:
Φα
X
k∈S
zk=
φα
X
k∈S
zk
1, . . . ,
φα
X
k∈S
zk
d.
The definition of the coalitional technology (CT) yields, under the Φα-
aggregator, the following aggregated technology for all S ⊆ K and |S| ≥ 1:
TS
α:=
Φα
X
k∈S
Tk.(5.11)
It enables different cases to be captured. When α= 0, a multi-output Cobb-
Douglas technology is designed. When α= 1, we retrieve the well-known
aggregation over sets, studied in Li and Ng (1995), Briec et al. (2003), F¨are
et al. (2008), Briec and Mussard (2014).
It has been shown in the previous sections that the distance function
is a good candidate to measure technical efficiency for a group of firms
with respect to the general aggregator Φ. The directional distance func-
tion introduced by Chambers, Chung and F¨are (1996, 1998) is homogeneous
of degree one. With respect to the Φα-aggregator, it is given by DTS
α:
Φ−1
α Mn+m×Φ−1
α Mn+m−→ R+involving a simultaneous input and out-
put variation in the direction of a pre-assigned vector g= (gi, go)∈Rn
+×Rm
+.
Following the firm game, Definition 4.2 and the Φα-aggregator, the direc-
tional distance function is expressed as,
vα(S) := DTS
α!Φα
X
k∈S
xk,
Φα
X
k∈S
yk;g",(5.12)
and the aggregation bias by,
ABα(S) := vα(S)−
φα
X
k∈S
vα({k}),∀S ⊆ K,|S| ≥ 2.(5.13)
15

By generalizing the technologies thanks to the Φαaggregator, the resulting
aggregated technology is enough flexible to cover positive aggregation bias
as well as negative ones. A negative bias indicates that some firm coalitions
improve their technical efficiency, whereas this result has been found to be
impossible under the standard sum of the firms’ technologies.
5.3 Negative technical bias: examples
We show that the transformation of the data thanks to Φαenables the tech-
nical bias inherent to the directional distance function to be designed as
negative (positive), i.e., the improvement (decline) of technical efficiency of
the firm group is due to the cooperation between firms.
Example 5.1 Suppose that K={1,2}and that Tk={(x, y)∈R3
+:yα−
(x1)α−(x2)α≤0}for k= 1,2. Assume moreover that for k= 1,2we have
(x1, y1) = (2,1,1) and (x2, y2) = (1,2,1). In the following, it is shown that
T1=T2, it is possible to find some αsuch that ABα(S;g)S0.
For all α > 0, let us consider the isomorphism φα:R−→ Rdefined by,
φα(λ) = λαif λ≥0
−(−λ)αif λ < 0
and its inverse is defined by,
φ−1
α(λ) = λ1
αif λ≥0
−(−λ)1
αif λ < 0.
We have by definition:
Φα
X
k=1,2
Tk
α=(T1)α+ (T2)α1
α.
Set T0=T1=T2. By construction T0is quasi linear and satisfies a constant
returns to scale assumption. We obtain from Briec, Dervaux et Leleu (2003):
Φα
X
k=1,2
Tk
α=T0.
Setting g= (1,1,0), we have DT0(2,1,1; 1,1,0) = 1 and DT0(1,2,1; 1,1,0) =
1. It follows that:
α
X
k=1,2
DTk
α(xk, yk; 1,1,0) = (DT0(2,1,1; 1,1,0))α+(DT0(1,2,1; 1,1,0))α1
α= 2 1
α.
Moreover,
Φα
X
k=1,2
(xk, yk) = (2α+1α)1
α,(1α+2α)1
α,(1α+1α)1
α=(1+2α)1
α,(1+2α)1
α,21
α.
16

