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Spatial equilibrium in the US coal industry
Spatial equilibrium in the US coal industry T. C. Campbell, M. J. Hwang and F. Shahrokh
Traditional general equilibrium theory has ignored distance, which is a costly factor, in its basic analytical framework. Spatial factors, however, have a significant effect at the theoretical level as well as on the interpretation of empirical data. In this paper a quadratic programming model of the US coal market is developed in which market boundaries are first delineated according to freight rates and mine prices. On the basis of these redefined market areas, supply and demand functions are estimated for each region and used to generate optimal prices and trade patterns. The optimal solutions are compared with actual recent trends in prices and trade.
The traditional conception of a market, as formulated by Cassel,’ WickseJJ,2 Walras,3 and Pareto4 in their general equilibrium theory, is one in which all sellers are located at one point and buyers at an adjacent point. This classical market type is, therefore, possible when distance cost is zero. Coal companies sell to buyers who are scattered widely over costly distances rather than to consumers who are essentially concentrated within a market centre. What one hypothesizes for a system in which distance is costless may not hold for a system in which such cost is significant. Economists such as Fetter,5 Hyson and Hyson, L&h,’ Beckman,* Baumol,’ Hoover,” Greenhut,” and others have contributed to the development of a general theory of market area analysis and spatial equilibrium. To transform the conceptual framework of spatial equilibrium into practical use, Enke12 demonstrates how competitive equilibrium prices and the corresponding pattern of flow of goods among spatially separated but independent markets can be solved by electric analogue. Samuelson’ then relates Enke’s formulation to a transport cost minimization problem. He shows how such a conceptual economic problem can be converted The authors are, respectively, Professor, Associate Professor, and Research Assistant in Economics and staff members of the Regional Research Institute, West Virginia University, Morgantown, WV 26506, USA. This paper is based on research funded of Transportation. Final manuscript
230
by the US Department
received 23 June 1980.
mathematically into an extreme problem by maximizing ‘net social payoff. By postulating linear demand and supply functions, Takayama and Judge14 show that it is possible to convert the Samuelson formulation into a quadratic programming problem. This paper contains an operational model developed through the work of Enke, Samuelson, and Takayama and Judge and applies it to the US coal industry. In particular, a spatial feature is incorporated in the delineation of markets in our model. The market boundaries in economic space are not predetermined, but should be determined on the basis of both mine prices and freight rates, as they are not only different in each of the regions, but also are the major sources that influence final delivered prices. It is our purpose to redefine the US coal markets given in the 1974 fiojecr Independence Report 15 * before empirical estimates ot parameters in each region and equilibrium results are made. Since transport cost within regions was minimized first through our market realignment, it is anticipated that our equilibrium results will be optimal.
A standard quadratic programming model The spatial equilibrium model for the coal industry is an application of a special type of programming problem formulated by Enke, Samuelson, and later by Takayama and Judge. An optimum solution is shown to maximize net social payoff (NSP) for all regions. NSP for all regions is defined by Samuelson as the sum of the n * The
Project Independence Report was published by the US Federal Energy Administration and was based on studies by several interagency task forces.
0140-9883/80/04Q230~7
302.00
0 1980
IPC Business Press
Spatial equilibrium in the US coal industry: T. C Gzmpbell, M. J. Hwang and F. Shahrokh separate payoffs minus the transport costs of all shipments. Social payoff in any region can be expressed as the algebraic area under the excess demand curve. NSP in the n regions for coal, according to Takayama and Judge, can be formulated as:
zqij
x4ij
NWQ) =
xi
s
P*bi-Z
dd i s
0
(1)
0
0
where: pi=ai
i
Figure 2. Net social payoff under transport cost constraints.
and: +
1 dik C qii i
k
The shaded area represents NSP. Yet NSP must be constrained by transport cost. If an export moves from region 2 to region 1, the demand price in region 1 must be greater than or equal to the supply price plus transport cost, r21. This provides:
are regional demand and supply functions, respectively. The demand price in the ith region is pi and pi stands for the jth regional supply price, while: Xi = C qii, and Xi = Cqii. i i Equation
-
Price, p
- 1 bikCqijk k
pi = cj
t
P,>P2+t21
(4)
PI -P2>t21
(4’)
(1) can further be reduced to: or:
NWQ) = C ai 1 qij - 1 Ci1 qii i
j
i
i
and:
-
P,,Pa>Q
- 112;
The problem of maximizing NSP subject to the transportation cost constraints is shown graphically in Figure 2, where ES1 is excess supply and ED2 is excess demand, and the subscripts refer to two regions, 1 and 2. The primal problem can therefore be generalized as:
dik(1qii)’ i
The following quadratic programming model would result if the domain of the integral were transformed from Q to P:
Maximize NSP(P) = c eipl - 1 q/pi
NWP) = Z: eipi - 1 qjpi - l/2 1 fi@i)2 i
(5)
i
i
i
- l/2 1 h,(p’)2 i For simplicity, if only two markets are considered, can be illustrated as in Figure 1. I
(3)
/
112cfrW2 - l/2 1
c $@‘I2 i
(6)
NSP subject to: Pi - Pj - tij 2 0
(7)
Pi20
(8)
pj > 0.
(9)
The Iagrangian expression subject to the above constraints is given as follows: L = C epi - X 4ipi - l/2 Cfi@r)2 i
price.P Figure 1. Net social payoff.
ENERGY ECONOMICS October 1980
i
i
+~ZWij@i-Pi-tU)+~YP+$ztpi i i
- 1/2 C h,(~‘)~ /
i
(10)
231
Spatial equilibrium in the US coal industry: T. C Campbell, M. J. Hwang and F. Shahrokh
Market area sizes of the coal industry
subject to: l+L -=e,-frpr ah
-
aL
tCWij+Y1=0 j
= e2 -f2pz
aP2
aL -=yl
-hIpI
W
There are seven coal producing regions identified by the Project Independence Report.? More than 98% of US coal production comes from regions 1,2,3, 5, and 6. For simplicity, only these five regions are included in this study and are shown in Table 1. The objective is to define market boundaries from these five market centres. The boundaries are not predetermined but should be determined on the basis of both mine prices and freight rates, as they not only differ in different regions, but also are the major sources that influence final delivered prices. For analytical simplicity, assume:
+ C wzi +y2 = 0 i
+C
Wil +Zl
=O
i
l
aL
-=q2-h2p2+xwi2+Z2=0
aP2
l
i l
(11) Notice that if fixed demand and supply are assumed, the above quadratic spatial equilibrium model can easily be reduced to a linear model: L=Ceipi-~qipj+C~wwijOli-pi-tij) i
+x
jipi +C Zjp’
i
(12)
subject to: Cwri+yr i
We will employ Li5sch’s concept of perfect intra-industry dispersion and general market equilibrium in economic space as a point of departure in defining the market areas. Following the analytical framework set forth by Fetter,” and Hyson and Hyson,” the impacts of freight rate and mine price on the size and shape of markets can be demonstrated. The market boundary between two market centres is determined by the same delivered prices where the difference between freight costs of the two markets is equal to the difference between mine prices. The delivered prices for.both markets are shown, respectively, as:
= -er
x:W2j+Y2=-e2
that the quality of coal is the same except in heating value; that buyers are evenly distributed over a plane and sellers are located at a market centre; that sellers sell on an fob mill basis.
