The standard reverse approach for decomposing economic inefficiency

Figures & data

Table 1. The traditional profit inefficiency decomposition based on the weighted additive model (Range Adjusted Measure, RAM). (w,p) = (1, 4).

Firm	Model (3)(3) $$\begin{array}{cccc}T{I}_{\mathit{\text{WA}}}\left(x_o,y_o;{\rho }^{-},{\rho }^{+}\right)=\hfill & \underset{s_{\mathit{\text{oWA}}}^{-},s_{\mathit{\text{oWA}}}^{+},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{m=1}^M{\rho }_m^{-}{s}_{\mathit{\text{oWAm}}}^{-}+\sum _{n=1}^N{\rho }_n^{+}{s}_{\mathit{\text{oWAn}}}^{+}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oWAm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{y}_{\mathit{\text{jn}}}+s_{\mathit{\text{oWAn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{-}=\left(s_{\mathit{\text{oWA}}1}^{-},\dots ,s_{\mathit{\text{oWAM}}}^{-}\right)\ge 0,\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{+}=\left(s_{\mathit{\text{oWA}}1}^{+},\dots ,s_{\mathit{\text{oWAN}}}^{+}\right)\ge 0,\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & {\lambda }_{\mathit{\text{oj}}}\ge 0,\hfill & j=1,\dots ,J.\hfill \end{array}$$(3)	WA Project. (3)(3) $$\begin{array}{cccc}T{I}_{\mathit{\text{WA}}}\left(x_o,y_o;{\rho }^{-},{\rho }^{+}\right)=\hfill & \underset{s_{\mathit{\text{oWA}}}^{-},s_{\mathit{\text{oWA}}}^{+},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{m=1}^M{\rho }_m^{-}{s}_{\mathit{\text{oWAm}}}^{-}+\sum _{n=1}^N{\rho }_n^{+}{s}_{\mathit{\text{oWAn}}}^{+}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oWAm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}{y}_{\mathit{\text{jn}}}+s_{\mathit{\text{oWAn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_{\mathit{\text{oj}}}=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{-}=\left(s_{\mathit{\text{oWA}}1}^{-},\dots ,s_{\mathit{\text{oWAM}}}^{-}\right)\ge 0,\hfill & m=1,\dots ,M,\hfill \\ \hfill & \hfill & s_{\mathit{\text{oWA}}}^{+}=\left(s_{\mathit{\text{oWA}}1}^{+},\dots ,s_{\mathit{\text{oWAN}}}^{+}\right)\ge 0,\hfill & n=1,\dots ,N,\hfill \\ \hfill & \hfill & {\lambda }_{\mathit{\text{oj}}}\ge 0,\hfill & j=1,\dots ,J.\hfill \end{array}$$(3)	Monetary (6)(6) $$\Pi I\left(x_o^{},y_o^{},w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o^{}-w\cdot x_o^{}\right)=T\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,w,p\right)+A\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,w,p\right).$$(6)			Normalized^(1) (8)(8) $$N\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,\tilde{w},\tilde{p};{\rho }^{-},{\rho }^{+}\right)=T{I}_{\mathit{\text{WA}}}\left(x_o,y_o;{\rho }^{-},{\rho }^{+}\right)+{\mathit{\text{NAI}}}_{\mathit{\text{WA}}}\left(x_o,y_o,\tilde{w},\tilde{p};{\rho }^{-},{\rho }^{+}\right).$$(8)
			Profit Ineff.	Tech. Profit Ineff.	Alloc. Profit Ineff.	Profit Ineff.	Tech. Ineff.	Alloc. Ineff.
$(x_o,y_o)$	Tech. status	$({\widehat{x}}_{\mathit{\text{oWA}}}^{},{\widehat{y}}_{\mathit{\text{oWA}}}^{})$	$\Pi I$	$T\Pi I_{\mathit{\text{oWA}}}$ =$p\cdot s_{\mathit{\text{oWA}}}^{+*}+w\cdot s_{\mathit{\text{oWA}}}^{-*}$	$A\Pi I_{\mathit{\text{WA}}}$	$N\Pi I_{\mathit{\text{WA}}}$	$T{I}_{\mathit{\text{WA}}}$	${\mathit{\text{NAI}}}_{\mathit{\text{WA}}}$
C1	C2	C3	C4	C5	C6= C4-C5	C7= C4/$N{F}_{\mathit{\text{WA}}}^{}$	C8	C9= C7-C8
A = (6,9)	Eff.	(6, 9)	9	0	9	0.75	0	0.75
B = (9,12)	Eff.	(9, 12)	0	0	0	0	0	0
C = (7,10)	Eff.	(7, 10)	6	0	6	0.5	0	0.5
D = (6,4)	Ineff.	(6, 9)	29	20	9	2.417	0.278	2.139
E = (8,8)	Ineff.	(6, 9)	15	6	9	1.250	0.222	1.028
F = (8,9)	Ineff.	(6, 9)	11	2	9	0.917	0.167	0.75
G = (9,9)	Ineff.	(6, 9)	12	3	9	1	0.25	0.75
H = (9,10)	Ineff.	(7, 10)	8	2	6	0.667	0.167	0.5
I = (10,8)	Ineff.	(6, 9)	17	8	9	1.417	0.389	1.028
J = (12,10)	Ineff.	(7, 10)	11	5	6	0.917	0.417	0.5
K = (12,3)	Ineff.	(6, 9)	39	30	9	3.25	0.833	2.417