The input set is by definition L0(2 1
α) = {(x1, x2)∈R2
+: (x1)α+ (x2)α≥21
α}.
It follows that:
DTS
αΦα
X
k=1,2
(xk, yk); 1,1,0=DTS
α(1+2α)1
α,(1+2α)1
α,21
α; 1,1,0= (1+2α)1
α−1.
Thus,
ABα= (1 + 2α)1
α−1−21
α.
•If α= 1, we have
ABα= 3 −1−2 = 0.
•If α= 1/2, we have
ABα= (1 + √2)2−1−22= (1 + √2)2−5>0.
•If α= 2, we have
ABα=√5−1−√2<0.
It is also possible to show that, when α= 0, that the technical bias is
either positive or negative.
Example 5.2 Let z1= 11dand Sa coalition of two firms, k= 1,2, such
that TS
α=T1
α. Set α= 0, we show in the following that AB0(S;g)S0. We
get, for α= 0:
AB0(S;g) = DTS
0 Y
k=1,2
zk;g!−DT1
0(z1;g)·DT2
0(z2;g).
Since TS
0=T1
0, it comes that
DTS
0 Y
k=1,2
zk;g!=DT1
0(z1;g).
By definition, DTS
0(·)≥0. If DT2
0(z2;g)T1then AB0(S;g)S0.
Those examples show, for some values of α, that the cooperation between
firms provides either an improvement of technical efficiency (negative bias),
a constant technical efficiency (null bias) or finally a decrease of technical
efficiency (positive bias). In what follows, the values of αare defined at the
neighborhood of infinity in order to capture negative biases associated with
semilattice technology sets.
6 Input/output fixed firm games: limit cases
We introduce input/output firm games in order to exhibit the conditions
allowing for negative biases to be conceived. Those games rely on semilattice
technologies. Solutions inside the core are characterized, so that the core is
partitioned with respect to either negative biases or positive ones.
17

6.1 Aggregated technologies
The negative bias represents the improvement of technical efficiency for any
given coalition of firms. The demonstration of its existence is made with the
directional distance function either output oriented or input oriented. We
investigate aggregate technologies that are either defined on the maximum
available input-output combination or on the minimum one, respectively. In
the maximum case, any coalition of firms takes benefit from cooperation since
its technical efficiency is improved, in other terms, the game is subadditive
(SUBΦ), defined from now onwards (SUBΦα) – see Definitions 4.3 and 5.1.
For that purpose, the φα-aggregator is defined for particular values of α.
First note that for all u∈Rd
+:
φα
X
ℓ∈[d]
uℓ:= 


min
ℓ∈[d]uℓif α=−∞
max
ℓ∈[d]uℓif α=∞.(6.1)
Notice that by construction
φα
Pℓuℓ=φ−1
αPℓφα(uℓ)when α /∈ {−∞,∞}.
In such a case, Blackorby et al. (1981) axiomatically characterize this aggre-
gator5. This notation is justified by the fact that:
limα−→−∞
φα
Pℓ∈[d]uℓ= min
ℓ∈[d]uℓif α=−∞
limα−→+∞
φα
Pℓ∈[d]uℓ= max
ℓ∈[d]uℓif α=∞.
(6.2)
Notice also that in the case where α=−∞, if there is some ℓwith uℓ= 0,
then φα(uℓ) = +∞and it follows that
φα
Pℓ∈[d]uℓ= min
ℓ∈[d]uℓ= 0.
Definition 6.1 –Semilattice Aggregators – For all collections Z={zk:
k∈ S} ∈ Rd
+an upper semilattice-aggregator is given by:
Φ∞
X
k∈S
zk=_
k∈S
zk= max{z1
1, . . . , z|S |
1}, . . . , max{z1
d, . . . , z|S |
d}.
A lower-semilattice aggregator is given by:
Φ−∞
X
k∈S
zk=^
k∈S
zk= min{z1
1, . . . , z|S |
1}, . . . , min{z1
d, . . . , z|S |
d}.
In the sequel, we define the notions of semilattice with respect to the
usual partial order defined over Rn+m
+.
5It is a generalized quasi-linear function that respects continuity, monotonicity, sepa-
rability and symmetry.
18

Definition 6.2 –Semilattices –A subset Aof Rn+mis an upper semilat-
tice if for all z, z′∈Awe have z∨z′∈A. A subset Bof Rn+mis a lower
semilattice if for all z, z′∈Bwe have z∧z′∈B.
Example 6.1 Let us recall the notion of B-convex (B−1-convex) sets. A
B-convex hull of a set A=z1, . . . , z|S| ⊂Rn+m
+is
B(A) = n_
k∈S
tkzk,max
k=1,...,|S| tk= 1, t >0o.
The B−1-convex hull of a set Ais given by:
B−1(A) = (^
k∈S
skzk,min
k=1,...,|S| sk= 1, s >0).
The B-convex and B−1-convex technologies are given by:
TS
max =n(x, y)∈Rm+n
+:x>_
k∈S
tkxk, y 6_
k∈S
tkyk,max
k∈S tk= 1, t >0o,
TS
min =n(x, y)∈Rm+n
+:x>^
k∈S
skxk, y 6^
k∈S
skyk,min
k∈S sk= 1, s >0o.
B-convex technologies belong to the class of Kohli technologies analyzed by
Kohli (1983). These technologies exhibit output complementarity in the pro-
duction. Inverse B-convex technologies are related to Leontief production
functions because they imply input complementarity of production factors.
They are, however, defined in a multi-output context. Let us remark that the
free disposal assumption can be represented thanks to the free disposal cone
K:= Rm
+×(−Rn
+). In this respect any technology respecting the free disposal
assumption may be rewritten as: T= (A+K)∩Rm+n
+. As a consequence,
B-convex and B−1-convex technologies are also given by:
TS
max = (B(A) + K)∩Rm+n
+;TS
min = B−1(A) + K∩Rn+m
+.
These are represented in Figures 1a and 1b.
x
y
z1
z3
0
z4
✻
✲
TS
max
z2
Figure 1.a B-convex Technology
x
y
z1
z3
0
z4
✻
✲
TS
min
z2
Figure 1.b Inverse B-convex Technology
19

