Turnpike Optimality in Environment-Input-Output System: Analysis on
the Adjustment Potential of China’s Industry Structure for Carbon
Emission Reduction*
Xue FU 1, Bo MENG2 Klaus Hubacek 3
1, State Innovative Institute for Public Management, Fudan University, Shanghai, China, 200433
2, International Input-Output Analysis Studies Group, Institute of Developing Economies, IDEJETRO, Chiba, 261-8545 Japan
3, University of Maryland, Maryland, USA
Abstract: This article works out the China’s potential in industry structure adjustment for carbon
emission reduction. We present turnpike optimality in dynamic input-output models and give the
reverse algorithm to get the turnpike Optimality of China’s industry adjustment considering
carbon emission reduction target on Copenhagen Conference based on a compiled EnergyCarbon-Emission-Economy Input-Output tables and modified technology coefficients and capital
formation matrixes. It finds that the turnpike line have better economy performance. Industry
structure is improved by increase in the service and decrease in manufacture.
Key words: carbon emission, optimal dynamic input-out model, industry restructuring
1 Aim
This article works out the China’s potential in industry structure adjustment toward Copenhagen
conference’s target. We present turnpike optimality in dynamic input-output models, with
comparing Leontief’s model, Von-Norman’s model, and Hua’s model. We give the reverse
algorithm of dynamic environment-input-output model to get the turnpike Optimality of China’s
industry adjustment considering carbon emission reduction target on Copenhagen Conference
based on a compiled Energy-Carbon-Emission-Economy Input-Output tables and modified
Supporting foundation: “Dynamic Programming on Industry Structure of Low carbon Economy based on
Multiregional Input-Output Model”, National Social Science Foundation, 2010, 7, No.10CJL033, (hosting);
“Research on the strategy of low carbon development during China’s urbanization”, Key grant of project
of National Social Science Foundation, 2010, 7, No. 10zd&032, (participating).”
*
technology coefficients and capital formation matrixes.
2 Turnpike thermo in dynamic input-output system
The turnpike thermo is first presented by Dorfman, Samuelson, Solow (1949)[1]. It is concerned
with efficient paths of accumulation of economic expansion. Efficiency is defined with reference
to the terminal stocks paths, namely, a path of accumulation lasting N periods is efficient if no
other path, starting from the same initial stocks and complying with the technical restrains,
reaches larger terminal stocks after N periods. We connect turnpike thermo to three models, von
Neumann model, Leontief’s model, and Hua’s model.
2.1 Three Models
von Neumann model is a closed linear model of production in which the output of one
period furnish the inputs of the next period, and combining with Turnpike theorem to get the path
of proportional expansion of stocks which achieve the largest possible rate of expansion, so
called Neumann rays [2,Lionel W. McKenzie].
Leontief’s model is a general form. Hicks[3,1961], Morishima[4,1961] and Mckenzie
[5,1963] make the turnpike theorem became the global for disaggregated industries production
with variable production coefficients and durable capital goods. Jinkichi Tusukui [6, 1966]
applied the turnpike theorem in a computable dynamic Leontief system to the planning of
economic growth. For example, planning for efficient accumulation in Japan[7, Jinkichi Tusukui,
1968, 8] and consumption considering international trade [Jinkichi Tusukui & Yasusuke
Mutakami, 1979].
Hua’s model considered a close model, in which the initial output should fit characteristic
vector. Otherwise, the system will collapse. Plan can made to adjust the output structure to fit
characteristic vector. In Leontief trajectory, if the initial output may not fit characteristic vector
of output, technical progress and structure adjustment make economy system to meet the Hua’s
theorem.
