The Measurement of Efficiency of Production
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Standard Notations and Mathematical Appendix A.1 Some Standard Notations Let A and B be two sets, we mean by
.=
A:= B A is defined by B;
E:
aE:A
a is an element in A;
ft-
aft-A
a is not an element in A;
c
A~B
A is a subset of B;
C
ACB
A is a true subset of B;
~
A =~
A is an empty set;
n
AnB
A intersection B;
\
A\B
A \ B: = Ix: x E: A, x
ft-
Bl;
201
202
THE MEASUREMENT OF EFFICIENCY OF PRODUCTION
+
A
+B
A
+ B: =
(z: x E: A, Y E: B, z
=
x
+ y);
Euclidean space of dimension I; If x and y E: Rl, then x 2; y if and only if Xi 2; yJor all i
I·,
= 1, 2, ... ,
x 2 Y if and only if x 2; y and x ¥- y;
>
x
>y
if and only if Xi
x ). y if and only if Xi I
11·11
> Yi for all i = 1, 2, ... , I; > Yi or Xi = Yi = 0, i = 1, 2, ... , I;
Ilxll: = (~ (Xi)2)\ Euclidean norm of x E: Rl; 1~1
=
R~:
(x: x E: Rl, X 2; 0);
= (x: x E: Rl, x> 0);
R~+: R~: =
(x: x E: Rl, X ;S 0);
[,]
[a, b]:
=
[,)
[a, b): = (x E: R: a ;S x
3:
there exists;
(x E: R: a ;S x ;S b);
< b);
the sequence Sl converges to so; closure of A;
= PIXI + P2X2 + ... + PnXn, P and x E: Rn;
px
px
ru
ru=rlul +r2u2
o =>
A, x E: Rl, A 0 x: = (AIXI, A2X2, ... , AIXI);
x E: A
+ ... +rmum, randuE:R m;
=> x E: B, x
x E: A
~
belongs to A only if x belongs to B;
x E: B, x belongs to A if and only if x belongs to
B; --+
+00 x
--+
+00, x tends to +00.
A.2 Mathematical Appendix
A.2.1
A set A ~ R! is bounded if and only if supUlx - yll: x E: A, y E: A)
A.2.2
A set F ~ Rl is closed if and only if for every sequence Sl with Sl E: F for alII = 1,2, ... , SO E: F.
< +00.
--+
SO
203
APPENDIX
A.2.3
A set K s;;: RI is compact if and only if it is closed and bounded.
A.2.4
A correspondence S :RI - S(x) s;;: Rk is closed if and only if for every sequence (Xl, /) E: RI X Rk such that Xl - XO and yl E: S(x l ), yO E: S(XO).
A.2.S
A function ¢: R"t - R+ is upper semicontinuous if and only if for each XU E: R"t and each sequence Xl - xO, lim sup ¢(Xl) ;£ ¢(XO). 1-+
00
A.2.6
A function! RI ---+ R is lower semicontinuous if and only if -fis upper semicontinuous.
A.2.7
A class of sets A has the finite intersection property if and only if every finite subclass has a nonempty intersection.
A.2.S
A correspondence S: R ---+ S(x) s;;: Rk is quasi-concave if and only if S(Ax + (l-A)X') ~ S(x) n S(x'), for all x, x' E: RI and A E: [0, 11.
204
THE MEASUREMENT OF EFFICIENCY OF PRODUCTION
Bibliography Afriat, S. N. (1972) "Efficiency Estimation of Production Functions," International Economic Review, 13:3 (October), pp. 568-98. Aigner, D. J., Amemiya, T., and Poirier, D. J. (1976) "On the Estimation of Production Frontiers: Maximum Likelihood Estimation of the Parameters of a Discontinuous Density Function," International Economic Review, 17:2 (June), pp. 377-96. ___ , and Chu, S. F. (1968) "On Estimating the Industry Production Function," American Economic Review, 58:4 (September), pp. 826-39. ~, Lovell, C. A. K., and Schmidt, P. (1977) "Formulation and Estimation of Stochastic Frontier Production Function
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