Jrl Syst Sci & Complexity (2006) 19: 76–87
A DYNAMIC APPROACH TO CALCULATE SHADOW
PRICES OF WATER RESOURCES FOR NINE MAJOR
RIVERS IN CHINA∗
Jing HE · Xikang CHEN · Yong SHI
Received: 8 August 2004
Abstract China is experiencing from serious water issues. There are many differences among the
Nine Major Rivers basins of China in the construction of dikes, reservoirs, floodgates, flood discharge
projects, flood diversion projects, water ecological construction, water conservancy management, etc.
The shadow prices of water resources for Nine Major Rivers can provide suggestions to the Chinese government. This article develops a dynamic shadow prices approach based on a multiperiod input–output
optimizing model. Unlike previous approaches, the new model is based on the dynamic computable general equilibrium (DCGE) model to solve the problem of marginal long-term prices of water resources.
First, definitions and algorithms of DCGE are elaborated. Second, the results of shadow prices of
water resources for Nine Major Rivers in 1949–2050 in China using the National Water Conservancy
input–holding–output table for Nine Major Rivers in 1999 are listed. A conclusion of this article is that
the shadow prices of water resources for Nine Major Rivers are largely based on the extent of scarcity.
Selling prices of water resources should be revised via the usage of parameters representing shadow
prices.
Key words Computable general equilibrium, dynamic, input–output analysis, nine major rivers,
shadow prices, water development.
1 Introduction
The Chinese government lists water resources as a top priority sector for infrastructure development and considers the management of water resources an important job. Global climate
variability scenarios generally portray declining water availability in China in the future. Flooding of river systems frequently displaces millions of people in China. Geographically, China has
abundant water in the south and scarce water in the north. China’s population will reach
about 1.6 billion people in 2030, but the country has limited new arable land and limited water
resources for future generations.
Jing HE
Research Center on Data Technology and Knowledge Economy, Chinese Academy of Sciences, Beijing 100080,
China. Email: [email protected].
Xikang CHEN
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China. Email:
[email protected].
Yong SHI
Research Center on Data Technology and Knowledge Economy, Chinese Academy of Sciences, Beijing 100080,
China. Email: [email protected].
∗ This work was supported in part by National Nature Science Foundation of China (No. 70472074, No. 70131002,
and No. 60474063) and China Postdoctoral Science Foundation.
A DYNAMIC APPROACH FOR SHADOW PRICES OF WATER RESOURCES
77
Water resources in China could be divided into the Nine Major Rivers basins: Yellow River,
Yangtze River, Songhuajiang River and Liaohe River, Haihe River and Luanhe River, Huaihe
River, Zhujiang River, Southeast Rivers, Southwest Rivers, and Inland Rivers shown in Fig. 1.
Graph 1 China’s drainage areas of nine big rivers
China’s Drainage Areas of Nine Big Rivers
Song and Liao River Basin
Inland River Basin
Haihe and Luanhe
Yellow River
Huaihe River Basin
Yantze River
Southwest Rivers
Southeast Rivers
Zhujiang River
Figure 1 China’s drainage areas of Nine Major Rivers
There are many differences among Nine Major Rivers in the construction of dikes, reservoirs,
floodgates, flood discharge projects, flood diversion projects, water ecological construction,
water conservancy management, etc. Because of the key role in managing water demand and
supply, shadow prices of Nine Major Rivers are the important policy instruments for creating
incentives to conserve and allocate water efficiently. Shadow prices can be calculated for goods
and services set by the government. The Chinese government still controls the prices of water
resources, so shadow prices can give useful decision support. As water resource moves from
least productive to most productive uses, places, and time points for efficient allocation, there
will be a convergence of the scarcity value, opportunity cost, and long-term marginal cost of
the resources. Unfortunately, such a convergence is rarely reflected in practice. Long-term
marginal cost of water resources is still not included in the calculation of water prices. Market
prices can be estimated much lower. The shadow prices can provide suggestions to the Chinese
government in water resource-saving national and regional economy systems, including water
resource-saving production and water resource-saving consumption systems.
There are four approaches to calculate shadow prices of water resources in China: (1) The
