Timber Production
Efficiency Analysis
Content
Defining the production technology and
technical efficiency
The measurement and decomposition of
cost efficiency
Empirical methods
Technical efficiency in southern U,S,
pulpwood harvesting
Summary
y
X
f(x)’
f(x)· ·
· · ·
·
· ·
· ·
Standard production economics analysis begins with
the production function y =f (x) where y is output
and x is input
··
··
1.Defining the production technology
and technical efficiency
All feasible input-output vector relationships are
described by the graph of the production
technology or production possibilities set,PT,
such that
PT=｛ (y,x), x can produce y｝ 7.1
Input and output sets
The input (output) sets of the production technology,
which describe the sets of all input (output)
vectors that are feasible for each output (input)
vector y (x),are defined as
L(y)=｛ x,(y,x) ∈ PT｝ 7.2
P(x)=｛ y,(y,x) ∈ PT｝ 7.3
Input and output isoquants
An input isoquant
Isoq L(y)=｛ x,(y,x) ∈ PT,λx ∈ L(Y),λ＜ 1｝ 7.4
An output isoquant
Isoq P(x)=｛ y,(y,x) ∈ PT,θy ∈ P(x),θ＞ 1｝ 7.5
Definiting the technical dfficiency
The definition for input-oriented technical efficiency
is given by the function
TE1 (y,x) = min{λ:y ≤ f (λx)} 0＜ TE1 (y,x) ≤ 1
7.6
The definition for input-oriented technical efficiency
is given by the function
TE0 (y,x) = [max{θ:θy ≤ f (λx)}]-1 0＜ TE0 (y,x) ≤ 1
7.7
2,The measurement and ecomposition
of cost efficiency
For a single output,the cost frontier is a function
7.8
The ratio of minimum to observed cost is termed cost
efficiency and given by the function
7.9
NN Rwwwxfwyc ++∈),.,,,,( p r i c ei n p u t )}(≤y:x w{m i n),( 1Tx ＝＝
1≤),,(0,/),(),,( wxyCEXwwycwxyCE T ＜＝
Two components
The overall level of cost inefficiency can be decomposed
into two components
The input-oriented technical component,λ
TE1 (y,x) = min{λ:y ≤ f (λx)} 0＜ TE1 (y,x) ≤ 1
7.10
The allocative component,θ
7.11
1),,(0),(/),,(),,( 111 ≤wxyAExyTEwxyCEwxyAE ＜＝
3,Empirical methods
SFA model
DEA model
CCR model
SFA model
The modified classical regression model by
proposing a composed error specification
7.12
where
7.13
ieXify i εβ );(＝
iii uv －＝ε
The log likelihood function
Technical efficiency is the ratio of observed to max or
frontier output
7.14
The log likelihood function for the composed error model
can be given as
7.15
7.16
i
i v
i
iu
i eXf
y
eTE );( β＝＝ －
∑ ∑
1 1
222 εσ2 1))σ γε(Φ1(σ)π/2(2)σ,γ,β|(
N
i
N
i i
iInN I nInNyI n l
＝ ＝
－－－＝ +
vu
vu
iii
iii
ii
Xfyi
σσγ
σσσ
βε
/)(
)(
);()(
222
＝
＋＝
－＝
The average or mean technical efficiency
The average or mean technical efficiency for the date sample
is given by
7.17
The observation or firm-specific technical efficiency can be
calculated as
7.18
1≤≤0))σ(Φ1(2)( 2/σ 2 TEeeETE uu u －＝＝ －
1≤≤0) ) ]σ
γε
()σ/γε(Φ1
)σ/γε(φ
(σ
σσ
[
ii TEeTE
i
i
ivu －
－－＝
DEA model
This is done by developing weights for each
firm’s inputs that solve the following problem
7.19
0,
,.,,,,,,,2,1,1
'
'
..
'
'
m a x
,
≥
≤
vu
nj
xv
yu
ts
xv
yu
j
j
i
i
vu
＝
CCR model
This nonlinear problem can be transformed into the
linear programming multiplier problem by
imposing the constraint,which yields
7.20
1' ＝ixv
0≥,μ
,.,,,,,2,10≤μ
1'
..
'μm a x
,μ
v
njvxy
xv
ts
y
ii
i
iv
＝－
＝
Alternatively,one can also solve a dual linear
envelopment program
7.21
o
Xx
Yy
ts
i
i
≥ω
0≥ωθ
0≥ω
..