2.2 Turnpike lane in Leontief’s dynamic input-Output system
We develop a dynamic input-output model to make the economy growth in the turnpike lane.
Comparing the previous model, we introduce the dynamic programming in the model with
optimal accumulation of consumption, under the constraints of dynamic input-output equation
including international trade, on the target of carbon emission reduction.
Table 1: present research on planning by meanings of input-output model
theorem
Methods
linear programming
turnpike theorem
筑井(1968); Tusukui &
Murakami(1979); J. He
(2005)
dynamic programming
no turnpike theorem
H. Zhang (2001);R. Na
(1998,a,b)
G. Liu (1998,2009)
Although there are few researches on the planning, most of them are simple linear
programming based on input-output model ( H. Zhang, 2001, R. Na,1998). G. Liu (1998, 2009)
gives some algorithms of dynamic input-output model, but it is not base on turnpike thermo. A
outstanding work on turnpike theorem was made by Jinkichi Tusukui & Yasusuke Mutakami
[1979], but it is still not involved in dynamic programming, and merely maximizes the capital of
final year. We develop the model with optimal accumulating of consumption during the planning
period.
It assumed that all economy structure grow at the same speed if economy develop along the
turnpike lane. So we modify the structure coefficient in our models. But we should take out of
the various structure coefficients at certain year, combining them with the analysis of
econometrics and experience of other countries, to modify them to reasonable coefficients.
Considering von-Norman model’s close characteristic, we give up this assumption because it
doesn’t fit the situation of China’s development, and take use of the solution of balance growth
in Leontief dynamic input-output model T to obtain the initial output structure.
3. Turnpike Optimality based on dynamic input-output model by dynamic programming
This section builds the optimization input-output model. The goal is to accumulation of
consumption of time 1 to time T. The process of resolving mode is divided into tows parts:
First, according to the dynamic input-output model’s structure coefficients, to obtain the output
structure x (T  1) on the growth balance lane. Secondary, we resolve the solution of linear
programming. From the economy output structure after (T+1), we get the output structure
backward step by step. For example, we make use of the model
3.1 Model characteristics
This model has several characteristics:
(1) It is an optimal accumulating of consumption, it make the consumer has maximal benefit.
Most turnpike model considers the capital maximal in the planning year. But considering
the environment factor, a social’s benefit is embodied in consumer actual benefit, but not
the capital. The model takes the objective as the maximal accumulated GDP from the
initial year until the planning year.
(2) It considers the environment factor. The model includes the constraint of carbon
emission as independent constraint equation. It is put into the production not only in the
household’s consumption because carbon dioxide emitted by industries is largest part of
source of carbon emission from fossil energy consumption.
(3) The algorithm of this model is dynamic programming, not linear programming. Most
optimal models are to make maximal GDP of one year (Zhang, 2001, Na Risa, 2000,
Global). Even involving in the turnpike lane, those models make maximal GDP or
capital at planning year as objective (He Jin, 2003). However this dynamic model makes
the sum of consumption maximal as objective. So it uses dynamic programming.
Reverse algorithm can be applied to get solution of this model.
(4) It is an international model. The model includes export and import.
3.2 Dynamic Model
The dynamic model is established in equation (4.1) as objective function, and equation (4.2) as
constraints.
ft[ x(t )]  max{ o[( I  A(t )) x(t )  B(t )u (t )]  f t 1 [ x(t )  u (t )]}
(4.1)
s.t.
 A(t ) x(t )  B(t  1)( x(t  1)  x(t ))  y c (t )  p (t )  m(t )  x(t )