Pr = mr +fldl
(14)
P2 = m2 +fd2
(15)
When mine prices ml #m2 and constant
i
freight rates
fi#f2,the boundary line defined by PI =P2 in Equations (14) and (15) is obtained
by:
mr +f,d, =m+f2d2 CWil
+zl
iwi2
+z2=92
(16)
It can be shown that, when fl=f2and nzr #m2. the market boundary is either an ellipse or a hyperbola; when fl #f2 and ml =m2, the market boundary is circular.rg If both ml fm2 and flff2 and transport rate, f, is no longer constant, the market boundary defined in
=91
Table 1. Regions included in the study.
Wij,Yi7
zj
20
(13)
As can be seen when demand and supply are fixed, the quadratic terms disappear and the model becomes linear. It is manifest that the quadratic programming model has the advantage of varying regional demand and supply. To find a spatial equilibrium result. regions must first be defined, and then demand and supply functions of each region can be estimated. Our next section will redefine regions designated by the 1974 Project lndependence Report.
232
No
Region
Market centre
1 2 3 5 6
Northern Appalachian Southern Appalachian Interior Northern Great Plains Rocky Mountains
Wheeling, West Virginia Nashville, Tennessee St Louis, Missouri Billings, Montana Albuquerque, New Mexico
t Details of the theory of market area analysis formulated here are given by Campbell and Hwang.16 Thus paper expands the research discussed in that article, details of the analytical framework must be left for other papers.
ENERGY ECONOMICS October 1980
Spati
equilibtium in the US coal indushy: T. C Gnpbell,
M. J. Twang and E Shuhrokh
Equation (16) is derived in the following quadratic form, since a freight rate is related to distance in the quadratic function: H,(x,y)=al(x2+y2)+a~ - bl f(h -x)’
7x
+y
+a3
+y2f
-b,&m--b3=0
(17)
where aI, 1z2,a3, bl, bz, and b3 are constant, and a3 and b3 are the sum of mine price and intercept value of the transport rate function. The folIowing relation can be established from the triangle FL1 L2 of Figure 3, where
LI and L2 are market centrev, ~&,y)
= (h -x)’
+y2 + h2
-2h~~cose
-x2-y2=0 (18)
Letting d, - Jxv
and & = d(w,
Equations
where the elements in the inverse matrix are obtained by
taking the partial derivatives of Hr and Hz with respect to 6% and dz. Tfie initial vaIues of dl’ and da’ can be found when the direct distance between two market centres is known with, 6 = 0. By adding another degree of angle, another set (dCllzfi,dzi”) can be found from Equation (19). A computer program has been derived from the above relation, and the boundary points between the two market centres can be generated. To derive the boundary between two markets, market centres are fust identified as above. Mine prices and freight rates in terms of cents/million Btu are then
x
Figure3.
Triangular relatian between market centres.
estimated as is indicated in Table 2. For analytical simplicity, only the difference in beating values is taken into consideration in our model as coal is used mainly as a fuel. The mine prices per ton in Appalachia are the highest and those of the Northern Great Plains are the lowest, about half those in Appalachia. On a Btu basis, App~achi~ coal continues to be the most expensive, but the si~~can~ of the difference is lessened by the higher calorific value of AppaIachian coal. On the basis of the mine price and freight rate data in Table 2 we have computed the market areas shown in Table 3. Given the market centres identified by Project ~~depe~~ce Report, the market boundaries can be defined on the basis of the same delivered prices which, in turn, are determined by miII price and freight rates in adjacent markets. Some cities are on the bo~da~es, for example, Cincinnati (Ohio) and Wilson (West Virginia) lie between regions 1 and 2, Lansing ~ic~~an~ and Fort Wayne (Indiana] Iie between regions I and 3, and Evoke (~nd~na~ is between regions 2 and 3. Between regions 3 and 5 are Wadena ~~neso~~ and Sioux Falls (South Dakota). Between regions 3 and 6 and 5 and 6 are, respectively, Perry (Okiahoma) and Craig (Colorado). The market’s boundary is not predetermined, as was shown in Project independence Report, but is determined after the structure of the markets is ascertained and their parameters estimated.
Table 2. Mine prices and transport raters.
Region
Average mine price wton)
Mine price kants/million
1
21.63
87.68
2
20.93
86.93
3
16.56
75.81
5
10.67
63.76
6
16.75
74.17
Btu)
transport rate (cents/million Btu transported)
a*
Rate = 9.05 &.!%I Rate = 6.34 (2.67) Rate = 7.05 15.08) Rate = 1.16 El.381 Rate = 4.22
0.8807
- 0.000024dz (-2.81) - 0.000026ds (-2.43) - 0.000026da (-3.391 - 0.000021d~ f-2.851 - O.~~dz
f 1.om 1-4.27)
+ 0.068d (5.81) f 6 05W (4.361 + 0.05Qd (7.10) +0.0706
0.6467 0.7923 0.8708
(2.24) + O.OS8d 15.07~
0.6483
Note: Figures in parenthesis are r-statistics. Sources: Mine price data are from Minerais Y~r~~ok, US De~rtment of the interior, Washi~on, DC; average Btu content of coal is from P. H. Mutschler, R. 4. Evans and G. M. Larvvood, ~~rnpamtj~ Tm~~~~tjo~ Cost of Supplying Low-Sulfur Fuels to ~jdwes~rn and Eastern Domastic Enaqy Markets, US Department of the Interior, W~hing~n, DC, 1974; and the transport rate is from lnvestig8tion of Railroad Freight Rate Structure-Cm/, US Interstate Commerce Commission, December 1974.