static computable general equilibrium (CGE) model regards the prices of water resources as
the output prices of the water resource sector and the input price of the nonwater resource
sector because the prices for the supply and the demand are equal[1] . (2) Equilibrium prices
in input–output table[2] . The supplying water resource sector is entered into the input–output
table as a sector or a commodity and then theoretical prices of the sector are regarded as
the shadow prices. The difficulty of this approach is in constructing the input–output table of
water resources, which is the basic work in water conservancy economics in China. (3) Marginal
prices[3] are the most popular calculation approach and the prices are equal to the derivative
prices of the production function. (4) In the linear programming (LP) model[4] , we can use the
optimal solution of the resources allocation and get shadow prices of the resources from the
optimal solution of the dual problem according to the water resources constraint line in the
78
JING HE · XIKANG CHEN · YONG SHI
linear programming. In nonlinear programming the shadow prices are equal to the Lagrange
multipliers. In dynamic programming the shadow prices are equal to the vector in the Hamilton
matrices[5] .
In a word, these four models show the different processes of resource allocation and economic
theory directing them.
When we apply the real-time data to the above models, we always face the same difficulty
of collecting the required interdependent data. Therefore we try to solve this problem by constructing the input–output table of water resources for Nine Major Rivers. Some experts have
applied the input–output analysis to water resources, such as research on Colorado River developed by California and Arizona in the United States[6] . This article lists the water resources
sector alone[7] . The water resources input–output table in Beijing of China[8] , the water resources input–output table of Shanxi Province[3], the water resources input–output table in
Huabei and Xinjiang regions of China[3] are good examples of applications in China. All of
these water input–output tables are for regional water resources and there previously have been
no input–output tables for the drainage area. In our model we construct and apply the water
conservancy economy input–holding–output tables for drainage areas of Nine Major Rivers in
China.
This article is arranged as follows: Section 2 reviews the basic concepts of dynamics input–
output analysis and Turnpike theory. Sections 3 and 4 elaborate the definitions and algorithms
of the proposed dynamic model for Chinese shadow prices calculated for Nine Major Rivers.
Section 5 brings forward the results of shadow prices for 1949–2050 in China for Nine Major
Rivers by using the water conservancy economy input–holding–output tables for drainage areas
of Nine Major Rivers for 1999 in 19 sectors. Further research problems and the lessons are
outlined in Section 6.
2 Extended Dynamics Input–Output Model and Turnpike Model
To understand the dynamic computable general equilibrium (DCGE) model for shadow
prices, the extended dynamics input–output model and Turnpike model are crucial. We also
develop the two models for the calculation based on separation of amounts and structure.
2.1 Dynamics Input–Output Model
The linear static and dynamic input–output models are well known in economic theory and
practice[9] . The dynamic extended input–holding–output model is shown as follows[10] :
n
X
j=1
Xij (t) +
n
X
Zij (t) + Y ci (t) = Xi (t),
(1)
j=1
Kij (t + 1) = Kij (t) + Zij (t) − Dij (t + 1),
i, j = 1, 2, · · · , n,
where Xij (t) is output in sector i from sector j in year t, Kij (t) is the available capital goods
in sector i from sector j in year t, Zij (t) is capital investment in sector i from sector j in year
t, Dij (t) is depreciation in sector i from sector j, and Kij (t) is the available capital goods in
sector i from sector j in year t.
The direct input coefficient in year t is defined as follows:
aij (t) =
Xij (t)
,
Xj (t)
(2)
where aij (t) is the direct input coefficient in sector i from sector j in year t, and Xj (t) is the
output in sector j in year t.
A DYNAMIC APPROACH FOR SHADOW PRICES OF WATER RESOURCES
79
The added investment coefficient in year t is defined as follows:
bij (t) =
Kij (t) − Kij (t − 1)
,
Xj (t) − Xj (t − 1)
(3)
where bij (t) is the depreciation coefficient in sector i from sector j in year t, Kij (t − 1) is the
available capital goods in sector i from sector j in year t − 1, Xj (t − 1) is the output in sector
j in year t − 1.
The depreciation coefficient in year t is defined as follows:
βij (t) =
Dij (t)
,
Xj (t)
(4)
where βij (t) is the depreciation coefficient in sector i from sector j in year t. Thus,
n
X