 A(T ) x(T )  B(T  1)( x(T  1)  x(t ))  y c (T )  p(T )  m(T )  x(T )
 f [ x(T )]  o ( I  A) x(T )
 T
~
 ic (t ) x(t )  c y (t )  TC (t )
 l
u
 x (t )  x(t )  x (t )
 l
u
 p (t )  p (t )  p (t )
m l (t )  m(t )  m u (t )

t  1,..., T
(4.2)
Here, A is the input-output coefficient matrix, B is the coefficient matrix of capital formation,
x is the output vector, u is the change of output, y c is the consumption of household, p is the
~
export vector, m is the import vector, ic  is the vector of carbon intensity of industry, c y is the
vector of carbon emission of household, TCT is the carbon emission at target year, o is unit
vector, the subscript u is the up bound of the variable, the low script is the low bound of variable,
t is the time variable, t=1,…,T, T is the planning year.
f T is the terminal constraint at time T, f t is the objective, i.e. cost function at time t,
x(t )  x(t  1)  u (t ) is the state variable, u (t ) is the control variable, control variable in this
model. We deduce a reverse algorithm to get solution of this dynamic IO model in the following
section 4.
3.3 The determination of total carbon emission of each year TC(t).
Target of carbon emission reduction is assumed to be achieved through linear decline. The
reduction of carbon emission intensity at t year to the based year is γt . Target of reduction of carbon intensity
decrease 40% for 15 years so the constraints of reduction of carbon emission is: the reduction of carbon
emission intensity in 2020 to that in 2005 γ15 ∈ (40%, 45%). The carbon emission intensity is denoted as
c t  TC t / GDPt
c15 
,
the
reduction
of
carbon
emission
intensity
between
15
years
is,
TC15 / GDP15  TC1 / GDP1
 100%  (40%,45%) . The reduction of carbon emission per year in
TC1 / GDP1
1
average from 2005 to 2020 is r  1  (1   15 ) 15 ,   (6.76%,5.93%) . Reduction of carbon emission for t
years is
 t  1  (1   ) t , for example,  3  (9.7%,11.3%) . The computation process is as follows (up
script l means low constraint, up script u means up constraint),
(1) Intensity reduction per year:
 l  1  15 (1  45%)  5.93% ,  u  1  15 (1  40%)  6.76% ,
3
(2) Intensity
reduction
for
three
years:
 3l  1  (1  40%) 15  9.712%
3
15
  1  (1  45%)  11.2696%
u
3
ct 
TC (t ) / GDP(t )  TC (1) / GDP(1)
 100%  (1  (1   l ) t ,1  (1   u ) t )
TC (1) / GDP(1)
The carbon emission constraint in time t is
GDP(t )
GDP(1)
4. Reverse Algorithm in Turnpike Optimality in Environment-Input-Output System
TC (t )  (1   t )TC(1)
,
Assumed the initial output as x(1) , the control variable is defined as the increase of output
u (t )  x(t )  x(t  1) , t  1,2,..., T . The output at time t is state variable, x(t )  x(t  1)  u (t ) , so
t
we denote x(t )  x(1)   u (k ) . Then we rewrite the equations (4.1) and (4.2) into equations
k 1
(4.3) and (4.4).
ft[ x(t )]  max{ o [( I  A(t )) x(t )  B(t )u (t )]  f t 1 [ x(t )  u (t )]}
(4.3)
 A(1) x(1)  B(2)u (1)  y c (1)  p (1)  m(1)  x(1)

t
t
 A(t )( x(1)  u (k ))  B(t  1)u (t )  Y (t )  p (t )  m(t )  x(1)  u (k )


c

k 1
k 1

 f T [ x(T )]  o ( I  A(T )) x(T )

t
~

i
(
t
)(
x
(
1
)

u (t )  c y (t )  TC (T )

s.t. c
(4.4)
k 1
 l
u (t )  u (t )  u u (t )
 l
u
 p (t )  p (t )  p (t )
 l
u
m (t )  m(t )  m (t )
t  1,2,..., T