In what follows, we analyze the properties of the aggregated technologies
TS
∞and TS
−∞, which are semilattice technologies:
TS
∞:=
φα
Pk∈S Tk=Wk∈S Tkif α= +∞
TS
−∞ :=
φα
Pk∈S Tk=Vk∈S Tkif α=−∞.
(6.3)
Proposition 6.1 We have the two following properties:
(i) Suppose that for all k∈ S,Tkis an upper semilattice. Then Wk∈S Tkis
an upper semilattice.
(ii) Suppose that for all k∈ S,Tkis a lower semilattice. Then Vk∈S Tkis a
lower semilattice.
Proof:
(i) Suppose that z, w ∈Wk∈S Tk. We need to prove that z∨w∈Wk∈S Tk.
By hypothesis, for all k∈ S there is some zk, wk∈Tksuch that z=Wk∈S zk
and w=Wk∈S wk. Therefore,
z∨w= (_
k∈S
zk)∨(_
k∈S
wk) = _
k∈S
(zk∨wk).
Since for all k,Tkis an upper semilattice, it follows that zk∨wk∈Tk. Thus
z∨w∈Wk∈S Tk, and Wk∈S Tkis an upper semilattice. The proof of (ii) is
similar.
Now, we prove that the aggregated technology respects the usual assump-
tions (T1)-(T5) – see Section 2. The first one (T1) displays the no free lunch
assumption, i.e. (0n,0m)∈Tkand (0n, yk)∈Tk=⇒yk= 0m.
Proposition 6.2 We have the two following properties:
(i) If for all k∈ S,Tksatisfies the no free lunch assumption (T1), then
Wk∈S Tksatisfies (T1).
(ii) If for all k∈ S,Tksatisfies the no free lunch assumption (T1), then
Vk∈S Tksatisfies (T1).
Proof:
Straightforward.
Now, we show that infinite outputs cannot be obtained from a finite input
vector (T2), i.e. the set A(xk) = (uk, yk)∈Tk:uk6xkof dominating
observations is bounded ∀xk∈Rn
+.
Proposition 6.3 We have the two following properties:
(i) If for all k∈ S,Tksatisfies (T2), then Wk∈S Tksatisfies (T2).
(ii) If for all k∈ S,Tksatisfies (T2), then Vk∈S Tksatisfies (T2).
20

