Pro duction Technology and Technical Efficiency in ( k, y) Sp ace

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1 15 5 Vol. 15 No OPERA TIONS RESEARCH AND MANA GEMEN T SCIENCE Oct. 006 ( ) : K L Y k ( L ) y ( K L Y) ( k y) ( k y) ( k y) ( K L Y) ( C R ) C R ( k y) : ; ;; : F4. 0 :A : (006) Pro ductio Techology ad Techical Efficiecy i ( k y) Sp ace KAN G Mei ( School of Ecoom ics ad M aagemet Harbi Istit ute of Techology Harbi Chi a) Abstract : Techical efficiecy ca be estimated by paramet ric or oparamet ric approach. As covetioal ef ficiecy aalyzig idexes capital K labor ad output Y ca t idetif y embodied techology. Some litera t ures adopt labor productivity ad capitallabor ratio k (or plus labor) i paramet ric efficiecy aalysis. I or der to explore t he oparamet ric approach uder t hese idexes t he covetioal costat ret urs to scale (RTS) productio techology is projected ito ( k y) so that a oicreasig RTS productio techology is obtaied. Techical efficiecy defied i ( k y) space just equals the covetioal C R techical efficiecy i ( K L Y) space. This result simplifies the calculatio of ( C R) techical efficiecy ad also assets physical techology ca be approached i ( k y) space which called asset s operatigf rotier techology. Key words :liear program ; dual liear program ; productio techology ; techical efficiecy 0 Farrell (1957) Aiger Chu (1968) [46 ] Elorata Holmstrgm [7 ] Agrell West (001) [8 ] K L Y [9 ] ( K L Y) K L : : (1970) Chia Academic Joural Electroic Publishig House. All rights reserved.

2 k ( L ) y 1 ( K L Y) ( k y) ( k y) ; ( k y) ( K L Y) C R ;3 1 ( k y) 0 K > 0 L > 0 KL Y ( K L Y) k y ( k y) ( K L Y) A 1 = { ( K i L i Y i ) : } ( k y) A = { ( k i y i ) : } i i DM U i ( K i L i Y i ) DM U i ( k i y i ) ( K L Y) A 1 C R S C R A 1 ( ) S C R (004) : S C R = { ( K L Y) : i K i K i L i L Chia Academic Joural Electroic Publishig House. All rights reserved. i Y i Y i 0 } (1) S C R S X 1 = { ( K L Y) : K > 0 L > 0 Y 0} ( K L Y) S X 1 ( k y) S X = { ( k y) : k > 0 y 0} k 0 = K 0 / L 0 y 0 = Y 0 / L 0 F :{ s ( K 0 L 0 Y 0 ) ; s R + < S X 1 ( k 0 y 0 ) S X } F S X 1 () ( k y) S X F S X 1 S X S C R : ( K 0 L 0 Y 0 ) S C R { s ( K 0 L 0 Y 0 ) : s R + } < S C R F ( k y) ( k 0 y 0 ) S C R F S 3 ( S C R ( k y) ) S 3 5 : 1 () A < S 3 ( ) ( ) ( k 1 y 1 ) S 3 ( k y ) S 3 ( k 1 y 1 ) + (1 - ) ( k y ) S S 3 (0 0) k 0 (1) 1 : k 1 > 0 k > 0 F ( k 1 y 1 ) S C R { L ( k 1 1 y 1 ) : L R + } F R + ( k 1 1 y 1 ) S C R R + ( k 1 y ) S C R S C R L R + L ( k 1 1 y 1 ) + (1 - ) L ( k 1 y ) S C R F ( k 1 + (1 - ) k y 1 + (1 - ) y ) S 3 () : k 1 > 0 ( k y ) = (0 0) ( k 1 y 1 ) S 3 k 1 > 0 L R + L ( k 1 1 y 1 ) S C R ( k 1 1 y 1 ) = ( k 1 y 1 ) S C R 1 S C R ( k 1 1 y 1 ) S C R S C R S C R F ( k 1 y 1 ) S 3 3 ( ( ) ) : (1) ( k 1 y 1 ) S 3 k > k 1 ( k y 1 ) S 3 () ( k 1 y 1 ) S 3 y < y 1 ( k 1 y ) S 3 (1) ( k 1 y 1 ) S 3 L R + L ( k 1 1 y 1 ) S C R k > k 1 S C R L ( k 1 y 1 ) S C R F ( k y 1 ) S 3 4 () :( k 1 y 1 ) S 3 a (0 1 ] a ( k 1 y 1 ) S 3