θm i n
ω,θ
－
－ +
4.Technical efficiency in southern U.S,
pulpwood harvesting
The southern U.S,IS one of the most important
timber-producing regions in the world
The southern pulpwood logging industry has
experienced substantial structural shifts in the
past
4.1 Methods and Data
Technical eficiency in southern pulpwood harvesting
is modelled using the SFA methodology
Data for this analysis stem from two comprehensive
American Pulpwood Association (APA) surveys of
southern U.S,pulpwood producers,
4.2 Results
The age of the owner and years of experience are
important explanatory factors impacting the level of
technical efficiency
The level of output is also important,suggesting that
economies of scale influence efficiency levels
It is ntable that the auxiliary regressions have low
explanatory power
Increasing the efficiency level of the industry can
reduce the costs of pulpwood production
Efficiency Analysis
Content
Defining the production technology and
technical efficiency
The measurement and decomposition of
cost efficiency
Empirical methods
Technical efficiency in southern U,S,
pulpwood harvesting
Summary
y
X
f(x)’
f(x)· ·
· · ·
·
· ·
· ·
Standard production economics analysis begins with
the production function y =f (x) where y is output
and x is input
··
··
1.Defining the production technology
and technical efficiency
All feasible input-output vector relationships are
described by the graph of the production
technology or production possibilities set,PT,
such that
PT=｛ (y,x), x can produce y｝ 7.1
Input and output sets
The input (output) sets of the production technology,
which describe the sets of all input (output)
vectors that are feasible for each output (input)
vector y (x),are defined as
L(y)=｛ x,(y,x) ∈ PT｝ 7.2
P(x)=｛ y,(y,x) ∈ PT｝ 7.3
Input and output isoquants
An input isoquant
Isoq L(y)=｛ x,(y,x) ∈ PT,λx ∈ L(Y),λ＜ 1｝ 7.4
An output isoquant
Isoq P(x)=｛ y,(y,x) ∈ PT,θy ∈ P(x),θ＞ 1｝ 7.5
Definiting the technical dfficiency
The definition for input-oriented technical efficiency
is given by the function
TE1 (y,x) = min{λ:y ≤ f (λx)} 0＜ TE1 (y,x) ≤ 1
7.6
The definition for input-oriented technical efficiency
is given by the function
TE0 (y,x) = [max{θ:θy ≤ f (λx)}]-1 0＜ TE0 (y,x) ≤ 1
7.7
2,The measurement and ecomposition
of cost efficiency
For a single output,the cost frontier is a function
7.8
The ratio of minimum to observed cost is termed cost
efficiency and given by the function
7.9
NN Rwwwxfwyc ++∈),.,,,,( p r i c ei n p u t )}(≤y:x w{m i n),( 1Tx ＝＝
1≤),,(0,/),(),,( wxyCEXwwycwxyCE T ＜＝
Two components
The overall level of cost inefficiency can be decomposed
into two components
The input-oriented technical component,λ
TE1 (y,x) = min{λ:y ≤ f (λx)} 0＜ TE1 (y,x) ≤ 1
7.10
The allocative component,θ
7.11
1),,(0),(/),,(),,( 111 ≤wxyAExyTEwxyCEwxyAE ＜＝
3,Empirical methods
SFA model
DEA model
CCR model
SFA model
The modified classical regression model by
proposing a composed error specification
7.12
where
7.13
ieXify i εβ );(＝
iii uv －＝ε
The log likelihood function
Technical efficiency is the ratio of observed to max or
frontier output
7.14
The log likelihood function for the composed error model
can be given as
7.15
7.16
i
i v
i
iu
i eXf
y
eTE );( β＝＝ －
∑ ∑
1 1
222 εσ2 1))σ γε(Φ1(σ)π/2(2)σ,γ,β|(
N
i
N
i i
iInN I nInNyI n l
＝ ＝
－－－＝ +
vu
vu
iii
iii
ii
Xfyi
σσγ
σσσ
βε
/)(
)(
);()(
222
＝
＋＝
－＝
The average or mean technical efficiency
The average or mean technical efficiency for the date sample
is given by
7.17
The observation or firm-specific technical efficiency can be
calculated as
7.18
1≤≤0))σ(Φ1(2)( 2/σ 2 TEeeETE uu u －＝＝ －
1≤≤0) ) ]σ
γε
()σ/γε(Φ1
)σ/γε(φ
(σ
σσ
[
ii TEeTE
i
i
ivu －
－－＝
DEA model
This is done by developing weights for each
firm’s inputs that solve the following problem
7.19
0,
,.,,,,,,,2,1,1
'
'
..
'
'
m a x
,
≥
≤
vu
nj
xv
yu
ts
xv
yu
j
j
i
i
vu
＝
CCR model
This nonlinear problem can be transformed into the
linear programming multiplier problem by
imposing the constraint,which yields
7.20
1' ＝ixv
0≥,μ
,.,,,,,2,10≤μ
1'
..
'μm a x
,μ
v
njvxy
xv
ts
y
ii
i
iv
＝－
＝
Alternatively,one can also solve a dual linear
envelopment program
7.21
o
Xx
Yy
ts
i
i
≥ω
0≥ωθ
0≥ω
..
θm i n
ω,θ
－
－ +
4.Technical efficiency in southern U.S,
pulpwood harvesting
The southern U.S,IS one of the most important
timber-producing regions in the world
The southern pulpwood logging industry has
experienced substantial structural shifts in the
past
4.1 Methods and Data
Technical eficiency in southern pulpwood harvesting
is modelled using the SFA methodology
Data for this analysis stem from two comprehensive
American Pulpwood Association (APA) surveys of
southern U.S,pulpwood producers,
4.2 Results
The age of the owner and years of experience are
important explanatory factors impacting the level of
technical efficiency
The level of output is also important,suggesting that
economies of scale influence efficiency levels
It is ntable that the auxiliary regressions have low
explanatory power
Increasing the efficiency level of the industry can
reduce the costs of pulpwood production