⁽¹⁾The normalization factor in expression (7(7) $$N\Pi I_{\mathit{\text{WA}}}\left(x_o,y_o,\tilde{w},\tilde{p};{\rho }^{-},{\rho }^{+}\right)=\underset{\mathrm{(}\text{Normalized}\mathrm{)}\mathrm{ }\text{Profit}\mathrm{ }\text{Inefficiency}\mathrm{(}\Pi I_{\mathit{\text{WA}}}\mathrm{)}}{\underbrace{\frac{\Pi I\left(x_o,y_o,w,p\right)}{\mathrm{\text{min}}\left\{\frac{w_1}{{\rho }_1^{-}},\dots ,\frac{w_M}{{\rho }_M^{-}},\frac{p_1}{{\rho }_1^{+}},\dots ,\frac{p_N}{{\rho }_N^{+}}\right\}}}}\ge \underset{\text{Technical}\mathrm{ }\text{Inefficiency}\mathrm{(}T{I}_{\mathit{\text{WA}}}\mathrm{)}}{\underbrace{T{I}_{\mathit{\text{WA}}}\left(x_o,y_o,{\rho }^{-},{\rho }^{+}\right)}},$$(7) ) is $\mathrm{\text{min}}\left\{w/{\rho }_{}^{-},p/{\rho }_{}^{+}\right\}$ = $\mathrm{\text{min}}\left\{1/(1/(2\times 6)),4/(1/(2\times 9))\right\}$ = $\mathrm{\text{min}}\left\{12,72\right\}$= 12.

Table 2. The standard reverse profit inefficiency decomposition. (w,p) = (1, 4).