ENERGY ECONOMICS Octaber 1980
233
Spatial equilibrium in the US coal industry: T. C. Gzmpbell, M. J. Hwang and F. Shahrokh Table 3. Market
areas redefined. Region as defined by Project Independence
Region
Report
Appalachian
Michigan, Pennsylvania, northern West Virginia
Southern
Appalachian
Southern West Virginia, eastern Kentucky, Tennessee, Virginia, Alabama
Kentucky, Alabama
Iowa, Illinois, Indiana, Missouri, Oklahoma, western Kentucky, northern Arkansas, northern Texas, eastern Nebraska, eastern Kansas
Iowa, Kansas, Arkansas, eastern Oklahoma, eastern Kansas, eastern Nebraska
North Dakota, South Dakota, Wyoming, Idaho
North Dakota, South Dakota, Montana, Washington, Oregon, Wyoming, western Nebraska, northern Utah, northern Nevada, northern California
Northern
Rockv
Great Plains
Mountains
Utah,
Colorado,
Arizona,
Maryland,
region
Northern
Interior
Ohio,
Redefined
Michigan, Ohio, Pennsylvania, West Virginia, Virginia, northern North Carolina, New England states
Montana,
New Mexico
Tennessee,
southern
North
Carolina,
Colorado, New Mexico, Texas, Arizona, western Oklahoma, western Kansas, western Nebraska, southern California, southern Nevada
where PD stands for demand price (cents/million Btu),$ CD for coal demand (million Btu), Y for per capita personal income (constant dollars), ED for electrical demand (million kWh), Ps for supply price (centsj
million Btu), CS for coal supply (million Btu),$ and for average productivity (output per man per day). The time series data for 1957 to 1977 were collected, and related regression equations for each region were estimated. We selected the estimated functions on the basis of the higher R* value from either the ordinary least squares method (OLS) or the two-stage least squares method (2SLS). The selected demand and supply functions are summarized in Tables 4 and 5. Table 6 summarizes the transport costs (cents/million Btu) among market centres and also shows average distances. These transport costs are computed directly from estimated transport functions in Table 2. By applying relevant data, along with the spatial constraints indicated in our model, to the quadratic computer program, the equilibrium optimal distribution
$ The
8
Spatial equilibrium among markets What are the optimal equilibrium prices and optimal patterns of trade? To find the solution, we follow the theoretical model and the regional definitions given above. To find a spatial equilibrium result, the demand and supply functions of each region must be identified and estimated first. They are identified, respectively, as: Po=ae+arCD+aaYtasED
(20)
Ps=b,,+b,CS+b2AF’
(21)
price in cents/million
P (S/ton) 2000
value is equal to:
Quantitv
in lOI
o (thousand
Btu is computed
tons) x 100 x 2000
x Btu (per lb)
Table 4. Estimated Region
Btu in constant
x 1 O6
AP
from the relation: x Btu
lO’$
regional demand functions.
Demand
P = 138.7
function
(2.68) P = 44.77 (4.02)
16.32CD (-3.23) 10.08CD (-6.00) P = 99.47 - 60.46CD (3.68) (-4.24) P = 46.49 - 49.05CD (6.51) i-0.63) P = 45.05 - 25.63CD (5.89) (-1.15)
- 0.003281 Y - 0.0003886ED (-7.19) (-1.63) - 0.006052Y - 0.0003152ED (-9.05) (-2.66) - 0.003144Y + 0.0008481 ED l-8.07) (3.63) - 0.00222Y - 0.007486ED (-4.89) (5.91) - 0.001771 Y - 0.0009796ED (-3.43) (3.21)
R’
Method
0.7074
2SLS
0.9020
2SLS
0.8819
OLS
0.9245
OLS
0.4741
OLS
used for estimation
Note: Figures in parenthesis are t-statistics. Sources: Minerals Yearbook, US Department of the Interior, Washington, DC;Survey of Current Business, US Department of Commerce, Bureau of Economic Analysis; and Edison Electric Institute Statistical Year Book of the Electric Utntry maustry, Edison Electric Institute, 1957-75.
234
ENERGY ECONOMICS October 1980
Spatial equilibrium in the US coal industry: T. C olmpbell, M. J. Hwang and F. Shahrokh Table 5. Estimated
regional supply functions.
Region
Supply function
R’
Method
1
P = 9.964 + 0.717OCS - 3.814AP (-6.00) (6.38) (1.04) P=6.104+6.104CS-3.184AP (-3.50) (3.73) (2.23) P = 3.513 + 8.802CS - 2.090AP (3.00) (2.47) (- 1.62) P = 5.62 + 21.3OCS - 0.1327AP (7.15) (- 2.94) (7.74) P = 7.327 + 7.232CS - 05462AP (-1.44) (4.51) (3.61)
0.7490
OLS
0.4996
2SLS
0.4499
OLS
0.7905
2SLS
0.4483
2SLS
2 3 5 6
Note: Figures in parenthesis are t-statistics. Sources: Minerals Yearbook. US Deoartment Association, Washington, DC, 1957-75.
of the Interior,
Washington,
Table 6. Transport costs among regional centres (cents/million Btu).
Region
Region 1
1
14.70
2
3
5
6
(171)
20.47 (306)
33.69 (550)
42.45 (1 470)
42.69 (1 490)
2
16.44 (306)
13.88 (154)
19.14 (260)
34.69 (1340)
28.80 (1 130)
3
32.39 (550)
21.15 (260)
11.80 (128)
41.82 (1035
40.90 (945)
5
58.68 (1470)
57.25 (1 340)
51.11 (1035)
15.04 (216)
42.80 (775)
6
51.62
52.78
(1490)
(1 130)
50.25 (945)
46.05 (775)
14.43 (201)
Note: Figures in parenthesis are average distances (miles) within regions and between regions.