We simplify the constraints (4.4) as the following equation
ST (t )u (t )  b(t ), t  1,..., T
s.t.
(4.5)
 f T [ x(T )]  o( I  A(T )) x(T )
We make use of reverse algorithm to obtain the solution of dynamic IO model. Computation
starts from the year T-1. Then cost function (4.3) and constraint (4.5) at time T-1 is written as
f T 1 [ x(T  1)]  max{ o[( I  A(t )) x(T  1)  B(t )u (T  1)]  f T [ x(T  1)  u (T  1)]}
ST (T  1)u (T  1)  b(T  1)
s.t.
 f T [ x(T )]  o( I  A(T )) x(T )
According to the deduction which combines the cost function at time t with terminal
condition1, equation (4.6) can be written
1
We know the index function (4.3) at time T-1 as
f T 1 [ x(T  1)]  max{ o[( I  A(T  1)) x(T  1)  B(T  1)u(T  1)]  f T [ x(T  1)  u(T  1)]}
(D-1)
From the terminal condition,
f T [ x(T )]  o( I  A(T )) x(T )
(D-2)
we get the index function at time T-1 as
fT 1[ x(T  1)]  max{ o[( I  A(T  1)) x(T  1)  B(T  1)u (T  1)]  fT [ x(T  1)  u (T  1)]}
 max{ o[( I  A(T  1)) x(T  1)  B(T  1)u (T  1)]  o( I  A(T  1))( x(T  1)  u (T  1))}
 max{ 2o( I  A(T  1)) x(T  1)  o( I  A(T  1)  B(T  1))u (T  1)}
Thus the index function (4.3) and constraints at time T-1 is
(D-3)
(4.6)
f T 1 [ x(T  1)]  max{ 2o[( I  A(T  1)) x(T  1)  o( I  A(T  1)  B(t )u (T  1)]}
(4.7)
ST (t )u (t )  b(t ), t  T  k , k  1
s.t.
From equations (4.7), we induce the control vector,
u* (T  1)
, and cost function at time T-1,
f T 1 [ x(T  1)]  2o[( I  A(T  1)) x(T  1)  f T*1}
*
T 1
(4.8)
 o( I  A(T  1)  B(T  1))u(T  1)
Here, f
Next, we come to the cost function and constraints at time T-2,
(4.9)
f T  2 [ x(T  2)]  max{ o[( I  A(t )) x(T  2)  Bu (T  2)  f T 1 [ x(T  2)  u (T  2)]}
s.t.
(4.10)
ST (t )u (t )  b(t ), t  T  2, T  1,
The equations (4.10) can be simplified, so that we can introduce control variable,
u* (T  2)
,
and the cost function at time T-2 in equation (4.11) 2
f T 2 [ x(T  2)]  max{ 3o[( I  A(t )) x(T  2)  f T* 2  f T*1 ]}
(4.11)
s.t. s.t. ST (t )u (t )  b(t ), t  T  2, T  1
*
Here fT 2  o[2( I  A(T  2))  B(T  2)]u (T  2)
Introducing control variable,
u * (T  3)
, we come to the cost function and constraint at time
T-3, and can obtain simplified version of cost function.3
f T 1 [ x(T  1)]  max{ 2o ( I  A(T  1)) x(T  1)  o ( I  A(T  1)  B(t ))u (T  1)}
(D-4)
ST (t )u (t )  b(t ), t  T  k , k  1
s.t.
2
The index function at time T-2 in equation (4.3) is simplified after
fT  2 [ x(T  2)]  max{ o[( I  A(T  2)) x(T  2)  B(T  2)u (T  2)  fT 1[ x(T  2)  u (T  2)]}
 max{ o[( I  A(T  2)) x(T  2)  B(T  2)u (T  2)]
 2o( I  A(T  2))( x(T  2)  u (T  2))  o( I  A(T  1)  B(T  1)u (T  1)}
introducing
equation
(4.8)as
 max{ 3o( I  A(T  2)) x(T  2)  o[2( I  A(T  2))  B(T  2)]u (T  2)  o( I  A(T  1)  B(T  1)u (T  1)} (D-5)
Hence, the index function at time T-2 and constraints in time T-2 are shown as.
fT  2[ x(T  2)]  max{ 3o[( I  A(t )) x(T  2)  fT* 2  fT*1 ]}
ST (t )u (t )  b(t ), t  T  2, T  1
*
Here f T*2  o[2( I  A(T  2))  B(T  2)]u (T  2)