Proof:
(i) From (T2) each set A(xk) is assumed to be bounded. Hence, A(Wk∈S xk)
is also bounded, and so, Wk∈S Tkrespects (T2). The same holds true for (ii).
The aggregated semilattice technology is shown to be closed (T3).
Proposition 6.4 For all k∈ S, let Tkbe (i) an upper semilattice or (ii) a
lower semilattice, then two following properties hold true, respectively.
(i) For all k∈ S, if Tksatisfies the closedness assumption (T3), then Wk∈S Tk
satisfies (T3).
(ii) For all k∈ S, if Tksatisfies the closedness assumption (T3), then
Vk∈S Tksatisfies (T3).
Proof:
(i) Assume by contradiction that Wk∈S Tkis open. Hence, there is some
w, z ∈Rn+m
+such that (w, z)/∈Wk∈S Tk. From Proposition 6.1, Wk∈S Tk
is an upper semilattice. Then, Wk∈S (wk∨zk)/∈Wk∈S Tk. Hence, the map
wk7→ Wk∈S wkis not defined and not continuous on some intervals, that is, it
exists some k∈ S and at least one i∈ {1, . . . , d}such that maxk∈S wk
iis not
defined. Then, for wk= (xk, yk) it exists xk
i∈[¯xk
i,¯xk
i+ǫ] or yk
i∈[¯yk
i,¯yk
i+ǫ]
with ǫ > 0 such that maxk∈S wk
iis not defined. In such a case, xk∨yk/∈Tk,
and Tkis open on some intervals [¯xk
i,¯xk
i+ǫ] or [¯yk
i,¯yk
i+ǫ]. (ii) Mutatis
mutandis (i).
We can also prove that the aggregated semilattice technology respects the
free disposal assumption (T4).
Proposition 6.5 We have the two following properties:
(i) If for all k∈ S,Tksatisfies a free disposal assumption (T4), then Wk∈S Tk
satisfies (T4).
(ii) If for all k∈ S,Tksatisfies a free disposal assumption (T4), then
Vk∈S Tksatisfies (T4).
Proof:
(i) Suppose that z= (x, y)∈Wk∈S Tkand let z′= (x′, y′)∈Rn+m
+such that
x′>xand y′6y. We need to prove that z′∈Wk∈S Tk. By hypothesis one
can find (z1, . . . , z|S |)∈Wk∈S Tksuch that z=Wk∈S zk. If y′6y, then there
is some v∈Rm
+such that y′=y−v. Moreover y−v= (Wk∈S yk)−v=
Wk∈S (yk−v). However, y−v>0m,y−v= (y−v)∨0m. Consequently,
y−v= _
k∈S
(yk−v)∨0m=_
k∈S (yk−v)∨0m].
For all k, since yk>0m, (yk−v)∨0m6yk∨0m=yk. Similarly, if x′>x, then
there is some u∈Rn
+such that x′=x+u= (Wk∈S xk) + u=Wk∈S (xk+u).
21

However, since each Tksatisfies a free disposal assumption the inequalities
(yk−v)∨0m6ykand xk+u>xkimplies that (xk+u, (yk−v)∨0m)∈Tk.
Hence z′= (x′, y′)∈Wk∈S Tk, which ends the proof. (ii) The proof is
similar except that one should use the distributivity of the operation ∨on
∧. Suppose that z= (x, y)∈Vk∈S Tkand let z′= (x′, y′)∈Rn+m
+such that
x′>xand y′6y. By hypothesis one can find (z1, . . . , z|S|)∈Vk∈S Tksuch
that z=Vk∈S zk. Paralleling the proof above, if y′6y, then there is some
v∈Rm
+such that y′=y−v. Moreover y−v= (Vk∈S yk)−v=Vk∈S (yk−v).
Since, y−v>0m,y−v= (y−v)∨0. Therefore,
y−v= ^
k∈S
(yk−v)∨0m=^
k∈S (yk−v)∨0m].
For all k, since yk>0mone has (yk−v)∨0m6yk∨0m=yk. Moreover, if
x′>x, then there is some u∈Rn
+such that x′=x+u= (Vk∈S xk) + u=
Vk∈S (xk+u). However, since each Tksatisfies a free disposal assumption
(yk−v)∨0m6ykand xk+u>xkimplies that z′= (x′, y′)∈Vk∈S Tk,
which ends the proof.
The constant returns to scale assumption (T5) is also respected by the
aggregated semilattice technology. Recall that the technology Tksatisfies
constant returns to scale if ∀β≥0, (xk, yk)∈Tkimplies (βxk, βyk)∈Tk.
Proposition 6.6 The two following properties hold true:
(i) If for all k∈ S,Tkis an upper semilattice and satisfies the constant
returns to scale assumption (T5), then Wk∈S Tkis an upper semilattice re-
specting (T5).
(ii) If for all k∈ S,Tkis a lower semilattice and satisfies the constant re-
turns to scale assumption (T5), then Vk∈S Tkis a lower semilattice respecting
(T5).
Proof:
(i) Suppose that z, w ∈Wk∈S Tk. We have to prove that βz ∨βw ∈Wk∈S Tk
for some β≥0. For all k∈ S choose some zk, wk∈Tksuch that z=Wk∈S zk
and w=Wk∈S wk. Therefore,
βz ∨βw = ( _
k∈S
βzk)∨(_
k∈S
βwk) = _
k∈S
(βzk∨βwk).
By (T5), since Tkis an upper semilattice for all k∈ S, we get that βzk∨
βwk∈Tk. Thus βz ∨βw ∈Wk∈S Tk, and so z∨w∈Wk∈S Tk. Consequently,
Wk∈S Tkis an upper semilattice satisfying (T5). (ii) Mutatis mutandis (i).
A last property of equal technology will be useful in the games defined
below. Inside a coalition, when the technologies of the firms are the same,
the aggregated technology of the coalition inherits the same technology.
22