aij (t)Xj (t) + Y ci (t) + Xi (t)Y ci (t)
j=1
+
n
X
[bij (t + 1)(Xj (t + 1) − Xj (t)) + βij (t + 1)Xj (t + 1)]
j=1
= Xi (t),
i, j = 1, 2, · · · , n,
(5)
where Y ci (t) is the final demand in year t in sector i.
Equation (5) can be written in matrix form:
[I − A(t) + B(t + 1)]X(t) − [B(t + 1) + β(t)]X(t + 1) = Y c(t),
(6)
where I is the unitary matrix, A(t) is the direct input coefficient matrix in year t, B(t + 1) is
added investment coefficient matrix in year t + 1, X(t + 1) is output matrix in year t, β(t) is
the depreciation coefficient matrix in year t, Yc (t) is the final demand matrix in year t, and
X(t + 1) is the output matrix in year t + 1.
2.2 Turnpike Model
The Turnpike model[11] is based on linear programming with the objective function of the
maximum of the accumulating capital at the end of the objective term. When we calculate the
balanced growth solution of the Turnpike model. The dynamic input–output model is used to
extend the original structure of the Turnpike model to avoid the constraint of the occluding
hypothesis in the Neumann model of Turnpike model[11] . The modified model without occluding
constraint can describe the real status in China. Recall the dynamic input–output model in
Eq. (6) as follows:
[I − A(t) + B(t + 1)]X(t) − [B(t + 1) + β(t)]X(t + 1) = Y c(t).
The Final demand coefficient in year t is defined as follows:
C(t) =
Y c(t)
,
X(t)
(7)
where C(t) is final demand depreciation coefficient. Then, Eq. (6) can be rewritten as follows:
X(t) = A(t)X(t) + B(t + 1)(X(t + 1) − X(t)) + β(t)X(t) + C(t)X(t),
(8)
80
JING HE · XIKANG CHEN · YONG SHI
where C(t) is the final demand coefficient matrix. Suppose the equilibrium growth rate is the
same in different sectors:
X(t + 1) = (1 + α)X(t),
(9)
where α is the balanced growth development rate. Then
1
X(t) = [I − A(t)β(t) − C(t)]−1 [B(t + 1) + β(t)]X(t).
α
(10)
1
X(t) = [I − A − β − C]−1 [B(t + 1) + β]X.
α
(11)
Thus,
We discuss the existence of balanced growth solutions in the extended model, Eq. (11). Before
discussing the existence of a balanced growth path for Eq. (11), we need to recall some basic
issues related to nonnegative systems. If a square matrix is nonnegative (all of whose elements
are nonnegative), then its spectral radius is an eigenvalue. Corresponding to this eigenvalue
there exists a nonnegative eigenvector[11]. Moreover, an irreducible matrix is characterized by
exactly one nonnegative eigenvector (up to scalar multiplication) and this eigenvector is positive.
Based on the hypothesis in the input–output model, Hawkins Simon condition, and Solow
condition in input–output analysis[9] , we can safely conclude that the modulus of eigenvalue of
the matrix [I − A − β − C] is below 1 and [I − A − β − C]−1 is a nonnegative matrix. Then
[I − A − β − C]−1 [B(t + 1) + β] = H
is a nonnegative matrix according to the Perron-Frobenius theorem. Therefore the balanced
growth rate is equal to the reciprocal of the Perron-Frobenius solution, and the eigenvector
is equal to the output structure on the balanced growth path[9] . This is an application of
Perron-Frobenius’s result to the economic model.
2.3 Separating Amount and Structure of Output
The basic style of input–output analysis is
X = AX + Y,
(12)
where X is output and Y is final demand.
To impose the structure of the economic system, we change Eq. (12) to
e = (I − A)−1 Ẏ Ye ,
Ẋ X
(13)
e Ye are structure vectors that are the output ratio vector of
where Ẋ, Ẏ are amount vectors, X,
the every sector occupying in the gross output vector.
3 Dynamic Computable General Equilibrium Model
for Calculation of Shadow Prices
The dynamics of change in shadow prices of water resources in China are consequently a
project with a long time span, distinctive character and complex structure. The DCGE assumes
that the level of production and resources controls the structures of the economic system on
the balanced growth path. The balanced growth path is instrumental to the maintenance and
A DYNAMIC APPROACH FOR SHADOW PRICES OF WATER RESOURCES
81
improvement of shadow prices. It also assumed that the increased input coefficient and the
decreased gross amount of water resources are synonymous with the increased shadow prices.
3.1
Basic Structure of DCGE
Our model has the fllowing advantages: (1) Shadow prices are consistent with the dynamic
global optimal solution, reflecting the dynamic order of the resources optimal allocation. (2)
The model can be modified to calculate the shadow prices of a specific year easily. The solution
sets of shadow prices are useful in analysis of the balanced development of an economic system.
The model for specific major rivers in which the parameters and constraints are limited in the
drainage area is as follows:
MaxZ = (I − A(T ))X(T )