of coal between regions is obtained and summarized in Table 7. The principal diagonal indicates the amount of production in each major market consumed within its own region. Other cells indicate cross-border movements. There is only one interregional coal flow, from region 2 to region 1. Otherwise, there is no movement among regions. Coal will move from one region to another if, and only if, differences in regional prices exceed transport costs between regions. The small number of crossborder movements of coal, as shown in Table 7, indicates Table 7. Optimal
flow of coal (1W5 BtuL
Demand region
Supply 1
region 2
1 2 3 5 6
5.89 0 0 0 0
Total aJPPlY
5.89
Total demand
3
5
6
0.65 1.53 0 0 0
0 0 1.23 0 0
0 0 0 0.28 0
0 0 0 0 0.71
6.54 1.53 1.23 0.28 0.71
2.18
1.23
0.28
0.71
10.29
ENERGY ECONOMICS October 1980
DC; and Bituminous
used for estimation
Coal Facts, National
Coal
that our regional markets are well defined with respect to demand and supply in each region. The theoretical assumption of all sellers being located at market centres indicates another reason for few interregional coal flows. Furthermore, the high cost of transport for bulky products such as coal makes the interregional flow even more difficult. The only movement of coal, from region 2 to region 1, is caused by the high demand in the highly industrial and populated region 1. Time series data were used to estimate demand and supply functions since cross section data were not available. The estimated equations should reflect attributes of data over time rather than the position of a single point in time. Mid-year data (for 1967) of actual coal flow are therefore computed and shown in Table 8. There are plenty of interregional coal flows since some mines in the real world are not located in market centres. The mines located at the periphery of the market find it easier to sell their coal to other regions because of lower transport costs compared with mines located in the market centre. However, because of the high transport cost for bulky products such as coal, most coal produced in a region is consumed within that region, and the actual interregional movements indicated in Table 8 are, therefore, small. The optimal results, shown in Table 7, can now be compared with the actual flows shown in Table 8. The principal coal consuming states have been those of the Table 8. Actual flow of coal (10”Btu). Demand region
Supply region 1 2
3
5
6
1 2 3 5 6
4.78 0.43 0.55 0.02 0.003
1.14 0.92 0.53 0.009 0.005
0.02 0.20 1.44 0.03 0.02
0 0 0.02 0.09 0.01
0 0 0 0.08 0.19
5.94 1.55 2.54 0.23 0.23
5.78
2.61
1.71
0.12
0.27
10.49
Total =PPly
Source: Mineral lndusrry Surveys, US Department Interior, Bureau of Mines, Washington, DC.
Total demand
of the
235
Spatial equilibrium
in the US coal industry:
T. C. Campbell, M. J. Hwang and F. Shahrokh
Table 9. Optimal supply and demand prices (cents/million Btul. Region 1
17.20 Optimal supply price Optimal demand price 31.90 Actual supply price 24.75
2
3
5
$
15.46 29.34 24.45
14.33 26.13 22.14
9.80 24.84 21.05
11.49 25.92 21.56
industrial Midwest, Pennsylvania, and West Virginia (the whole state of West Virginia is now defined as our region 1) in the east and the populated northeast. However, major production comes mainly from region 1 and, subsequently, from regions 2 and 3. Low demand in region 2 makes it possible to export coal to high demand in region 1. Our optimal results are, therefore, consistent with the actual coal flow figures of 1967. The optimal supply prices and demand prices given by our model are shown in Table 9. As was indicated above, demand price must be equal to or greater than the supply price plus transport cost. When most of the coal produced is consumed within the producing region, demand prices are equal to supply prices plus average transport costs within their regions (eg in region 1, the demand price of 3 1.90 cents/million Btu equals the supply price of 17.20 cents/million Btu plus the average transport cost of 14.70 cents/million Btu). When coal moves from region 2 to region 1, demand price, 3 1.90 cents/ million Btu, is equal to supply price, 15.46 cents/million Btu, plus transport cost, 16.44 cents/million Btu. As expected, the equilibrium supply prices are relatively low in comparison with actuaI supply prices (see the third row of Table 9). Conclusions The principal focus of this paper has been a redefinition of coal markets and the sources of supply for each market. Coal is a heavy commodity with low value per unit of weight. To transport it long distances frequently costs as much or more than the price at the mines. With expansion in output of mines in western USA and the long distances over which this and other coal is moving, minimizing the distances and cross-boundary movements will assure both global optimal flows among coal markets and optimal transport costs. The paper represents part of a research effort to redefme the markets, to estimate demand and supply functions, and to generate optimal spatial equilibrium results. Careful theoretical analysis is presented as a foundation for further investigation of a more empirical
236
nature. From this sounder theoretical base, our purpose is to assist in the determination of the location of power generating plants and the supply regions that can supply these plants at minimum costs. Because of the large tonnages and the relatively long periods associated with plant location, design, and capacity changes in the industry, investigation that will assist in the selection of plant locations and the scale of each plant will minimize unit energy costs in specific regions and for the USA as a whole. The analysis in redefining the supply regions and the markets in this paper is part of a broader investigation.
References Gustav Cassel, Fundamental Thoughts in Economics, T. F. Unwin, London, 1925. Knut Wicksell, Lectures on Political Economy, Routledge, London, 1935. Leon Walras, Ekments of Pure Economics. George Allen and Unwin, Londdn, 1954. Vilfredo Pareto,. ‘Mathematical economics’, International Economic Papers, No 5, Macmillan, New York, 1955. 5 F. Fetter, ‘The economic law of market areas’, Quarterly Journal of Economics, 1924, pp 520-529. 6 C. D. Hyson and W. P. Hyson, ‘The economic law of market areas’, The Quarterly Journal of Economics, 1950, pp 319-327. August LSsch, The Economics of Location, Yale University Press, New Haven, CT, 1954. Martin Beckman, ‘A continuous model of transportation’, Econometica, 1952, pp 643-660. W. J. Baumol, Spatial Equilibtium with Supply Points Separated from Markets with Supplies Predetermined, Bureau of Agricultural Economics,
USDA, Washington, DC, 1952. E. M. Hoover, An Introduction to Regional Economics, Alfred A. Knopf, New York, 1971. 11 M. L. Greenhut, A Theory of the Firm in Economic Space, Lone Star Publishing, Austin, TX, 1974. 12 S. Enke, ‘Equilibrium among spatially separated markets: solution by electric analogues’, Econometrica, 195 1, pp 40-48. 13 P. A. Samuelson. ‘Suatial urice eauilibrium and linear programmhg’, Ame&an Eionomic Review, 10
1952, pp 283-303.
14 T. Takayama and G. G. Judge, ‘Spatial equilibrium and quadratic programming’, Journal of Farm Economics, 1964, pp 67-93. 15 Federal Energy Administration? Project Independence Report, Federal Energy Admimstration, Washington, DC, 1974. 16 T. C. Campbell and M. J. Hwang, ‘Spatial analysis and the identification of coal markets’, The Journal of Energy and Development, Autumn 1978, pp 104125. Fetter, op tit, Ref 5. ;: Hyson and Hyson, op tit, Ref 6. 19 Greenhut, op tit, Ref 11.