and f T 1  o ( I  A(T  1)  B(T  1)u (T  1)
s.t.
3
The index function at time T-2 in equation (4.3) is simplified after introducing equation (4.11) as
f T  3 [ x(T  3)]  max{ o [( I  A(T  3)) x(T  3)  B(T  3)u (T  3)]  f T  2 [ x(T  3)  u (T  3)]}
 max{ o [( I  A(T  3)) x(T  3)  B(T  3)u (T  3)]
 3o ( I  A(T  3))( x(T  3)  u (T  3))  o [2( I  A(T  2))  B(T  2)]u (T  2)
 o ( I  A(T  1)  B(T  1))u (T  1)}
Let
(D-6)
(D-7)
 max{ 4o ( I  A(T  3)) x(T  3)  o [3( I  A(T  3))  B(T  3)]u (T  3)
 o [2( I  A(T  2))  B(T  2)]u (T  2)  o ( I  A(T  1)  B(T  1))u (T  1)}
*
fT  3  o[3( I  A(T  3))  B(T  3)]u (T  3) , f T*2  o[2( I  A(T  2))  B(T  2)]u (T  2) , and f T*1  o( I  A(T  1)  B(T  1))u (T  1)
Then
f T 3 [ x(T  3)]  max{ 4o( I  A(T  3)) x(T  3)  fT*3  fT* 2  fT*1
,
(D-8)
f T 3 [ x(T  3)]  max{ 4o( I  A(T  3)) x(T  3)  f T*3  f T*2  f T*1
(4.12)
ST (t )u (t )  b(t ), t  T  3, T  2, T  1
s.t.
Here,
fT*3  o[3( I  A(T  3))  B(T  3)]u (T  3)
(4.13)
,
From above deduced equations, we further get the model at k step. Given k  T 1 , the
cost function and constraints at time T  k is shown as in the equation (4.14)4.
f T  k [ x(T  k )]  max{( k  1)o [( I  A(T  k )) x(T  k )  o [k ( I  A(T  k )  B(T  k )]u (T  k ) 
s.t.
ST (t )u (t )  b(t ), t  T  k , ..., T - 1, k  T - 1
Here, f
*
T k
T 1
f
i T  k 1
*
i
(4.14)
 o[k ( I  A(T  k ))  B(T  k )]u (T  k )
Cost function with constraints at time T  k ( k  T 1 ) as format of (4.14) satisfies all
models at time T  k ,..., T  1 .
f1 [ x(T  k )]  max{( k  1)o [( I  A(1)) x(1)  o [k ( I  A(1)  B(1)]u (1) 
s.t.
T 1
f
i T  k 1
*
i
]}
(4.15)
ST (t )u (t )  b(t ), t  T  k ,..., T - 1
*
*
From equation (4.15), we determine the optimal strategy u (1),..., u (T  1) . Given the
initial output x(1) , we know the vector of output at each year, x(2),..., x(T  1), x(T ) ,
We can get the general version of the model at k step, the index function at time T-k.
From the above deduction, we further know the cost function
4
fT  4 [ x(T  4)]  max{ o[( I  A(T  4)) x(T  4)  B(T  4)u (T  4)]  fT 3[ x(T  4)  u (T  4)]}
 max{ o[( I  A(T  4)) x(T  4)  B(T  4)u (T  4)]
(D-9)
 o[4( I  A(T  4)]( x(T  4)  u (T  4)  fT*3  ...  fT*1}
Hence, f T  4 [ x(T  4)]  max{ o[5( I  A(T  4)]( x(T  4)  f T* 4  f T*3  ...  f T*1 }
Here, f T*4  o[4( I  A(T  4)]( x(T  4)
From (D-4),(D-6), (D-8) and (D-9), we can assumed the cost function at k  1 step as
f T ( k 1) [ x(T  (k  1)]  max{ ko( I  A(T  (k  1))) x(T  (k  1))  f T*( k 1)  f T*( k 2)  ...  f T*1
Here, fT*(k 1)  o[(k  1)( I  A(T  (k  1))]( x(T  (k  1))
When k=T-1,
From equation (4.3) we get the cost function at
(D-10)
k step as
fT  k [ x(T  k )]  max{ o[( I  A(T  k )) x(T  k )  B(T  k )u (T  k )]  fT  k 1[ x(T  k )  u (T  k )]}
 max{ o[( I  A(T  k )) x(T  k )  B(T  k )u (T  k )]
 o[k ( I  A(T  k )  B(T  k )]( x(T  k ]  u (T  k )  fT* ( k 1)  fT* ( k  2)  ...  fT* (T 1) }
 max{ o[( k  1)( I  A(T  k )) x(T  k )  o[k ( I  A(T  k )  B(T  k )]u (T  k )  fT* ( k 1)  fT* ( k  2)  ...  fT*1}
Let fT* k  o[k ( I  A(T  k ))  B(T  k )]u (T  k ) , then the model at k  1 step take the version as
fT  k [ x(T  k )]  max{( k  1)o[( I  A(T  k )) x(T  k )  fT* k  fT* ( k 1)  ...  fT*1
 max{( k  1)o[( I  A(T  k )) x(T  k )  o[k ( I  A)  B]u (T  1) 
s.t.ST (T  k )u (T  k )  b(T  k ), t  T  k ,..., T  1, k  T  1
]}
T 1
f
i
i T  k 1
*
]}
(D-11)
x(k  1)  x(k )  u (k )  x(1)  u (1)  ...,u (k ) , k  1,..., T  1 . The detail format is shown in
equation (4.16).
f 1 [ x(1)]  max{ To [( I  A(1)) x(1)  o [(T  1)( I  A(1)  B(1)]u (1) 
T 1
f
i T  k 1
*
i
]}
 A(1) x(1)  B(2)u (1)  y c (1)  p (1)  m(1)  x(1)