A(t)X(t) + B(t + 1)(X(t + 1) − X(t)) + β(t)X(t) + C(t)X(t)






≤ X(t) t = 1, 2, · · · , T − 1;






e + 1)Ẋ(T + 1) − X(T )) + β(T )X(T )

A(T )X(T ) + B(T + 1)(X(T






+C(T )X(T ) ≤ X(T );


s.t. Aw (T )X(T ) ≤ W (T );




H(t)X(t) ≤ h(t), t = 1, 2, · · · , T ;






L(t)X(t) ≤ l(t), t = 1, 2, · · · , T ;






X(t) ≥ 0, t = 1, 2, · · · , T ;





Ẋ(T + 1) ≥ 0,
(14)
where X(t) is output in year t in a specific drainage area, A(t) is direct input coefficients in
year t in a specific drainage area, Ẋ(t) is the amount vector in year t in a specific drainage
e
area, X(t)
is the structure vector in year t in the specific drainage area, C(t) is final consuming
demand coefficient in year t in the specific drainage area, Aw (t) is direct water input coefficient
in year t in the specific drainage area, W (t) is total water input in year t, B(t) is investment
coefficient in year t in the specific drainage area, β(t) is depreciation coefficient in year t in the
specific drainage area. H(t), h(t), L(t), and l(t) are other constraints and parameters in the
specific drainage area.
3.2 Notes for DCGE
3.2.1 Basic constraint
DCGE suggests a new method that can be used to overcome the difficulties of dynamic
shadow prices in water conservancy projects evaluation. The model proposed in this article
differs from traditional analysis and has been based on a large linear programming in discrete
time. The shadow prices are calculated as well as a balance of economic system results.
During the whole development period, the time factor has been included to reflect the
relationship between different time points. Stagnant of time has been selected as 1 year. The
constraint combines the dynamic input–output model and Turnpike theorem, together with
the separation of amount and structure. The objective function is the maximum of the Gross
Domestic Product (GDP) in the objective year of the plan in the specific drainage area.
Data of the Nine Major Rivers input–output table used in this calculation demonstrates
a practical approach to balance economic development. Beyond the input–output equilibrium
82
JING HE · XIKANG CHEN · YONG SHI
relations, it is not limited in the fixed technical coefficient and can have a nonlinear character.
Multiplying the gross amount with the structure vector produces the vector of the output. The
separation can indicate the exact practical work. The shadow prices of the water resources in
balanced development path are the dual solution according to the constraint line of the water
resources in the objective year.
3.2.2 Other constraints
The aforementioned constraint is the basic one in the DCGE model. If we want to establish
the general equilibrium model we should add some other constraints to construct large-scale
systems avoiding the limit of the assumption of input–output analysis[9]. The following is the
other equilibrium and resources constraint, shortened as H(t) ≤ X(t) ≤ h(t) and L(t) ≤ X(t) ≤
l(t).
1) Constraint of production capacity is as follows:
Xi ≤ ϕi ,
(15)
where ϕi is the maximum production capacity in sector i in the specific drainage area.
2) Constraint of labor capacity is as follows:
Pn
i=1 Xi
≤ L,
Ti
(16)
where Ti is the labor-working rate in sector i in the specific drainage area, and L is the available
labor amount in the specific drainage area.
3) Other constraints of resources are as follows:
n
X
gki Xi ≤ hk ,
(17)
i=1
where gki is the input coefficient of the k resources in sector i in the specific drainage area, and
hk is the available resources amount of the k resources in sector i in the specidifc drainage area.
4) Constraint of the equilibrium between the import and export is as follows:
n
X
i=1
ei Xi =
n
X
Fi ,
(18)
i=1
where ei is input of the import commodity in sector i in the specifc drainage area, and Fi is
export of the i commodity in the specific drainage area.
5) Constraint of the equilibrium between the income reward and the consumable is as follows:
n
X
avj Xj + V ∗ − U ≤ Yw + W ,
(19)
j=1
P
where nj=1 avj Xj is income of dwellers in the property sector in the specific drainage area,
V ∗ is income of nonproperty sector in the specific drainage area, U is noncommercial payout
of dwellers in the specific drainage area, Yw is supply of the consumable, and W is the import
consumable in the specific drainage area.
6) An example of equilibrium in supply and demand of the main consumable is shown for
food supplies as follows:
X
αβ + F +
akj Xj ,
(20)
j6=k
A DYNAMIC APPROACH FOR SHADOW PRICES OF WATER RESOURCES
83
where α is average input of the food supplies in the nonfood supplies sector per person in the
specific drainage area, β is the sustained labor coefficient in the nonfood supplies sector in the
specific drainage area, F is population in nonfood supplies sector in the specific drainage area,
X
akj Xj
j6=k
is direct input in the nonfood supplies sector in the specific drainage area, γ is commodity rate
in the nonfood supplies sector in the specific drainage area, Xk is output in the food supplies
sector in the specific drainage area, and Ek is net export of the food supplies in the specific
drainage area.
7) Equilibrium between accumulation and consumption is as follows
n
X
Yiw + µ ≥ Yw ,
(21)
j=1
where Yiw is consumption amount in sector i in the previous term in the specific drainage area,
Yw is consumption amount in sector i in the present term in the specific drainage area, and µ
is other factors in the previous term in the specific drainage area.
n
X
Yk ≥
i=1
n
X
i=1
b i − Li ),
k̃i (L
(22)
where Yk is capital forming amount in the previous term in the specific drainage area, k̃i is
b i is number
average fund occupying amount per person in sector i in the specific drainage area, L
of employees in the previous term in the specific drainage area, and Li is number of employees
in the present term in the specific drainage area.
3.3 Notes for the Long-Term Marginal Cost in the DCGE
The Lagrange multipliers are given the names “shadow prices” and “dual activity” in linear
programming where these changes are analyzed by sensitivity analysis. Why are shadow prices
of the water resources in balance development path the dual solution according to the constraint
line of the water resources? Equations (14) can be shortened as
MaxZ = CX