ENERGY ECONOMICS October
1980
Traditional general equilibrium theory has ignored distance, which is a costly factor, in its basic analytical framework. Spatial factors, however, have a significant effect at the theoretical level as well as on the interpretation of empirical data. In this paper a quadratic programming model of the US coal market is developed in which market boundaries are first delineated according to freight rates and mine prices. On the basis of these redefined market areas, supply and demand functions are estimated for each region and used to generate optimal prices and trade patterns. The optimal solutions are compared with actual recent trends in prices and trade.
The traditional conception of a market, as formulated by Cassel,’ WickseJJ,2 Walras,3 and Pareto4 in their general equilibrium theory, is one in which all sellers are located at one point and buyers at an adjacent point. This classical market type is, therefore, possible when distance cost is zero. Coal companies sell to buyers who are scattered widely over costly distances rather than to consumers who are essentially concentrated within a market centre. What one hypothesizes for a system in which distance is costless may not hold for a system in which such cost is significant. Economists such as Fetter,5 Hyson and Hyson, L&h,’ Beckman,* Baumol,’ Hoover,” Greenhut,” and others have contributed to the development of a general theory of market area analysis and spatial equilibrium. To transform the conceptual framework of spatial equilibrium into practical use, Enke12 demonstrates how competitive equilibrium prices and the corresponding pattern of flow of goods among spatially separated but independent markets can be solved by electric analogue. Samuelson’ then relates Enke’s formulation to a transport cost minimization problem. He shows how such a conceptual economic problem can be converted The authors are, respectively, Professor, Associate Professor, and Research Assistant in Economics and staff members of the Regional Research Institute, West Virginia University, Morgantown, WV 26506, USA. This paper is based on research funded of Transportation. Final manuscript
230
by the US Department
received 23 June 1980.
mathematically into an extreme problem by maximizing ‘net social payoff. By postulating linear demand and supply functions, Takayama and Judge14 show that it is possible to convert the Samuelson formulation into a quadratic programming problem. This paper contains an operational model developed through the work of Enke, Samuelson, and Takayama and Judge and applies it to the US coal industry. In particular, a spatial feature is incorporated in the delineation of markets in our model. The market boundaries in economic space are not predetermined, but should be determined on the basis of both mine prices and freight rates, as they are not only different in each of the regions, but also are the major sources that influence final delivered prices. It is our purpose to redefine the US coal markets given in the 1974 fiojecr Independence Report 15 * before empirical estimates ot parameters in each region and equilibrium results are made. Since transport cost within regions was minimized first through our market realignment, it is anticipated that our equilibrium results will be optimal.
A standard quadratic programming model The spatial equilibrium model for the coal industry is an application of a special type of programming problem formulated by Enke, Samuelson, and later by Takayama and Judge. An optimum solution is shown to maximize net social payoff (NSP) for all regions. NSP for all regions is defined by Samuelson as the sum of the n * The
Project Independence Report was published by the US Federal Energy Administration and was based on studies by several interagency task forces.
0140-9883/80/04Q230~7
302.00
0 1980
IPC Business Press
Spatial equilibrium in the US coal industry: T. C Gzmpbell, M. J. Hwang and F. Shahrokh separate payoffs minus the transport costs of all shipments. Social payoff in any region can be expressed as the algebraic area under the excess demand curve. NSP in the n regions for coal, according to Takayama and Judge, can be formulated as:
zqij
x4ij
NWQ) =
xi
s
P*bi-Z
dd i s
0
(1)
0
0
where: pi=ai
i
Figure 2. Net social payoff under transport cost constraints.
and: +
1 dik C qii i
k
The shaded area represents NSP. Yet NSP must be constrained by transport cost. If an export moves from region 2 to region 1, the demand price in region 1 must be greater than or equal to the supply price plus transport cost, r21. This provides:
are regional demand and supply functions, respectively. The demand price in the ith region is pi and pi stands for the jth regional supply price, while: Xi = C qii, and Xi = Cqii. i i Equation
-
Price, p
- 1 bikCqijk k
pi = cj
t
P,>P2+t21
(4)
PI -P2>t21
(4’)
(1) can further be reduced to: or:
NWQ) = C ai 1 qij - 1 Ci1 qii i
j
i
i
and:
-
P,,Pa>Q
- 112;
The problem of maximizing NSP subject to the transportation cost constraints is shown graphically in Figure 2, where ES1 is excess supply and ED2 is excess demand, and the subscripts refer to two regions, 1 and 2. The primal problem can therefore be generalized as:
dik(1qii)’ i
The following quadratic programming model would result if the domain of the integral were transformed from Q to P:
Maximize NSP(P) = c eipl - 1 q/pi
NWP) = Z: eipi - 1 qjpi - l/2 1 fi@i)2 i
(5)
i
i
i
- l/2 1 h,(p’)2 i For simplicity, if only two markets are considered, can be illustrated as in Figure 1. I
(3)
/
112cfrW2 - l/2 1
c $@‘I2 i
(6)
NSP subject to: Pi - Pj - tij 2 0
(7)
Pi20
(8)
pj > 0.
(9)
The Iagrangian expression subject to the above constraints is given as follows: L = C epi - X 4ipi - l/2 Cfi@r)2 i
price.P Figure 1. Net social payoff.
ENERGY ECONOMICS October 1980
i
i
+~ZWij@i-Pi-tU)+~YP+$ztpi i i
- 1/2 C h,(~‘)~ /
i
(10)
231
Spatial equilibrium in the US coal industry: T. C Campbell, M. J. Hwang and F. Shahrokh
Market area sizes of the coal industry
subject to: l+L -=e,-frpr ah
-
aL
tCWij+Y1=0 j
= e2 -f2pz
aP2
aL -=yl
-hIpI
W
There are seven coal producing regions identified by the Project Independence Report.? More than 98% of US coal production comes from regions 1,2,3, 5, and 6. For simplicity, only these five regions are included in this study and are shown in Table 1. The objective is to define market boundaries from these five market centres. The boundaries are not predetermined but should be determined on the basis of both mine prices and freight rates, as they not only differ in different regions, but also are the major sources that influence final delivered prices. For analytical simplicity, assume:
+ C wzi +y2 = 0 i
+C
Wil +Zl
=O
i
l
aL
-=q2-h2p2+xwi2+Z2=0
aP2
l
i l
(11) Notice that if fixed demand and supply are assumed, the above quadratic spatial equilibrium model can easily be reduced to a linear model: L=Ceipi-~qipj+C~wwijOli-pi-tij) i
+x
jipi +C Zjp’
i
(12)
subject to: Cwri+yr i
We will employ Li5sch’s concept of perfect intra-industry dispersion and general market equilibrium in economic space as a point of departure in defining the market areas. Following the analytical framework set forth by Fetter,” and Hyson and Hyson,” the impacts of freight rate and mine price on the size and shape of markets can be demonstrated. The market boundary between two market centres is determined by the same delivered prices where the difference between freight costs of the two markets is equal to the difference between mine prices. The delivered prices for.both markets are shown, respectively, as:
= -er
x:W2j+Y2=-e2
that the quality of coal is the same except in heating value; that buyers are evenly distributed over a plane and sellers are located at a market centre; that sellers sell on an fob mill basis.