t
t
 A(t )( x(1)  u (i ))  B(t  1)u (t )  Y (t )  p (t )  m(t )  x(1)  u (i )


c

i 1
i 1

t
~
i  (t )( x(1)   u (t )  c y (t )  TC (t )
c
k 1
 l
s.t. u (t )  u (t )  u u (t )
 l
u
 p (t )  p (t )  p (t )
 l
u
m (t )  m(t )  m (t )
t  T  k , ..., T - 1. k  T - 1



*
Here, f T l  o[k ( I  A(T  l ))  B(T  l )]u (T  l ), l  1,..., T  1
(4.16)
The process of resolving solution is shown in Appendix 1.
5 Data
We take the date from the Energy-carbon-economy input-output table, which is shown as
appendix 2.
The core of environment dynamic input-output model is non-linear coefficients. So we
require modifying the direct input coefficient, capital coefficient and carbon emission coefficient.
Leontief dynamic input-output model requires capital coefficient matrix, while input-output table
issued by the Statistic Bureau in various countries merely reflect the capital formation vector. So
the dynamic model was studied theoretically, but not applied in general [Duchin Fayer]. Some
application is in Ten Raa (1986a, b) and education (Zhang, H.,2003, He, J., 2005, Fu, X.,2006).
In general, input-output table issued in various countries merely reflect the capital formation as
column vector. Chinese Statistic Bureau investigated the capital formation matrix in 2000, which
is also fit the requirement of Leontief dynamic model. Moreover, Chen X(2005) also built inputoccupied-output model, which investigate the fixed capital occupied matrix. The capital
formation is flow while the fix capital is the stock. The capital formation coefficient bkjt is the kth
capital product at time t for the jth output increased at time t
bkjt  f ijt /( xt 1  xt )
. The fixed
capital occupied coefficient bkjt is the kth capital occupied at time t for the jth output at time t
bkjt  f ijt / xt . Fixed capital occupied f t is the fix capital is acuminated capital, so bkjt the
capital coefficient in average. Take the occupied capital coefficient to adjust or replace capital
formation coefficient, it will help to establish the dynamic model.
Because of capital formation data is available in 2000, so we take the data from 2000 to
2002 as an example for the new model, and then find the turnpike optimality in environmentinput-output system. The indirect coefficient and capital coefficient are fix in this model, and the
initial output and final demand is the real data from 2000, 2001, 2002 and 2003 input-output
table. The 2000 and 2002 input-output table are obtain from the Chinese Statistics Bureau. 2001
and 2003 input-output table is extended table we compile according to combination the date from
national accounting.
6 Turnpike Optimality in Environment-Input-Output System
We find the turnpike optimality from the environment-input-output system. Table 2 gives the
adjusted output, which is compared to original output, table 3 gives the adjusted industry
structure is compared to original output.
The optimal structure in 2001 to 2003 is different with the structure in 2000 on the aspects:
(1) On the turnpike line, optimal structure of manufacture in 2001 (60%) is less than the
actual structure in 2001 (68%). However, optimal structure of service in 2001 (28%) is
larger than the actual structure in 2001 (22%).
(2) Optimal Consumption in 2001 and 2002 are 0.40 billion and 0.4 billion larger than the