Proposition 6.7 We have the two following properties:
(i) Suppose that for all k∈ S,Tk=Tis an upper semilattice. Then
Wk∈S Tk=T.
(ii) Suppose that for all k∈ S,Tk=Tis a lower semilattice. Then
Vk∈S Tk=T.
Proof:
(i) By hypothesis (0n,0m)∈Tkfor all k. Therefore, for all k,Tk=T⊂
Wk∈S Tk. Let us show the converse. Assume that z∈T. It follows that
for all kthere are some zk∈Tk, such that z=Wk∈S zk. Since T=Tk,
zk∈Tfor all k. Since Tis an upper semilattice it follows that Wk∈S zk∈T.
Therefore Wk∈S Tk⊂T. Consequently, Wk∈S Tk=T.
(ii) Assume that z∈T. It follows that for all kthere are some zk∈Tk,
such that z=Vk∈S zk. Since T=Tk,zk∈Tfor all k. Since Tis a lower
semilattice it follows that Vk∈S zk∈T. Therefore Vk∈S Tk⊂T. Let us
prove the converse inclusion, Suppose that, z∈T. Since T=Tkfor all k,
z∈Tk. Obviously z=Vk∈S zk. Hence z∈Vk∈S Tkand it follows that
T⊂Vk∈S Tkwhich proves the converse inclusion.
Finally, those properties indicate that the semilattice structure respects
the traditional assumptions (T1)-(T5) used in the literature.
6.2 Games
Since the Φ∞- and Φ−∞-aggregators provide semilattice aggregated technolo-
gies with desirable assumptions, we can now investigate their implications on
firm games. Some restrictions may be imposed on inputs and outputs in order
to clearly identify negative and positive technical biases.
Definition 6.3 –Input/Output fixed firm games –Let the firm game
be {K, vf(S) : S ⊆ K,Φα}.
(i) An input fixed firm game is given by,
I(K, vα(S),¯x) := {K, vf(S) : S ⊆ K,Φα},
where for all k∈ S,xk= ¯x.
(ii) An output fixed firm game is given by,
O(K, vα(S),¯y) := {K, vf(S) : S ⊆ K,Φα},
where for all k∈ S,yk= ¯y.
(i) In input fixed firm games I(K, vα(S),¯x), coalitions of firms are based on
possible outputs to be produced whereas the amount of inputs is limited to
¯xfor all possible coalitions. This may arise when the firms of specific sectors
are constrained by the amount of their inputs, for instance a maximum is
imposed to respect environmental norms.
23

(ii) In output fixed firm games O(K, vα(S),¯y), coalitions are formed on the
basis of inputs only, whereas the amount of outputs for each coalition is
constrained by ¯y, which may represent a production quota.
The core of those firm games is:
Cα:= (ϕ∈M|K| :X
k∈S
ϕk≤vα(S),∀S ⊂ K)\(X
k∈K
ϕk=vα(K)).(6.4)
The technical biases inherent to the input fixed firm game are the follow-
ing.
Proposition 6.8 If Tkis an upper semilattice (lower semilattice respec-
tively) respecting (T1)-(T4) such that Tk=Tfor all k∈ S ⊆ K, then
we have, respectively:
(i) [I(K, vα(S),¯x)∧(α= +∞)] =⇒[ABα(S;g)≤0].
(ii) [I(K, vα(S),¯x)∧(α=−∞)] =⇒[ABα(S;g)≥0].
Proof:
(i) If α=∞, then
Φ∞
Pk∈S Tk=Wk∈S Tkis an upper semilattice (see Proposi-
tion 6.1). From Proposition 6.5 this set satisfies the free disposal assumption
(T4), and moreover T=Wk∈S Tk(see Proposition 6.7). It follows that the di-
rectional distance function is weakly monotonic on T, that is, (x, y),(u, v)∈
Tsuch that u6yand v>ximply that DT(u, v;g)≥DT(x, y;g). Since this
is an input fixed game, we have xk= ¯x, for all k∈ S. Hence, for all S ⊆ K,
we have Wk∈S xk= ¯x. Moreover, Wk∈S yk>ykfor all k. From weak mono-
tonicity, we have DT(¯x, Wk∈S yk;g)≤DT(xk, yk;g) = DT(¯x, yk;g). However,
since Tk=T, we have for all k∈ S:
DT _
k∈S
xk,_
k∈S
yk;g!=DT"¯x, _
k∈S
yk;g≤DT(xk, yk;g) = DTk(xk, yk;g).
This implies that DT(Wk∈S xk,Wk∈S yk;g)≤max
k∈S DTk(xk, yk;g) which proves
(i).
(ii) If α=−∞, then
Φ−∞
Pk∈S Tk=Vk∈S Tk=Tis a lower semilattice by
Proposition 6.1. This set satisfies the free disposal assumption (T4) from
Proposition 6.5. It follows that the directional distance function is weakly
monotonic on Vk∈S Tk=Tby Proposition 6.7. Then (x, y),(u, v)∈T, such
that u6yand v>ximply that DT(u, v;g)≥DT(x, y;g). Since this is an
input fixed game, we have xk= ¯x, for all k∈ S. Hence, for all S ⊆ K, we have
Vk∈S xk= ¯x. Moreover, Vk∈S yk6ykfor all k. From weak monotonicity,
we have DT(¯x, Vk∈S yk;g)≤DT(xk, yk;g) = DT(¯x, yk;g). Moreover, since
Tk=T, we have for all k∈ S:
DT ^
k∈S
xk,^
k∈S
yk;g!=DT(¯x, ^
k∈S
yk;g)≥DT(xk, yk;g) = DTk(xk, yk;g).
24