Pr = mr +fldl
(14)
P2 = m2 +fd2
(15)
When mine prices ml #m2 and constant
i
freight rates
fi#f2,the boundary line defined by PI =P2 in Equations (14) and (15) is obtained
by:
mr +f,d, =m+f2d2 CWil
+zl
iwi2
+z2=92
(16)
It can be shown that, when fl=f2and nzr #m2. the market boundary is either an ellipse or a hyperbola; when fl #f2 and ml =m2, the market boundary is circular.rg If both ml fm2 and flff2 and transport rate, f, is no longer constant, the market boundary defined in
=91
Table 1. Regions included in the study.
Wij,Yi7
zj
20
(13)
As can be seen when demand and supply are fixed, the quadratic terms disappear and the model becomes linear. It is manifest that the quadratic programming model has the advantage of varying regional demand and supply. To find a spatial equilibrium result. regions must first be defined, and then demand and supply functions of each region can be estimated. Our next section will redefine regions designated by the 1974 Project lndependence Report.
232
No
Region
Market centre
1 2 3 5 6
Northern Appalachian Southern Appalachian Interior Northern Great Plains Rocky Mountains
Wheeling, West Virginia Nashville, Tennessee St Louis, Missouri Billings, Montana Albuquerque, New Mexico
t Details of the theory of market area analysis formulated here are given by Campbell and Hwang.16 Thus paper expands the research discussed in that article, details of the analytical framework must be left for other papers.
ENERGY ECONOMICS October 1980
Spati
equilibtium in the US coal indushy: T. C Gnpbell,
M. J. Twang and E Shuhrokh
Equation (16) is derived in the following quadratic form, since a freight rate is related to distance in the quadratic function: H,(x,y)=al(x2+y2)+a~ - bl f(h -x)’
7x
+y
+a3
+y2f
-b,&m--b3=0
(17)
where aI, 1z2,a3, bl, bz, and b3 are constant, and a3 and b3 are the sum of mine price and intercept value of the transport rate function. The folIowing relation can be established from the triangle FL1 L2 of Figure 3, where
LI and L2 are market centrev, ~&,y)
= (h -x)’
+y2 + h2
-2h~~cose
-x2-y2=0 (18)
Letting d, - Jxv
and & = d(w,
Equations
where the elements in the inverse matrix are obtained by
taking the partial derivatives of Hr and Hz with respect to 6% and dz. Tfie initial vaIues of dl’ and da’ can be found when the direct distance between two market centres is known with, 6 = 0. By adding another degree of angle, another set (dCllzfi,dzi”) can be found from Equation (19). A computer program has been derived from the above relation, and the boundary points between the two market centres can be generated. To derive the boundary between two markets, market centres are fust identified as above. Mine prices and freight rates in terms of cents/million Btu are then
x
Figure3.
Triangular relatian between market centres.
estimated as is indicated in Table 2. For analytical simplicity, only the difference in beating values is taken into consideration in our model as coal is used mainly as a fuel. The mine prices per ton in Appalachia are the highest and those of the Northern Great Plains are the lowest, about half those in Appalachia. On a Btu basis, App~achi~ coal continues to be the most expensive, but the si~~can~ of the difference is lessened by the higher calorific value of AppaIachian coal. On the basis of the mine price and freight rate data in Table 2 we have computed the market areas shown in Table 3. Given the market centres identified by Project ~~depe~~ce Report, the market boundaries can be defined on the basis of the same delivered prices which, in turn, are determined by miII price and freight rates in adjacent markets. Some cities are on the bo~da~es, for example, Cincinnati (Ohio) and Wilson (West Virginia) lie between regions 1 and 2, Lansing ~ic~~an~ and Fort Wayne (Indiana] Iie between regions I and 3, and Evoke (~nd~na~ is between regions 2 and 3. Between regions 3 and 5 are Wadena ~~neso~~ and Sioux Falls (South Dakota). Between regions 3 and 6 and 5 and 6 are, respectively, Perry (Okiahoma) and Craig (Colorado). The market’s boundary is not predetermined, as was shown in Project independence Report, but is determined after the structure of the markets is ascertained and their parameters estimated.
Table 2. Mine prices and transport raters.
Region
Average mine price wton)
Mine price kants/million
1
21.63
87.68
2
20.93
86.93
3
16.56
75.81
5
10.67
63.76
6
16.75
74.17
Btu)
transport rate (cents/million Btu transported)
a*
Rate = 9.05 &.!%I Rate = 6.34 (2.67) Rate = 7.05 15.08) Rate = 1.16 El.381 Rate = 4.22
0.8807
- 0.000024dz (-2.81) - 0.000026ds (-2.43) - 0.000026da (-3.391 - 0.000021d~ f-2.851 - O.~~dz
f 1.om 1-4.27)
+ 0.068d (5.81) f 6 05W (4.361 + 0.05Qd (7.10) +0.0706
0.6467 0.7923 0.8708
(2.24) + O.OS8d 15.07~
0.6483
Note: Figures in parenthesis are r-statistics. Sources: Mine price data are from Minerais Y~r~~ok, US De~rtment of the interior, Washi~on, DC; average Btu content of coal is from P. H. Mutschler, R. 4. Evans and G. M. Larvvood, ~~rnpamtj~ Tm~~~~tjo~ Cost of Supplying Low-Sulfur Fuels to ~jdwes~rn and Eastern Domastic Enaqy Markets, US Department of the Interior, W~hing~n, DC, 1974; and the transport rate is from lnvestig8tion of Railroad Freight Rate Structure-Cm/, US Interstate Commerce Commission, December 1974.