AX ≤ b,



s.t. Aw X ≤ bw ,



X ≥0
(23)
where Aw X ≤ bw is the water resources constraint in the original linear programming. The optimal basis B is given, then the optimal solution of the dual problem is Y ∗ = (y1∗ , y2∗ , · · · , yk∗ ) =
CB B −1 . The input water resources
of bi is equal to the optimal solution of the dual problem
P
m
y
yi∗ , then Z ∗ = y ∗ b, and Z ∗ = m
i=1 i bi , and thus
yi∗ =
Ym∗ =
∂Z ∗
,
∂bi
∂Z ∗
.
∂bw
i = 1, 2, · · · , m,
(24)
84
JING HE · XIKANG CHEN · YONG SHI
∗
In a word, the shadow prices of water resources ym
are the changed amount of the objective
function when the resources bw are changed, as well as the marginal long-term value. If the
water resources increase one m3 , the objective function (GDP) will increase one Yuan. So the
unit of shadow prices for water resources is Yuan/m3 .
4 Computer-Based Algorithm of DCGE
A heuristic DCGE algorithm can be outlined as follows:
Step 1 Use the Turnpike model in Eq. (11) to calculate the structure of the output in T + 1
e + 1) in the specific drainage area.
year X(T
e + 1) in the specific drainage area. During the process
Step 2 Solve the LP beginning with X(T
of calculation, perhaps there is no feasible solution in the model in some time. Then
we can adjust the upper and the lower limits. We also add or reduce the constraint
and balance conditions to get the right results.
Step 3 Get the shadow prices along with the structure and the amount of each year in the
specific drainage area. The process of calculation can be divided into short terms. For
example, if we get the result shadow prices of 2005 by using the 1999–2005 model in the
specific drainage area, the shadow prices denote the equilibrium shadow prices during
the five years in 1999–2005. Based on Nine Major Rivers water conservancy economy
input–holding–output tables for 1999, we can get the 1949–1999 prices backwards and
calculate the 1999–2050 prices forward as a whole system. When we face a long time
span, the stagnant of the time can be prolonged to 2 years or more to reduce the
complexity of the algorithm.
4.1 Notes for the Data and the Parameter
4.1.1 Input–holding–output tables
Many appraisals presented in Section 1 for calculation of shadow price are tentative and
have suffered from data inadequacies. To solve this question, we will describe the design and
implementation of Nine Major Rivers water conservancy economy input–holding–output tables.
The research group lead by Xikang Chen consisted of about 22 researchers and professors, who
spent more than one and a half years to construct water conservancy economy input–holding–
output tables of China Nine Major Rivers for 1999[3]. The table shorten to 19 sectors is the
main data resource for our calculation. The style of the table can be shown in Table 1.
It is a great systems engineering work to constructing the table. The following seven steps are
involved in constructing water conservancy economy input–holding–output tables for drainage
areas of Nine Major Rivers[3] :
Step 1 Construct 1999 input–output tables with 40 sectors for 31 provinces, autonomous regions, and municipalities of China.
Step 2 Divide the province input–output table into basin input–output tables for each
province.
Step 3 Construct Nine Major Rivers basin input–output tables on the basis of basin input–
output tables in each province.
Step 4 Construct 1999 basin input–output tables with 12 water conservancy sectors.
Step 5 Collect water use data and estimate the amounts of water used by each sector in every
basin. The water amount is in physical units.
Step 6 Construct holding part of the basic table.
Step 7 Revision work.
A DYNAMIC APPROACH FOR SHADOW PRICES OF WATER RESOURCES
85
Table 1 Water conservancy economy input–holding–output Table in the specific drainage area
Intermediate demands
Nonwater
conservancy
sectors
1, 2, · · · , S
Nonwater
conservancy
sectors
I
N
P
U
T
O
C
C
P
A
N
C
Y
Water
