数据包络分析CCR、BCC、SBM、TFP等

14(0571-86971892)50(DataEnvelopmentAnalysisDEA)(StochasticFrontierAnalysis,SFA)(DEA)CharnesCooporRhodes1978(DecisionMakingUnitsDMU)DEADEADEA--DEADEA(SFA)1977Aigner,Lovell,Schmidt,MeeusenvandenBroeck(TFP)Färeetal(1994)Malmquist214.1Farrell(1957)Debreu(1951)andKoopmans(1951)(01):,Farrell/Farrell(x1x2)(y)(14-1)14-114-1SS'()PQPQP0PTE=0Q0P=1-QP0P01114-1Q14-1SS'14-1AA(Allocativeefficiency,AE)(EconomicEfficiency,EE)P(AE)AE=0R0QEE=0R0P(0Q0P)×(0R0P)=0R0PEE=TE×AE14-114.13Farrell(a)(14-2)(b)Cobb-Douglas14-2:“??”(14-3)14-314-3(a)f(x)PFarrellAB/AP,CP/CD(FareandLovell1978).14-3(b)PAB/AP=CP/CD(y1y2)(x1)4(14-4)14-4,ZZA(A)ZZ14-4Farrell14-4ABTEO=0A/0BDD,AEO=0B/0CEEO=(0A/0C)=(0A/0B)(0B/0C)=TEOAEO0~1()14.2CCRFarrell(1957)20FarrellBoles(1966)Afriat(1972)CharnesCooperRhodes(1978)(DEA)CCRCharnes,CooperRhodes(1978)(CCR)DEA,DEABanker,CharnesCooper(1984)(VRS)BCCDEACCR14.2CCR514.2.1CCRN()MSxiyiMNX,SNY,NDEA12(14-2)14-2,uyi/vxi,uS1vM1:(),max1,1,2,,,0uviijjuyvxstuyvxjNuv′′′′≤=≥uv,i1vxi=1:(),max10,1,2,,,0iijjystxyxjNµνµνµνµν′′=′′−≤=≥uvµν:,min00,0iistyYxXθλθλθλλ−+≥−≥≥(14.1)θλN1(MSN1)θiFarrell(1957)θ≤1,1614.2.2DEA(14-2)(14-1)14-5CDABFarrell(1957)AB0A/0A0B/0BAx2(CA),DEAFarrell(θ)iYλ-yi=0,0xi-Xλ=0(θλ)014-514-5Ax2CA,(C)(AliSeiford1993)(14-5A(C):,,max0,0,0ssiiwesesstsxXsYyssλθλλλ−+−+−+−+=+=−=−≥≥≥(14.2)14.2CCR7s+,s(DEA)/FerrierLovell(1990)DEA14.2.3DEAFarrellCCR(VRS)(14-3)DEADUM(CoelliPerelman1996)14.2.4θ1DEAθ1θ1DEA814-614-65(ABCDE)5DEABCD331DEA31CCBDCCC0C/0C1.220%DEA(AE)14.2.5CCRCCRDEADPS14-72114-7CCR14.2CCR914-7--CCR14-7DEA_CCRiX1X2Y110.50000.00000.500020.10000.20000.500030.05560.11110.277840.14290.28570.714350.10000.20000.5000S*10.00000.50000.00000.00000.000020.00001.00000.00000.00000.000030.00001.00000.00000.00000.500040.00000.21430.00000.00000.285750.00000.00000.00000.00001.0000iix1x2y111.00002.00001.000035.00005.00003.000042.14291.42861.0000S*-12345x10.00000.00000.00000.00000.0000x20.50000.00000.00000.00000.0000S*+12345y10.00000.00000.00000.00000.0000iDEA10.5000DEA21.0000DEA30.8333DEA40.7143DEA51.0000DEA3TEI=0.833,,316.7%14-8332525325331014-8CCR4TEI=0.71431TEI=0.5211x2(Koopman)0.5x2125TEI1.0CCRDEADEA1DEA*=1iDEA2DEA*=1s-*=0s*=0()jDEA*1iDEADEADEAθ∗DEA,,,ssλθ∗+∗−∗∗*=1s-*=0s*=0*=1s-*=0s*=0iDEADEAs-*=0s*=0CCRεCCRDEACCR14.311DEACCRDEADEA11yixiii2=1yixiyiiDEADEA3=1yixiiDEA14.314.3.1(VRS)BCC(CRS)Banker,CharnesCooper(1984)DEA(VRS)(TE)(Scaleefficiency,SE)VRS(VRS)DEA(CCR)14.2I1λ=1BCC,min00,110iistyYxXIθλθλθλλλ−+≥−≥′=≥(14.3)12I1I11(SE)CCRBCCDEACCRBCCBCCCCR(14-9)14-9CRSCCRVRSBCCDEACRSPPPC,VRSPPVPCPV,14-9DEA()14-9TECRS=APC/APTEIVRS=APV/APSE=APC/APV0~1TECRS=TEVRS×SEAPC/AP=(APV/AP)×(APC/APV),CRS14.313(NIRS)DEACCRBCCBCC:,max00,110iistyYxXIφλφφλλλλ−+≥−≥′=≥(14.4)1≤φ∞,φ-11/φ0-1(TE)DEA14-1014-10DEAPPy1APAPy1BCCBCCDEADPS14-111414-11BCC14-11--BCC14-11VRSVRS()iDEA11.00001DEA20.62505DEA31.00001DEA40.90004DEA51.00001DEACRS14-11CCR0.5000.5001.0000.8000.833TECRS=TEVRS×SESE=TECRS/TEVRS(SE)0.5000.8001.0000.8890.83314-1214.31514-12VRS(BCC)DEA14-12CRS3(DEA)VRS135(1DEA)22CRSVRSCRS(TE)2/4=0.5;VRSTE2.5/4=0.625CRSTEVRSTE0.5/0.625=0.814.3.2(VRS)IRSDRS()()(Non-IncreasingReturnstoScale,NIRS)DEADEADecreasingReturnstoScale,DRSBCC11Iλ′=11Iλ′≤,min00,110iistyYxXIθλθλθλλλ−+≥−≥′≤≥(14.5)14-9NIRSDEA()NIRSVRS(14-9P)16(14-9Q)CCRBCCDRSDEAIRSDRS11Iλ′≤11Iλ′≥114.3.3DEASBMSBM(Slacks-basedmeasure,SBM)DEA(profit)SBMCCRCCRSBM(profit)SBM()()01010011//min11//0,0,0miiisrrrmsxssystxXsyYsssρλλλ−=+=−+−+−=−=+=−≥≥≥∑∑Charnes-CooperTone()()010100min1//11//(14.6)0,0,0,0miiisrrrtmsxsttssytxXstyYssstτ−=+=−+−+=−=+=Λ+=Λ−Λ≥≥≥∑∑SBMCCRSBM/(profit)CCR(Ratio)SBM14.317CCRSBMSBMDEA336DPS14-1314-13SBM14-13--SBM14-13SBMSBMCCRiCCR11.0000120.8333130.6250140.6000150.4286160.15000.25SBMCCRSBMCCRSBMCCR14.3.3DEA(Non-discretionary)DEABCCDEA18(XD)(XND)BCC,min000110iiDDNDNDistyYxXxXIθλθλθλλλλ−+≥−≥−≥′=≥(14.7)DEAθDEA211214-1414-14DEA14-14--14-14x1x2…√x1√iD1D2D3D4D5D6D7D8D9D10D11D1214.4DEA190.5781.0000.7760.9360.5190.7880.8180.6440.6800.8300.7990.76914.4DEA14.4.1DEACCR(TE)DEA*'*,*min00,110iiiixiwxstyYxXIλλλλλ−+≥−≥′=≥(14.8)wi*ixiwiyii(CE)*/iiiiCEwxwx′′=AE=CE/TEAECETE01114.4.2BCC(TE)DEA20*'*,*min00,110iiiiyipystyYxXIλλλλλ−+≥−≥′=≥(14.9)pi*iyipixii(RE)*/iiiiREpypy′′=AE=RE/TEAERETE01114.4.3DEADEA**'**,,**min00110iiiiiiixyipywxstyYxXIλλλλλ′−−+≥−≥′=≥(14.10)DEA010014.4.4DEADPS()Ctrl14.4DEA2114-15DEA221214-1514-15--i11.00001.00001.000021.00001.00001.000030.88270.93660.826841.00000.91460.914650.76350.97990.748160.83480.69820.582870.90200.99800.900280.79630.94220.750390.96040.98380.9448100.87060.79830.6951110.95510.82420.7872120.95820.99460.95312214.5Malmquist(Paneldata)MalmquistTFPMalmquist(TFP)()MalmquistCaves,ChristensenandDiewert(1982)FäreShephard(1953)(distancefunctions)()(output-oriented)(input-oriented)Shephard(1970)Färe(1988)(){}0(,)inf:,/()DxyxyPxδδ=∈xyδFarrellP(x)yP(x)1y1yP(x)1CavesChristensenDiewert(1982)Malmquist11(,)(,)ttottotottDxyMDxy++=(14.11)11011010(,)(,)tttttttDxyMDxy+++++=(14.12)0(,)tttDxy1011(,)tttDxy+++(tt+1)011(,)tttDxy++10(,)tttDxy+Färe(1989)MalmquistMalmquist14.5Malmquist23112011011011100(,)(,)(,,,)(,