This implies that DT(Vk∈S xk,Vk∈S yk;g)≥min
k∈S DTk(xk, yk;g) which proves
(ii).
The technical biases inherent to the output fixed firm game are the fol-
lowing.
Proposition 6.9 If Tkis an upper semilattice (lower semilattice respec-
tively) respecting (T1)-(T4) such that Tk=Tfor all k∈ S ⊆ K, then
we have, respectively:
(i) [O(K, vα(S),¯y)∧(α= +∞)] =⇒[ABα(S;g)≥0]
(ii) [O(K, vα(S),¯y)∧(α=−∞)] =⇒[ABα(S;g)≤0].
Proof:
(i) If α=∞, then
Φ∞
Pk∈S Tk=Wk∈S Tk=Tis an upper semilattice re-
specting the free disposal assumption (T4), see Propositions 6.1, 6.5 and
6.7. Thereby, the directional distance function is weakly monotonic on
T, that is, (x, y),(u, v)∈T, such that u6yand v>ximply that
DT(u, v;g)≥DT(x, y;g). Since this is an output fixed firm game, we
have yk= ¯y, for all k∈ S, and so Wk∈S yk= ¯yfor all k∈ S. More-
over, Wk∈S xk>xkfor all k∈ S. From weak monotonicity, we have
DT(Wk∈S xk,¯y;g)≥DT(xk, yk;g) = DT(xk,¯y;g). Since Tk=T, we have
for all k∈ S:
DT _
k∈S
xk,¯y;g!≥DT(xk, yk;g) = DTk(xk,¯y;g).
Consequently, DT(Wk∈S xk,¯y;g)≥max
k∈S DTk(xk, yk;g) which proves (i).
(ii) If α=−∞, then
Φ−∞
Pk∈S Tk=Vk∈S Tk=Tsatisfies the free disposal
assumption, see Propositions 6.1, 6.5 and 6.7. The directional distance func-
tion is then weakly monotonic on T, in other words (x, y),(u, v)∈Tsuch
that u6yand v>ximply that DT(u, v;g)≥DT(x, y;g). Since this is an
output fixed firm game, we have yk= ¯yand so Vk∈S yk= ¯yfor all k∈ S ⊆ K.
Moreover, Vk∈S xk6xkfor all k∈ S. From weak monotonicity, we have
DT(Vk∈S xk,¯y;g)≤DT(xk, yk;g). Since Tk=T, we have for all k∈ S:
DT ^
k∈S
xk,¯y;g!≤DT(xk, yk;g) = DTk(xk,¯y;g).
This yields DT(Vk∈S xk,¯y;g)≤min
k∈S DTk(xk, yk;g) which proves (ii).
The previous results indicate that the data aggregation process Φ∞(re-
spectively Φ−∞) is relevant with a negative bias that embodies an improve-
ment of technical efficiency in the input fixed firm game (respectively in the
output fixed firm game).
25