conservancy
sectors
Fresh
water
W
Recycle
A
water
T
E
Waste
R water
emission
Water
conservancy
sectors
S + 1, S + 2,
··· ,n
Final
demands
Total output
and total
water
1, 2, · · · , t
1, 2, · · · S
S + 1, S + 2,
··· ,n
Xij
Yij
Xi
Fij
Zij
Wi
Pj
R
Ww
1, 2, · · · , k
k + 1, k + 2,
··· ,m
Primary
input
Total
input
1, 2, · · · , S
Fixed
assets
Circulating
capital
Labor
force
1, 2, · · · , n
Dij
1, 2, · · · , n
Cij
1, 2, · · · , g
Lij
Vj
Xj
4.1.2 Other parameters
Some parameters for DCGE should be estimated and emended for each year, such as input coefficient, added value coefficient, final demand, water input coefficient, added capital
coefficient, and depreciation capital coefficient by using the nonlinear and the key emendation
method in statistics and economics. A study of the computational efficiency and complexity
through a series of empirical tests is also important in DCGE.
5 Computer-Based Result of the DCGE Model for Shadow Prices
5.1 Result and Analysis for Shadow Prices of Water Resources
in 1949–2050 for Nine Major Rivers in China
During calculation of DCGE in the specific drainage area, the basic variable of output is
19 ∗ 19 ∗ 100 = 36100 for one major river, not including the slack variable and the others. The
scale of calculation is so enormous that most of the common software can not work it out, such
86
JING HE · XIKANG CHEN · YONG SHI
Table 2 Shadow prices of water resources for 1949–2050 in China
for Nine Major Rivers(Unit: Yuan/m3 )
1949
1959
1965
1980
1993
1994
1995
1996
1997
1998
1999
2000
2005
2008
2010
2015
2020
2025
2030
2040
2050
Southeast Yangtze Zhujiang Southwest Huaihe Songhuajiang Inland Yellow Haihe
Reivers
River
River
River
River & Liaohe
Rivers River River
Rivers
& Luanhe
River
0.26
0.36
0.33
0.26
0.32
0.38
0.40
0.57
0.65
0.06
0.08
0.08
0.06
0.07
0.09
0.09
0.13
0.15
1.11
1.50
1.37
1.10
1.33
1.59
1.67
2.38
2.72
1.59
2.16
1.97
1.58
1.91
2.29
2.39
3.42
3.91
1.91
2.59
2.36
1.90
2.28
2.74
2.87
4.09
4.68
1.97
2.67
2.44
1.96
2.36
2.83
2.96
4.22
4.83
2.03
2.76
2.52
2.02
2.43
2.92
3.06
4.36
4.99
2.09
2.84
2.60
2.08
2.51
3.01
3.15
4.49
5.14
2.16
2.93
2.68
2.15
2.59
3.11
3.25
4.63
5.30
2.09
2.84
2.59
2.08
2.50
3.00
3.14
4.48
5.13
2.12
2.88
2.63
2.11
2.54
3.05
3.19
4.55
5.21
2.13
2.89
2.64
2.12
2.55
3.06
3.20
4.56
5.22
2.20
2.98
2.73
2.19
2.63
3.16
3.31
4.72
5.40
2.26
3.07
2.80
2.25
2.70
3.25
3.40
4.85
5.55
2.36
3.21
2.93
2.35
2.83
3.40
3.55
5.07
5.81
2.46
3.34
3.05
2.45
2.95
3.54
3.70
5.28
6.05
2.52
3.43
3.13
2.51
3.02
3.63
3.79
5.41
6.20
2.63
3.57
3.26
2.61
3.15
3.78
3.95
5.64
6.45
2.71
3.69
3.37
2.70
3.25
3.91
4.08
5.83
6.67
2.81
3.82
3.49
2.80
3.37
4.05
4.23
6.04
6.92
2.97
4.03
3.68
2.95
3.56
4.27
4.47
6.37
7.29
as Matlab and Excel. We use the “Lingo” software (see http://www.lindo.com) to get the fitful
result. The shadow prices shown for Nine Major Rivers are based on the unchanged price index
in 1999 and do not reflect the inflation of average market prices, the exchange rate, etc.
6 Conclusions and Recommendations
These conclusions suggest that Southeast Rivers, Yangtze River, and Zhujiang River belongs
to the area of abundant water resources and thus the average shadow prices are lower. The
shadow prices of Haihe River and Yellow River are higher and we can safely conclude that the
water supply in the two areas is insufficient. Based on these empirical findings of the difference
between the shadow prices in Nine Major Rivers there is an alternative perspective result, which
supports the hypothesis that the present economic efficiency alone in estimating the value of
the water resources is incomplete. The main objective of this article is to propose a new method
that can be used as an indicator in evaluating the selling prices for Nine Major Rivers in China.
The government still controls the prices of water resources, so the result of shadow prices can
provide useful decision support.
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