Malmquist指数与Malmquist-Luenberger指数的比较研究

JournalofXianUniversityofEngineeringScienceandTechnology192(74)20056Vol.19,No.2(SumNo.74):1671850X(2005)02015205MalmquistMalmquistLuenberger崔文田,高宇,张博(,710049):提出了基于减少有害产出同时增加有益产出的MalmquistLuenberger生产力指数评价方法.分析了该指数与Malmquist指数联系与差异,并通过美国电力产业的数据(19962000)对两种指数进行了实证对比.与Malmquist生产效率评价方法相比较,本文中所提出的方法可用于企业面对环境约束下的生产效率评价,并且在评价中不需要有关投入和产出的价格信息,评价指数也可以通过求解一个简单的线性规划模型来计算.:生产力;直接距离函数;生产力指数:F224:A().,,,[1,2].,,[3~6].,.(TornqvistFisher[7]),.,().,Pittman[8]Fare[9].,,Malmquist[10].,,.,MalmquistLuenberger.,,,.Malmquist,,.,49,.1有益有害两种产出的数学描述,,:20040910:,Email:[e-mail]::(1958),,,.SO2.,,.y!RM+,b!RI+,x!Rn+;x,,p(x):p(x)={(y,b):x(y,b)}.(1):(1)有害产出的弱随意处置性(y,b)!p(x)0∀∀1,(y,b)!p(x);(2)有益产出的随意处置性(y,b)!p(x)y∀y,(y,b)!p(x);(3)有害产出与有益的关联性(y,b)!p(x),b=0,y=0.1,(1),;(2).(3)#∃,,.(1)(3),1[4].MalmquistShephard[11]:D0(x,y,b)=inf{:((y,b)/!p(x)}.(2)Shephard,,,Farrell[12].,(3)Shephard,.D0(x,y,b;g)=sup{:((y,b+g)!p(x)}.(3)g#∃.y=(y,-b),,.Shephard,,.Shephard1.C,ShephardA,OC/OA.OA/OC,.,B.,,g.,CB(),.,g=(y,b),(3)D0(x,y,b;y,b)=sup{:D0(x,(y,b)+(y,b))∀1}=sup{:(1+)D0(x,y,b)∀1}=sup{:∀D0(x,y,b)-1-1}=1/D0(x,y,b-1).(3)(3)Shephard.D0(x,y,b;y,b)=(1/D0(x,y,b))-1D0(x,y,b)=1/(1+D0(x,y,b;y,b)).2带有有害产出的生产率评价方法,.Fare[10]Shephard.1532MalmquistMalmquistLuenbergerStenMalmquist(1953),Malmquist.:t=1,2,%,T,Fare,GrosskopfLindgrenMalmquistMt+1t=Dt0(xt+1,yt+1,bt+1)Dt0(xt,yt,bt)Dt0(xt+1,yt+1,bt+1)Dt+10(xt,yt,bt)t/2.(4)(4),MalmquistMEFFCHt+1t=Dt0(xt+1,yt+1,bt+1)/Dt0(xt,yt,bt)(5)MTECHt+1t=Dt0(xt+1,yt+1,bt+1)Dt+10(xt+1,yt+1,bt+1)Dt0(xt,yt,bt)Dt+10(xt,yt,bt)1/2(6),Mt+1t=MEFFCHt+1t&MTECHt+1t.Malmquist:,..FisherTornqvist,FisherTornqvist,,.Malmquist,,,,.,Malmquist,g=(y,-b),Malmquist.MalmquistLuenberger,ML:MLt+1t=(1+Dt0(xt,yt,bt;yt,-bt))(1+Dt0(xt+1,yt+1,bt+1;yt+1,-bt+1))(1+Dt+10(xt,yt,bt;yt,-bt))(1+Dt+10(xt+1,yt+1,bt+1;yt+1,-bt+1))1/2.(7),g=(y,b)(y,-b),MalmquistLuenbergerMalmquist.Malmquist,MLEFFCHt+1t=1+Dt0(xt,yt,bt;yt,-bt)1+Dt+10(xt+1,yt+1,bt+1;yt+1,-bt+1),(8)MLTECHt+1t=(1+Dt+10(xt,yt,bt;yt,-bt))(1+Dt0(xt,yt,bt,yt,-bt))(1+Dt+10(xt+1,yt+1,bt+1;yt+1,bt+1))1+Dt0(xt+1,yt+1,bt+1;yt+1,-bt+1))1/2;(9)MLt+1t,MLt+1t=MLEFFCHt+1t&MLTECHt+1t.MalmquistLuenbergerMalmquist,,..t,K(xt,k,yt,k,bt,k),k=1,2,%,K;t=1,2,%,T,(1)(3)P(xt)={(yt,bt):Kk=1zkyt,km∋ytm,m=1,2,%,M;Kk=1zkbt,ki=bti,i=1,2,%,I;Kk=1zkxt,kn=xtn,n=1,2,%,N;zk∋0,k=1,2,%,K}(10):(1),p(x)=p(x),0;(2),x∋x!p(x)∀p(x).(10),r(xt,r,yt,r,bt,r),t=1,2,%,T,MalmquistMalmquistLuenberger,(11)(12).(1)Malmquist:(Dt0(xt,r,yt,r,bt,r))-1=max;s.t.Kk=1zkyt,km∋yt,rm,m=1,2,%,M;Kk=1zkbt,ki=bt,ri,i=1,2,%,I;Kk=1zkxt,kn∀xt,rn,n=1,2,%,N;zk∋0,k=1,2,%,K.