ENERGY ECONOMICS Octaber 1980
233
Spatial equilibrium in the US coal industry: T. C. Gzmpbell, M. J. Hwang and F. Shahrokh Table 3. Market
areas redefined. Region as defined by Project Independence
Region
Report
Appalachian
Michigan, Pennsylvania, northern West Virginia
Southern
Appalachian
Southern West Virginia, eastern Kentucky, Tennessee, Virginia, Alabama
Kentucky, Alabama
Iowa, Illinois, Indiana, Missouri, Oklahoma, western Kentucky, northern Arkansas, northern Texas, eastern Nebraska, eastern Kansas
Iowa, Kansas, Arkansas, eastern Oklahoma, eastern Kansas, eastern Nebraska
North Dakota, South Dakota, Wyoming, Idaho
North Dakota, South Dakota, Montana, Washington, Oregon, Wyoming, western Nebraska, northern Utah, northern Nevada, northern California
Northern
Rockv
Great Plains
Mountains
Utah,
Colorado,
Arizona,
Maryland,
region
Northern
Interior
Ohio,
Redefined
Michigan, Ohio, Pennsylvania, West Virginia, Virginia, northern North Carolina, New England states
Montana,
New Mexico
Tennessee,
southern
North
Carolina,
Colorado, New Mexico, Texas, Arizona, western Oklahoma, western Kansas, western Nebraska, southern California, southern Nevada
where PD stands for demand price (cents/million Btu),$ CD for coal demand (million Btu), Y for per capita personal income (constant dollars), ED for electrical demand (million kWh), Ps for supply price (centsj
million Btu), CS for coal supply (million Btu),$ and for average productivity (output per man per day). The time series data for 1957 to 1977 were collected, and related regression equations for each region were estimated. We selected the estimated functions on the basis of the higher R* value from either the ordinary least squares method (OLS) or the two-stage least squares method (2SLS). The selected demand and supply functions are summarized in Tables 4 and 5. Table 6 summarizes the transport costs (cents/million Btu) among market centres and also shows average distances. These transport costs are computed directly from estimated transport functions in Table 2. By applying relevant data, along with the spatial constraints indicated in our model, to the quadratic computer program, the equilibrium optimal distribution
$ The
8
Spatial equilibrium among markets What are the optimal equilibrium prices and optimal patterns of trade? To find the solution, we follow the theoretical model and the regional definitions given above. To find a spatial equilibrium result, the demand and supply functions of each region must be identified and estimated first. They are identified, respectively, as: Po=ae+arCD+aaYtasED
(20)
Ps=b,,+b,CS+b2AF’
(21)
price in cents/million
P (S/ton) 2000
value is equal to:
Quantitv
in lOI
o (thousand
Btu is computed
tons) x 100 x 2000
x Btu (per lb)
Table 4. Estimated Region
Btu in constant
x 1 O6
AP
from the relation: x Btu
lO’$
regional demand functions.
Demand
P = 138.7
function
(2.68) P = 44.77 (4.02)
16.32CD (-3.23) 10.08CD (-6.00) P = 99.47 - 60.46CD (3.68) (-4.24) P = 46.49 - 49.05CD (6.51) i-0.63) P = 45.05 - 25.63CD (5.89) (-1.15)
- 0.003281 Y - 0.0003886ED (-7.19) (-1.63) - 0.006052Y - 0.0003152ED (-9.05) (-2.66) - 0.003144Y + 0.0008481 ED l-8.07) (3.63) - 0.00222Y - 0.007486ED (-4.89) (5.91) - 0.001771 Y - 0.0009796ED (-3.43) (3.21)
R’
Method
0.7074
2SLS
0.9020
2SLS
0.8819
OLS
0.9245
OLS
0.4741
OLS
used for estimation
Note: Figures in parenthesis are t-statistics. Sources: Minerals Yearbook, US Department of the Interior, Washington, DC;Survey of Current Business, US Department of Commerce, Bureau of Economic Analysis; and Edison Electric Institute Statistical Year Book of the Electric Utntry maustry, Edison Electric Institute, 1957-75.
234
ENERGY ECONOMICS October 1980
Spatial equilibrium in the US coal industry: T. C olmpbell, M. J. Hwang and F. Shahrokh Table 5. Estimated
regional supply functions.
Region
Supply function
R’
Method
1
P = 9.964 + 0.717OCS - 3.814AP (-6.00) (6.38) (1.04) P=6.104+6.104CS-3.184AP (-3.50) (3.73) (2.23) P = 3.513 + 8.802CS - 2.090AP (3.00) (2.47) (- 1.62) P = 5.62 + 21.3OCS - 0.1327AP (7.15) (- 2.94) (7.74) P = 7.327 + 7.232CS - 05462AP (-1.44) (4.51) (3.61)
0.7490
OLS
0.4996
2SLS
0.4499
OLS
0.7905
2SLS
0.4483
2SLS
2 3 5 6
Note: Figures in parenthesis are t-statistics. Sources: Minerals Yearbook. US Deoartment Association, Washington, DC, 1957-75.
of the Interior,
Washington,
Table 6. Transport costs among regional centres (cents/million Btu).
Region
Region 1
1
14.70
2
3
5
6
(171)
20.47 (306)
33.69 (550)
42.45 (1 470)
42.69 (1 490)
2
16.44 (306)
13.88 (154)
19.14 (260)
34.69 (1340)
28.80 (1 130)
3
32.39 (550)
21.15 (260)
11.80 (128)
41.82 (1035
40.90 (945)
5
58.68 (1470)
57.25 (1 340)
51.11 (1035)
15.04 (216)
42.80 (775)
6
51.62
52.78
(1490)
(1 130)
50.25 (945)
46.05 (775)
14.43 (201)
Note: Figures in parenthesis are average distances (miles) within regions and between regions.