6.3 Core of firm games and core partitions
The negative bias or equivalently the α-subadditivity (SUBα) is not sufficient
to avoid the vacuity of the core. Briec and Mussard (2014) investigate the
submodularity of the technical bias inherent to allocative firm games in order
to find super-efficient firm groups. The submodularity of the firm game is
defined as follows.
Definition 6.4 –Submodularity –For all firm games {K, vf(S) : S ⊆
K,Φα}, such that S1,S2⊆ K with S1∩ S26=∅, the game is submodular (or
concave) if:
vα(S1∪ S2)≤vα(S1) + vα(S2)−vα(S1∩ S2).
In the same manner, the submodularity of the aggregation bias is, for S1,S2⊆
Ksuch that S1∩ S26=∅,
ABα(S1∪ S2)≤ABα(S1) + ABα(S2)−ABα(S1∩ S2).(6.5)
The submodularity displays the following interpretation: the loss of technical
efficiency due to the cooperation between two coalitions is no higher than
the aggregated loss of the coalitions taking into account the loss of the joint
cooperation ABα(S1∩ S2). It is shown below that the submodularity of the
aggregation bias is closely related to that of the game vα(·).
Proposition 6.10 If Tkis an upper semilattice (lower semilattice respec-
tively) respecting (T1)-(T4) such that Tk=Tfor all k∈ S ⊆ K, then we
have, respectively:
(i) [(AB+∞is submodular)] =⇒[v+∞is submodular] =⇒[ϕ∈◦
C∞6=∅].
(ii) If min
k∈S1∩S2
v−∞({k})= min
k∈S1∪S2
v−∞({k})then:
[(AB−∞ is submodular)] =⇒[v−∞ is submodular] =⇒[ϕ∈◦
C−∞6=∅].
Proof:
(i) Let S1,S2⊆ K such that S1∩ S26=∅. Since Tk=Tfor all k∈ S, then:
T=TS1=TS2=TS1∩S2=TS1∪S2.
Let zk= (xk, yk)∈Rn+m
+, the submodularity of the technical bias is given
by:
ABα(S1∪ S2)≤ABα(S1) + ABα(S2)−ABα(S1∩ S2).
This entails that:
DT _
k∈S1∪S2
zk;g!−_
k∈S1∪S2
DT(zk;g) + DT _
k∈S1∩S2
zk;g!−_
k∈S1∩S2
DT(zk;g)
≤DT _
k∈S1
zk;g!−_
k∈S1
DT(zk;g) + DT _
k∈S2
zk;g!−_
k∈S2
DT(zk;g).
26

Note that for R=S1,S2:
max
k∈S1∩S2
DT(zk;g)≤max
k∈R DT(zk;g)≤max
k∈S1∪S2
DT(zk;g).
Hence, the game vα(·) represented by the characteristic function DT(·) is
concave, that is, the distance function is submodular:
DT _
k∈S1∪S2
zk;g!≤DT _
k∈S1
zk;g!+DT _
k∈S2
zk;g!−DT _
k∈S1∩S2
zk;g!.
To find the previous relation, note that three cases have to be considered:
either the maximum distance vα({k}) is such that k∈ {S1\ S2}, or k∈
{S2\ S1}or finally k∈ S1∩ S2. By definition, the pay-off vector ϕsatisfies
linearity,symmetry, and efficiency (see Section 2). By Shapley (1972), the
core interior ◦
C+∞is therefore non empty.
(ii) In the lower semilattice case, we have for R=S1,S2:
min
k∈S1∩S2
DT(zk;g)≥min
k∈R DT(zk;g)≥min
k∈S1∪S2
DT(zk;g).
As a consequence, it is easy to show that the previous condition is not suf-
ficient to ensure the submodularity of v−∞. However, if min
k∈S1∩S2
v−∞({k}) =
min
k∈S1∪S2
v−∞({k}) then the submodularity of v−∞ follows,
DT ^
k∈S1∪S2
zk;g!≤DT ^
k∈S1
zk;g!+DT ^
k∈S2
zk;g!−DT ^
k∈S1∩S2
zk;g!.
Then, the game v−∞(·) is concave (submodular). Moreover, since by defi-
nition the pay-off vector ϕsatisfies linearity,symmetry, and efficiency (see
Section 2), then by Shapley (1972), the core interior ◦
C−∞ is non void.
The last proposition is interesting since it allows to get the non vacuity
of the core. However, it does not tell us the whole story of the sign of the
aggregation bias. Indeed, from the previous result, it is clear that the core
of the firm game may be non void and in the same time the aggregation bias
may be positive or negative as can be seen in Propositions 6.8 and 6.9. In
order to get a clear result about the non vacuity of core and about the sign
of the technical bias, we introduce a partition of the core. The first core
displays the set of imputations ϕinherent to a positive bias,
CABα≥0:= ϕ∈M|K| Pk∈S ϕk≤vα(S),∀S ⊂ K
ABα(S)≥0,∀S ⊂ K \(X
k∈K
ϕk=vα(K)).
The second core yields the set of pay-off vectors ϕinherent to a negative
bias,
CABα≤0:= ϕ∈M|K| Pk∈S ϕk≤vα(S),∀S ⊂ K
ABα(S)≤0,∀S ⊂ K \(X
k∈K
ϕk=vα(K)).
27