(11)15419(2)MalmquistLuenberger:(Dt0(xt,r,yt,r,bt,r;yt,r,-bt,r)=max;s.t.Kk=1zkyt,km∋(1+)yt,rm,m=1,2,%,M;Kk=1zkbt,ki=(1-)bt,ri,i=1,2,%,I;Kk=1zkxt,kn∀(1-)xt,rn,n=1,2,%,N;zk∋0,k=1,2,%,K.(12)3应用实例,.4919962000.(Capacity)(HeatInput);;(CO2)(SO2)(NOx).1,19962000.MalmquistMalmquistLuenberger(5),(6),(8),(9)2.1TCO213717968.2221496531.003002327.004334341.25SO281105.67273838.0024.0052631.321996NOx40503.24146446.002046.0029245.64(K=49)12450441.1221081179.002296061.003742157.122157.723969.10349.20664.38134645752.10211137057.0027271227.0041306558.86CO214045107.2623591360.002597264.844709730.11SO281084.16271920.6925.0055160.471997NOx41065.08161505.002026.2931860.87(K=49)12718955.4721279900.002241127.004043606.852158.433969.10349.20664.91137507485.10229935264.0025256344.0044848526.14CO214064864.8222730427.232749795.604601483.88SO280971.48284616.3029.0054560.851998NOx38282.40119059.802145.8024689.59(K=49)12942695.8220823017.002067714.003955720.002171.263969.10349.20654.16137855152.90232719168.0025295690.0044111993.61CO213684966.1023032323.712619156.144667651.83SO275123.23245243.2029.7052688.591999NOx33482.95104356.622536.0018972.73(K=49)12802306.4320387721.001818732.004075683.262168.653969.10349.20656.84135469831.50236094333.0024102110.0044604985.72CO213843726.2722337062.002708874.304537771.08SO264417.80171505.5026.8043766.622000NOx29278.1561701.782533.5312171.96(K=49)13153510.7821681930.002215215.004204065.652169.583969.10349.20657.95135675143.60217707990.0024881613.0043260789.911552MalmquistMalmquistLuenberger2MalmquistMalmquistLuenbergerM(ML)MEFFCH(MLEFFCH)MTECH(MLTECH)1996/1997(k=36)1.00267241.00546211.0015461.0038251.0011246591.001630921997/1998(k=34)0.98523431.00275970.989550.99654270.9956384271.0062385651998/1999(k=37)0.98723670.99643570.98619740.98630691.0010538461.0102693511999/2000(k=38)1.00136721.01640530.98536230.99579051.0162427261.020701903(k=145)0.99430271.0053930.99057540.99554171.0037821391.00991353549Malmquist,1/3,,(4)Malmquist(6).,MalmquistLuenberger.MalmquistLuenberger,Malmquist,2.,Malmquist,MalmquistLuenberger,MLM.4结束语MalmquistMalmquistLuenberger,,.,,.MalmquistLuenberger19962000.:[1]BRANNLUNDR,CHUNGYH,FARER,etal.Emissiontradingandprofitability:TheSwedishPulpandPaperIndustry.DiscussionPaperNo.95,SouthernIllinoisUniversityatCarbondale,1995.[2]BRICEW.Agraphtypeextensionoffarrelltechnicalefficiencymeasure[M].Memo:GREQAM,1995.[3]BALLVE,LOVELLCAK,NEHRINGRF.Incorporatingundesirableoutputsintomodelofproductionanapplic