3 5 : 77 ( ) 5 () : S 3 ( k y) A ( ) S 3 ( k y) A ( ) S 3 ( k y) S 3 ( k y) ( k y) ^S 3 () : ^S 3 < S 3 (1) A Α ^S 3 () (3) ( ) (4) ^S 3 F ^S 3 = F - 1 ( ^S 3 ) ^S ^S < S C R : (i) A 1 Α ^S ( ii) ( iii) ( iv) ( ) ^S 3 < S 3 ^S = F - 1 ( ^S 3 ) F () ^S < S C R ( i ) ( iv) :A Α ^S 3 ( k i y i ) ^S 3 { L ( k i 1 y i ) : L R + } ^S L i ( k i 1 y i ) = ( K i L i Y i ) ^S (ii) [0 1 ] ( k 1 y 1 ) ^S ( k y ) ^S F - 1 ( k 1 y 1 ) ^S 3 ( k y ) ^S 3 ( k 1 y 1 ) + (1 - ) ( k y ) = ( k 1 + (1 - ) k + (1 - ) y 1 + (1 - ) y ) (1 - ) (1 - ) = k 1 + k + (1 - ) 1 y + (1 - ) + y + (1 - ) + (1 - ) ( + (1 - ) ) ^S 3 (1 - ) (1 - ) k 1 + k + (1 - ) 1 y + (1 - ) + y + (1 - ) ^S + (1-3 ) F - 1 ( k 1 y 1 ) + (1 - ) ( k y ) ^S (iii) ( K 1 ) ^S ^S ( K 1 R + } = { a ( K 1 ) : a R + } < ^S ^S ) ^S 3 : F - 1 ( K 1 ) = { L ( K 1 1 ) : L (iv) ( K 1 ) ^S ( K 1 ) (0 0 0) K K 1 ^S ( K 1 ) ^S 3 ^S 3 ( K ) ^S 3 (0 0) ^S 3 ^S 3 K 1 ( K - (0 0) ^S 3 ( K ) ^S 3 F - 1 ( K 1 ) = ( K Y ) ^S ) + ^S ( K L Y ) A 1 ( ) ^S < S C R S C R ( K L Y) S 3 ( k y) A ( ) : 1 F : F :{ s ( K 0 L 0 Y 0 ) : s R + } < S X 1 ( k 0 y 0 ) S X k 0 = K 0 / L 0 y 0 = Y 0 / L 0 S 3 S 3 ( k y) FG ( ) S 3 = { ( k y) : i k i k i y i y i 1 i 0 } () S 3 FG S 3 ( k y) FG () (004 P. 73) ( k y) A = { ( k i y i ) : } FG Chia Academic Joural Electroic Publishig House. All rights reserved.

4 S 3 k 0 S 3 ^y 0 k 0 O FT ( k 0 ) = ^y 0 ; DM U 0 ( k 0 y 0 ) y 0 k 0 ^y 0 DM U 0 ( k 0 y 0 ) T E ( k 0 y 0 ) = y 0 / ^y 0 ; ^y 0 ;y 0 DM U 0 ( k 0 y 0 ) D T ( k 0 y 0 ) = ^y 0 / y 0 O FT ( k 0 ) = ^y 0 k 0 S 3 k 0 D EA (004) : ( k y) A = { ( k i y i ) : } DM U 0 ( k 0 y 0 ) A FG ^y 0 = 3 y 0 3 : max i k i k 0 i y i i Chia Academic Joural Electroic Publishig House. All rights reserved. y 0 i 0 DM U 0 ( k 0 y 0 ) T E ( k 0 y 0 ) = 1/ 3 3 DM U 0 ( k y) T E ( k 0 y 0 ) = y 0 / ^y 0 DM U 0 ( K L Y) C R T E ( K 0 L 0 Y 0 ) T E ( k 0 y 0 ) = T E C R ( K 0 L 0 Y 0 ) T E ( k 0 y 0 ) (3) C R T E C R ( K 0 L 0 Y 0 ) (4) (3) (4) : max s. t i K i K 0 i L i L 0 (4) i Y i Y 0 i 0 mi (k 0 + ) s. t. k i - vy i + 0 vy 0 = 1 0 v 0 0 (5) (3) (5) (5) (6) (6) (7) (7) (4) ((7) L 0 = L 0 i = i ) (3) (5) (6) (7) (4) mi 1 L 0 (K 0 + L 0 ) s. t K i - v Y i + L i 0 v Y 0 = L 0 0 v 0 0 (6) (3)

5 5 : max L 0 i K i K 0 / L 0 i L i 1 i Y i Y 0 i 0 79 (7) 3 ( K L Y) K L () ( K L Y ) ( k y) ( k y) ( k y) ( K L Y) ( K L Y) C R ( k y) C R - : [ 1 ] Flores J P Simar L. Parametric approximatios of oparametric frotiers[j ]. Joural of ecoometrics 005 l (14) : [ ] Sickles R C. Pael estimators ad the idetificatio of firmspecific efficiecy levels i parametric semiparametric ad oparametric settigs [J ]. Joural of ecoometrics 005 (16) : [ 3 ] Dorfma J H Koop G. Curret developmets i productivity ad efficiecy measuremet[j ]. Joural of ecoometrics 005 (16) : [ 4 ] MurilloZamorao L R VegaCervera J R The use of parametric ad oparametric frotier methods to measure the productive efficiecy i the idustry sector : A comparative study[j ]. Iteratioal joural of productio ecoomics 001 (69) : [ 5 ] Piesse J Thirtle C. A Stochastic frotier approach to firm level efficiecy techological chage ad productivity durig the early trasitio i Hugary[J ]. Joural of comparative ecoomics 000 (8) : [ 6 ] Griliches Z Regev H. Firm productivity i Israeli idustry [J ]. Joural of ecoometrics 1995 (65) : [7 ] Elorata E J. Holmstr om. Productivity recosidered : Critical assessmet of ivestmets[j ]. Iteratioal joural of productio ecoomics 1998 (5657) : [ 8 ] Agrell P J West B M. A caveat o the measuremet of productive efficiecy[j ]. Iteratioal joural of productio ecoomics 001 (69) : 114. [9 ] Jighai Z xiaoxua L Bigste A. Efficiecy techical progress ad best practice i Chiese state eterprises ( ) [J ]. Joural of comparative ecomomics 003 (31) : [ 10 ] Dhawa R. Firm size ad productivity differetial : theory ad evidece from a pael of US firm[j ]. Joural of ecoomic behavior & orgaiza tio 001 (44) : [11 ] Becchetti L Sierra J. Bakruptcy risk ad productive efficiecy i maufacturig firms[j ]. Joural of bakig & fiace 003 ( 7) : [ 1 ]. [ M ]. : Chia Academic Joural Electroic Publishig House. All rights reserved.