It is obvious that Cα=CABα≤0∪ CABα≥0. Consequently, either in the input
or the output fixed firm game, whenever the game vαis submodular, it is
always possible to find a solution in the core interior with improvement of
technical efficiency.
Corollary 6.1 If Tkis an upper semilattice (lower semilattice respectively)
respecting (T1)-(T4) such that Tk=Tfor all k∈ S ⊆ K, then we have,
respectively:
(i) I(K, vα(S),¯x)∧(α= +∞)∧(vαis submodular)=⇒[ϕ∈◦
CAB+∞≤0].
(ii) [I(K, vα(S),¯x)∧(α=−∞)∧(vαis submodular)=⇒[ϕ∈◦
CAB−∞≥0].
Corollary 6.2 If Tkis an upper semilattice (lower semilattice respectively)
respecting (T1)-(T4) such that Tk=Tfor all k∈ S ⊆ K, then we have,
respectively:
(i) [O(K, vα(S),¯y)∧(α= +∞)∧(vαis submodular)=⇒[ϕ∈◦
CAB+∞≥0].
(ii) [O(K, vα(S),¯y)∧(α=−∞)∧(vαis submodular)=⇒[ϕ∈◦
CAB−∞≤0].
Proof:
Both corollaries are deduced from Propositions 6.8, 6.9 and 6.10.
7 Conclusion
The data transformation suggested by Post (2001), in order to gauge tech-
nical efficiency, is of real interest to improve the accuracy of the mathemat-
ical tools usually employed in the literature. In the same manner, we first
characterize an aggregator Φ relevant with general distance functions, and
subsequently relevant with directional distance functions. This aggregator
reduces, under the homogeneity property, to the power mean Φα, which is a
good candidate to deal with aggregated technologies.
The improvement of (the aggregated) technical efficiency of a group of
firms has been shown to be no more impossible. The aggregation bias may
be negative and the result belongs to the core interior of the firm game.
This approach generalizes the firm game, introduced by Briec and Mussard
(2014), defined under the standard sum of technology sets. The aggregation
bias, issued from the Φα-aggregator, takes different values: positive, nega-
tive or zero. Then, the cooperation between firms entails all possible cases,
especially with aggregated semilattice technologies that respect all desirable
assumptions.
Our result can be extended to DEA frameworks in order to check the
convergence rate of the Φα-aggregator. It is possible to show that the aggre-
gation bias may be negative for technology subsets associated with values of
28

αthat do not necessarily tend to infinity, as in Example 5.1. Indeed, the di-
rectional distance functions may be computed by solving the following linear
program for all S ⊆ K and for all (xS, yS)∈Rn+m
+:
ˆvα(S) := max δS
s.t. : xS−δSgi>
Φα
Xk∈S θkxk
yS+δSgo6
Φα
Xk∈S θkxk
Φα
X
k∈S
θk= 1 , θk≥0.
Choosing a scalar benough large, the determination of the optimal value α∗
allowing for the improvement of the technical efficiency in at least one [resp.
for all] coalition[s] is the following.
Algorithm for the convergence of α
•Loop to α∈[−b, b] ;
֒→Compute ˆvα(S) for all S ⊆ K ;
֒→Compute ABα(S) = ˆvα(S)−
φα
P
k∈S
ˆvα({k}) ;
7→ If ABα(S)≤0 for at least one given S[for all S] then α∗=α:
End α;
7→ Else : change α;
•End α.
Finally, the aggregated technology inherent to the data transformation,
embodied by the Φα-aggregator, is closely connected to the best input/output
realizations of the firms of the sector (group), i.e., the so-called input/output
fixed firm games. This result is in line with the literature on firm concentra-
tions such as monopolization of industries. In our findings, the improvement
of technical efficiency due to cooperation is, by duality between cost func-
tions and distance functions, a cost reduction. The fusion of the firms may be
viewed as the purchase of the firms of the group realized by the most efficient
one. Empirically, the purchase of the firms may be allowed by the Compe-
tition Authority insofar the cost reduction implied by the fusion improves
the well-being of the consumers. In this case, the well-known cost-benefit
trade-offs are of interest, as in the celebrated result of Williamson (1968).
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