of coal between regions is obtained and summarized in Table 7. The principal diagonal indicates the amount of production in each major market consumed within its own region. Other cells indicate cross-border movements. There is only one interregional coal flow, from region 2 to region 1. Otherwise, there is no movement among regions. Coal will move from one region to another if, and only if, differences in regional prices exceed transport costs between regions. The small number of crossborder movements of coal, as shown in Table 7, indicates Table 7. Optimal
flow of coal (1W5 BtuL
Demand region
Supply 1
region 2
1 2 3 5 6
5.89 0 0 0 0
Total aJPPlY
5.89
Total demand
3
5
6
0.65 1.53 0 0 0
0 0 1.23 0 0
0 0 0 0.28 0
0 0 0 0 0.71
6.54 1.53 1.23 0.28 0.71
2.18
1.23
0.28
0.71
10.29
ENERGY ECONOMICS October 1980
DC; and Bituminous
used for estimation
Coal Facts, National
Coal
that our regional markets are well defined with respect to demand and supply in each region. The theoretical assumption of all sellers being located at market centres indicates another reason for few interregional coal flows. Furthermore, the high cost of transport for bulky products such as coal makes the interregional flow even more difficult. The only movement of coal, from region 2 to region 1, is caused by the high demand in the highly industrial and populated region 1. Time series data were used to estimate demand and supply functions since cross section data were not available. The estimated equations should reflect attributes of data over time rather than the position of a single point in time. Mid-year data (for 1967) of actual coal flow are therefore computed and shown in Table 8. There are plenty of interregional coal flows since some mines in the real world are not located in market centres. The mines located at the periphery of the market find it easier to sell their coal to other regions because of lower transport costs compared with mines located in the market centre. However, because of the high transport cost for bulky products such as coal, most coal produced in a region is consumed within that region, and the actual interregional movements indicated in Table 8 are, therefore, small. The optimal results, shown in Table 7, can now be compared with the actual flows shown in Table 8. The principal coal consuming states have been those of the Table 8. Actual flow of coal (10”Btu). Demand region
Supply region 1 2
3
5
6
1 2 3 5 6
4.78 0.43 0.55 0.02 0.003
1.14 0.92 0.53 0.009 0.005
0.02 0.20 1.44 0.03 0.02
0 0 0.02 0.09 0.01
0 0 0 0.08 0.19
5.94 1.55 2.54 0.23 0.23
5.78
2.61
1.71
0.12
0.27
10.49
Total =PPly
Source: Mineral lndusrry Surveys, US Department Interior, Bureau of Mines, Washington, DC.
Total demand
of the
235
Spatial equilibrium
in the US coal industry:
T. C. Campbell, M. J. Hwang and F. Shahrokh
Table 9. Optimal supply and demand prices (cents/million Btul. Region 1
17.20 Optimal supply price Optimal demand price 31.90 Actual supply price 24.75
2
3
5
$
15.46 29.34 24.45
14.33 26.13 22.14
9.80 24.84 21.05
11.49 25.92 21.56
industrial Midwest, Pennsylvania, and West Virginia (the whole state of West Virginia is now defined as our region 1) in the east and the populated northeast. However, major production comes mainly from region 1 and, subsequently, from regions 2 and 3. Low demand in region 2 makes it possible to export coal to high demand in region 1. Our optimal results are, therefore, consistent with the actual coal flow figures of 1967. The optimal supply prices and demand prices given by our model are shown in Table 9. As was indicated above, demand price must be equal to or greater than the supply price plus transport cost. When most of the coal produced is consumed within the producing region, demand prices are equal to supply prices plus average transport costs within their regions (eg in region 1, the demand price of 3 1.90 cents/million Btu equals the supply price of 17.20 cents/million Btu plus the average transport cost of 14.70 cents/million Btu). When coal moves from region 2 to region 1, demand price, 3 1.90 cents/ million Btu, is equal to supply price, 15.46 cents/million Btu, plus transport cost, 16.44 cents/million Btu. As expected, the equilibrium supply prices are relatively low in comparison with actuaI supply prices (see the third row of Table 9). Conclusions The principal focus of this paper has been a redefinition of coal markets and the sources of supply for each market. Coal is a heavy commodity with low value per unit of weight. To transport it long distances frequently costs as much or more than the price at the mines. With expansion in output of mines in western USA and the long distances over which this and other coal is moving, minimizing the distances and cross-boundary movements will assure both global optimal flows among coal markets and optimal transport costs. The paper represents part of a research effort to redefme the markets, to estimate demand and supply functions, and to generate optimal spatial equilibrium results. Careful theoretical analysis is presented as a foundation for further investigation of a more empirical
236
nature. From this sounder theoretical base, our purpose is to assist in the determination of the location of power generating plants and the supply regions that can supply these plants at minimum costs. Because of the large tonnages and the relatively long periods associated with plant location, design, and capacity changes in the industry, investigation that will assist in the selection of plant locations and the scale of each plant will minimize unit energy costs in specific regions and for the USA as a whole. The analysis in redefining the supply regions and the markets in this paper is part of a broader investigation.
References Gustav Cassel, Fundamental Thoughts in Economics, T. F. Unwin, London, 1925. Knut Wicksell, Lectures on Political Economy, Routledge, London, 1935. Leon Walras, Ekments of Pure Economics. George Allen and Unwin, Londdn, 1954. Vilfredo Pareto,. ‘Mathematical economics’, International Economic Papers, No 5, Macmillan, New York, 1955. 5 F. Fetter, ‘The economic law of market areas’, Quarterly Journal of Economics, 1924, pp 520-529. 6 C. D. Hyson and W. P. Hyson, ‘The economic law of market areas’, The Quarterly Journal of Economics, 1950, pp 319-327. August LSsch, The Economics of Location, Yale University Press, New Haven, CT, 1954. Martin Beckman, ‘A continuous model of transportation’, Econometica, 1952, pp 643-660. W. J. Baumol, Spatial Equilibtium with Supply Points Separated from Markets with Supplies Predetermined, Bureau of Agricultural Economics,
USDA, Washington, DC, 1952. E. M. Hoover, An Introduction to Regional Economics, Alfred A. Knopf, New York, 1971. 11 M. L. Greenhut, A Theory of the Firm in Economic Space, Lone Star Publishing, Austin, TX, 1974. 12 S. Enke, ‘Equilibrium among spatially separated markets: solution by electric analogues’, Econometrica, 195 1, pp 40-48. 13 P. A. Samuelson. ‘Suatial urice eauilibrium and linear programmhg’, Ame&an Eionomic Review, 10
1952, pp 283-303.
14 T. Takayama and G. G. Judge, ‘Spatial equilibrium and quadratic programming’, Journal of Farm Economics, 1964, pp 67-93. 15 Federal Energy Administration? Project Independence Report, Federal Energy Admimstration, Washington, DC, 1974. 16 T. C. Campbell and M. J. Hwang, ‘Spatial analysis and the identification of coal markets’, The Journal of Energy and Development, Autumn 1978, pp 104125. Fetter, op tit, Ref 5. ;: Hyson and Hyson, op tit, Ref 6. 19 Greenhut, op tit, Ref 11.
ENERGY ECONOMICS October
1980