actual comsumption in 2001 and 2002. It means that China should increase the
consumption instead of merely dependency on the investment and export.
(3) Optimal total output in 2001 and 2002 are 0.13 billion and 0.7 billion larger than the
actual total output in 2001 and 2002. It means the turnpike lane will improve the
economy much better.
(4) Because of we take the fix structure coefficient, the optimal structure is fixed. For
example, agriculture as 10%, manufacture as 60% and service as 28%. If we modify the
technical coefficient and capital coefficient. The optimal structure will be changed.
Table 2: The adjusted output comparing to original output
actual data
agriculture
manufactures
services
output
consumption
adjustment
agriculture
manufactures
services
output
consumption
output difference
consumption difference
2000
264482669.9
1729696512.3
653753201.9
2647932384.0
375538540.9
2001
258565469.3
1889139358.0
622103604.2
2769808431.0
638339569.1
2002
285787423.0
1905590585.0
942927008.7
2191378008.0
792857986.4
2003
357949104.5
2386754590.0
1181017257.0
3925720951.0
264482669.9
1729696512.3
653753201.9
2647932384.0
375538540.9
330603337.5
1755862433.0
817191502.4
2903657273.0
1127383296.0
133848841.2
489043727.1
330933940.8
1757618295.0
818008693.9
2906560930.0
1195709282.0
715182921.7
402851296.1
331264874.8
1759375913.0
818826702.6
2909467491.0
Note: The actual data of output by industry and consumption is from the Statistic Year book in
2001, 2002 and 2003, and the input-output table from 2000 to 2003.
Table 2: The adjusted industry structure comparing to original output
actual data
agriculture
manufactures
2000
10%
65%
2001
9%
68%
2002
13%
87%
2003
9%
61%
services
output
adjustment
agriculture
manufactures
25%
1.0
22%
1.0
43%
1.0
30%
1.0
0.1
65%
0.1
60%
0.1
60%
0.1
60%
services
output
25%
1.0
28%
1.0
28%
1.0
28%
1.0
Note: The actual data of output by industry and consumption is from the Statistic Year book in
2001, 2002 and 2003, and the input-output table from 2000 to 2003.
6 Conclusion
This article works out the China’s potential in industry structure adjustment for carbon emission
reduction. We present turnpike optimality in dynamic input-output models and give the reverse
algorithm to get the turnpike Optimality of China’s industry adjustment considering carbon
emission reduction target on Copenhagen Conference based on a compiled Energy-CarbonEmission-Economy Input-Output tables and modified technology coefficients and capital
formation matrixes. It finds that the turnpike line have better economy performance. Industry
structure is improved by increase in the service and decrease in manufacture. The optimal
consumption and total output should increase much more than the actual data. If we modify the
structure coefficient, the results will be improved.
Appendix
We will resolve the solution of the following model,
f1 [ x(T  k )]  max{( k  1)o [( I  A(1)) x(1)  o [k ( I  A(1)  B(1)]u (1) 
s.t.
f
*
T k
T 1
f
i T  k 1
*
i
]}
ST (t )u (t )  b(t ), t  T  k ,..., T - 1
 o[k ( I  A(T  k ))  B(T  k )]u (T  k )
As for the dynamic IO model, we can compile the matlab program to resolve the solution of
the model (4.16). The method is realized through some transformation of this model.
 z(t )  x(t ) p(t ) m(t )