Firm	Model (11)(11) $$\begin{array}{cccc}T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}+\sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oSRm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}-s_{\mathit{\text{oSRn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{\mathit{\text{oSR}}1}^{-},\dots ,s_{\mathit{\text{oSRM}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{\mathit{\text{oSR}}1}^{+},\dots ,s_{\mathit{\text{oSRN}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(11)	SR Project. (11)(11) $$\begin{array}{cccc}T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}+\sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oSRm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}-s_{\mathit{\text{oSRn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{\mathit{\text{oSR}}1}^{-},\dots ,s_{\mathit{\text{oSRM}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{\mathit{\text{oSR}}1}^{+},\dots ,s_{\mathit{\text{oSRN}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(11)	Monetary (10)(10) $$\Pi I\left(x_o^{},y_o^{},w,p\right)=\Pi \left(w,p\right)-\left(p\cdot y_o^{}-w\cdot x_o^{}\right)=T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)+A\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right).$$(10)			Normalized (15)(15) $$N\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,\tilde{w},\tilde{p}\right)=\frac{\Pi I\left(x_o,y_o,w,p\right)}{U_{\mathit{\text{SR}}}}=T{I}_{\mathit{\text{SR}}}\left(x_o,y_o\right)+\frac{\Pi I\left({\widehat{x}}_{\mathit{\text{oSR}}},{\widehat{y}}_{\mathit{\text{oSR}}},w,p\right)}{U_{\mathit{\text{SR}}}}.$$(15)
			Profit Ineff.	Tech. Profit Ineff.	Alloc. Profit Ineff.	Profit Ineff.	Tech. Ineff.	Alloc. Ineff.
$(x_o,y_o)$	Tech. status	$({\widehat{x}}_{\mathit{\text{oSR}}}^{},{\widehat{y}}_{\mathit{\text{oSR}}}^{})$	$\Pi I$	$T\Pi I_{\mathit{\text{SR}}}$ =$p\cdot s_{\mathit{\text{oSR}}}^{+*}+w\cdot s_{\mathit{\text{oSR}}}^{-*}$	$A\Pi I_{\mathit{\text{SR}}}$	$N\Pi I_{\mathit{\text{SR}}}$	$T{I}_{\mathit{\text{SR}}}$	${\mathit{\text{NAI}}}_{\mathit{\text{SR}}}$
C1	C2	C3	C4	C5	C6 = C4-C5	C7 = C4/$N{F}_{\mathit{\text{SR}}}^{}$	C8 = C5/$N{F}_{\mathit{\text{SR}}}^{}$	C9 = C6/$N{F}_{\mathit{\text{SR}}}^{}$
A = (6,9)	Eff.	(6, 9)	9	0	9	0.231	0	0.231
B = (9,12)	Eff.	(9, 12)	0	0	0	0	0	0
C = (7,10)	Eff.	(7, 10)	6	0	6	0.154	0	0.154
D = (6,4)	Ineff.	(6, 9)	29	20	9	0.744	0.513	0.231
E = (8,8)	Ineff.	(8, 11)^(1)	15	12	3	0.385	0.308	0.077
F = (8,9)	Ineff.	(8, 11)^(1)	11	8	3	0.282	0.205	0.077
G = (9,9)	Ineff.	(9, 12)	12	12	0	0.308	0.308	0
H = (9,10)	Ineff.	(9, 12)	8	8	0	0.205	0.205	0
I = (10,8)	Ineff.	(9, 12)	17	17	0	0.436	0.436	0
J = (12,10)	Ineff.	(9, 12)	11	11	0	0.282	0.282	0
K = (12,3)	Ineff.	(9, 12)	39	39	0	1	1	0
–	–	–	–	$N{F}_{\mathit{\text{SR}}}^{}$ = 39 = max C5	–	–	–	–

⁽¹⁾Model (11(11) $$\begin{array}{cccc}T\Pi I_{\mathit{\text{SR}}}\left(x_o,y_o,w,p\right)=\hfill & \underset{s_{\mathit{\text{oSR}}}^{+},s_{\mathit{\text{oSR}}}^{-},\lambda }{\mathrm{\text{max}}}\hfill & \sum _{n=1}^N{p}_n^{}{s}_{\mathit{\text{oSRn}}}^{+}+\sum _{m=1}^M{w}_m^{}{s}_{\mathit{\text{oSRm}}}^{-}\hfill & \hfill \\ \hfill & s.t.\hfill & \hfill & \hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{x}_{\mathit{\text{jm}}}+s_{\mathit{\text{oSRm}}}^{-}=x_{\mathit{\text{om}}},\hfill & m=1,\dots ,M\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j{y}_{\mathit{\text{jn}}}-s_{\mathit{\text{oSRn}}}^{+}=y_{\mathit{\text{on}}},\hfill & n=1,\dots ,N\hfill \\ \hfill & \hfill & \sum _{j=1}^J{\lambda }_j=1,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{-}=\left(s_{\mathit{\text{oSR}}1}^{-},\dots ,s_{\mathit{\text{oSRM}}}^{-}\right)\ge 0_M,\hfill & \hfill \\ \hfill & \hfill & s_{\mathit{\text{oSR}}}^{+}=\left(s_{\mathit{\text{oSR}}1}^{+},\dots ,s_{\mathit{\text{oSRN}}}^{+}\right)\ge 0_N,\hfill & \hfill \\ \hfill & \hfill & \lambda =\left({\lambda }_1,\dots ,{\lambda }_J\right)\ge 0_J.\hfill & \hfill \end{array}$$(11) ) yields (8, 11) as projection, which is a linear combination of firms A and B, with ${\lambda }_{\mathrm{A}}^{*}=0.333$ and ${\lambda }_{\mathrm{B}}^{*}=0.667$ as optimal solutions.