  (t )  u(t )  (t )  (t )

G (t )  o[k ( I  A (t )  B(t )],
~
G (t )  [G (t ) 0 0]
H (t )  B(t ) I  I 

Q(t )   (I  A (t )) 0 0

 (t )  (I  A (t )) x(1)  y (t )
Set

~
n( t )  ic ( t ) 0 0

I 0 0 



 W  I 0 0 

I 0 0

t  T  k , k  T  1...,0


Equation (4.16) is rewritten as

f1  max G (1) G (2)  G (T )z (1) z (2)  z (T )
  H (1)
 
  Q(2)
  
 
 Q(T )

  n(1)
  n(2)
 
  
  
 n (T )

 W
   l W


   l W
 

 W
  u W


  u W
 
0
0
0    (1)    (1) 
H (2) 
    (2)   (2) 



0      

 

 Q(T ) H (T )   (T )  (T )
0
n(2)

n(T )







 input - output arrays


0    (1)   TC (1)  c y (1)  cz (1) x(1)  


    (2)   TC (2)  c y (2)  cz (2) x(1)  

CO2 constraint s


0    


 
n(T )   (T ) TC (T)  c y (T)  cz (T) x(1) 
0

0    (1)    l z (1)


W

    (2)    l z (1)




0      


 
l
   W W    (T )   l z (1)
0

0    (1)    u z (1)


W 
    (2)    u z (1)


 
0      


 
u
  W W    (T )   u z (1)
Taking T  3 , n=3 for example, the model is shown as follows:



 low bound






 up bound



f1  max G (1) 0
 B(1)
 (I  A(2))

 (I  A( 3))
 ~
 ic  (1)
 ~
ic  ( 2)
 ~
 ic  (3)
I


I


I
  ρl I

  ρl I

  ρl I

l
 ρI
  ρl I

  ρl I

I


I

I


ρu I


ρu I

ρu I


ρu I


ρu I


ρu I


0 G (2) 0 0 G (3) 0 0u (1)  (1)  (1) u (2)  (2)  (2) u (3)  (3)  (3)
I I
0
0 0
0
0 0
0 0
B(2)
I I
0
0 0 
0 0  (I  A( 3)) 0 0
B(3) I  I 

0 0
0
0 0
0
0 0
~
0 0
ic  ( 2)
0 0
0
0 0

~
~
0 0
ic  (3)
0 0 ic  (3) 0 0 
0 0
0
0 0
0
0 0


(I  A(1)) x1  y1

0 0
0
0 0
0
0 0


(
I

A
(
2
))
x

y
1
2

0 0
0
0 0
0
0 0   u (1)  



(
I

A
(
3
))
x

y
1
3
0 0
I
0 0
0
0 0   (1)  

~


TC (1)  c y (1)  ic  (1) x(1) 
0 0
I
0 0
0
0 0    (1)  
~
TC (2)  c (2)  i  (2) x(2)

y
c
0 0
I
0 0
0
0 0   u (2)  


  TC (3)  c (3)  ~

i

y
c (3) x (3) 
l

ρI

0 0
0 0
I
0 0    (2)  
 l z (1)


l
l
0 0
ρI
0 0
I
0 0   (2)  

 z (1)



l
0 0
0 0
I
0 0   u (1)  
 ρl I

z
(
1
)

   (3)  
u
0 0
0
0 0
0
0 0




z
(
1
)



u
0 0
0
0 0
0
0 0   (3)  


z
(
1
)



u
0 0
0
0 0
0
0 0




z
(
1
)



0 0
I
0 0
0
0 0
0 0
I
0 0
0
0 0

0 0
I
0 0
0
0 0

ρu I
0 0
0 0
I
0 0
0 0
ρu I
0 0
I
0 0

u
0 0
0 0
I
0 0
ρI

Table A.1 Energy-carbon emission-economy input-output table
Intermediate usage
output
energy
input
Coal producing
104 Yuan
and selection
Coal(usage/consu ton/ton standard
mption)
coal
Carbon
metric ton
……
……
Energy industry
input
Intermediate input
Gas production
104 Yuan
and supplying
3
Gas(usage/consu M /ton standard
mption)
coal
Carbon
metric ton
Total Usage of
ton standard
energy
coal
Total consumption ton standard
of
coal
energy
CO2 emission
metric ton
Value
added
Fix capital depreciation
Remuneration of labor
Tax and profit
Value added in total
Total input
energy
Non-enery …
Total
output
X EE X EN
yE
xE
X uEE X uEN
y uE
xuE
X cEE X cEN
y cE
xcE
X cCE X cCN
yc
xc
eue
eun
e uf
eut
ece
ecn
ecf
ect
cze
czn
cy
ct
104 Yuan
104 Yuan
The third industry
Intermediate input in total
NonConsumption
energy
of household
…
… …
Non energy
industry
agriculture
Final usage
X NE X NN
yN
F NE
F NN
xN
104 Yuan
104
104
104
104
Yuan
Yuan
Yuan
Yuan
VE
VN
104 Yuan
xE
xN
The energy unit of value is 104 Yuan. The energy unit of volume is measured both as physic quantity and transferred standard unit. Physic
quantity unit is denoted for raw coal, crude oil, Petroleum products, Coking products as ton, for natural gas, oven gas, LPG as cube meter, for
hydro power and nuclear, and thermal power respectively as kW•h, heat kJ. Consistent unit, energy of various types can be measured as standard
coal. The carbon dioxide unit is metric ton.