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Rising Economic Rent Does Not Signal a Coal Resource Shortage
Copyrighl © 1f AC [n e rgv Svs lem s. and Economics. Toho . .J a pan 1989
~lanagemenl
OIL AND GAS; PROSPECTS AND MANAGEMENT
RISING ECONOMIC RENT DOES NOT SIGNAL A COAL RESOURCE SHORTAGE W. D. Watson Alillis!r), of Fillallct alld Natiollal Economy, Kingdom of Saudi Arabia, Riya dh 11452, Saudi Arabia
Abstract. This paper analyzes the relationship between rent and resource depletion occurring as a result of increasing costs (i .e., economic exhaustion). Economic exhaustion is used as the framework because it is more representative of natural resources markets than a Hotelling-type physical exhaustion model . Rent is measured as the discounted present value of future marginal cost increases resulting from today's extraction. A simple model with exhaustion rents and smoothly increasing costs shows that there is no systematic relationship between rents and exhaustion . Empirical analysis, in a simulation mode, is then undertaken to examine rents, prices, and exhaustion in the U.S. coal market over the period 1985 to 2075 . These empirical results also show that rising rent is not a leading indicator of exhaustion . Thus, detailed investigations of both geologic and economic conditions are required for successful forecasts of an impending resource shortage. Keywords. Economics; time-varying systems; optimization; programmed control; mathematical programming. INTRODUCTION
basis of these results, Dale suggested that it may be possible to use rents as leading indicators of exhaustible resource shortage .
The economic theory of exhaustible resources distinguishes between two models that have somewhat different frameworks and different theoretical conclusions. The first model, originally presented by Hotelling (1931), is one of physical exhaustion. There is a large base of an exhaustible resource available at a constant extraction cost (an oft-cited example is Middle East oil). The resource is completely physically exhausted along an extraction path that maximizes consumer and producer net present value . User cost or in situ rent to the resource owner rises at the rate of interest. Using this model as a basis, some researchers (Smith (1978) and Brown and Field (1979» have examined market data for systematic evidence of increasing rents . In general, this research has not yielded any empirical findings that rents (or prices) for exhaustible resources are rising at rates equal to or close to real interest rates.
This paper analyzes the relationship between rent and resource depletion occurring as a result of increasing costs (i . e . , economic exhaustion) . An economic exhaustion model is used because it is more representative of natural resource markets than a Hotelling-type physical exhaustion model . A simple mathematical model with exhaustion rents is presented first, to show the basic relationships between rent and depletion. Empirical analysis, in a simulation mode, is then undertaken to examine rents and depletion in the U.S. coal market over the period 1985 to 2075 . In this time period, according to the simulation, many U.S. regional coal supply basins would have all of their currently reported coal resources extracted . The rent paths for these regions are examined to see if rising rents or rising rent proportions (rent over price) precede and, thereby, forecast the shortages that occur. The results presented in this paper extend those provided by Slade and Dale.
A second model assumes that an exhaustible resource is heterogeneous, with a wide range of different qualities and increasing extraction costs. In this framework. rent is the present discounted value of future marginal cost increases that result from today ' s extraction (Levhari and Liviatan (1977». Exhaustion is economic exhaustion (not physical exhaustion), occurring as the market supplies the resource in order of increasing cost. Using this model as a basis, Slade (1982) presents empiri cal results that show a rising path for many natural resource prices . In particular, Slade finds that coal prices (f .o .b . per short ton) reached a minimum in 1910 and have been on an accelerating path since then . Also using an economic exhaustion model, Dale (1984) has examined the U.S. anthracite market and found that rents (as a percentage of price) for anthracite have an inverted V-shape (plotted against time) and that they peaked before anthracite production (also moving along an inverted V-shaped curve). On the
A SIMPLE MODEL Economic theory of increasing cost exhaustible resources states that the price of an exhaustible resource such as coal is equal to conventional marginal costs (for example, mining, transportation, and sulfur dioxide (SOz) scrubbing costs) plus static rents associated with transportation and S02 constraints plus opportunity costs of economic coal depletion. The latter term is the present value of future increases in extraction, transportation, and scrubbing costs that result as a consequence of choosing to extract certain tonnages at certain locations today. If increasing costs due to today ' s extraction can be approximated as
393
W. D. Watson
394
(1)
(where Co is current cost, t is time, e = 2.718+, and a is the rate of cost increase) then the current user cost or rent (R o )' which is the present value of the stream of increases in future marginal costs, is represented by T
Ro = a
f Coeate
-rtd t
(2)
o
where r is the real discount rate (r>a) and T is the terminal year of the discount period . Assuming T is very large then Rc
= (a/(r-a»C c
(3)
and (4)
From (4) it is seen that the growth rate for rent (= SRt/st + Rt = a) equals the growth rate of costs. Price (Pt) is the sum of rent and cost and , hence, Pt = Coe at
+
(a/(r-a»Coe
at
(5)
Using equations (4) and (5), the ratio of rent over price is air. The expression, air, is a constant that does not vary over time . From equation (4), it has already been noted that rent grows at a constant rate . These two results have an important implication . If the transition to a higher cost resource (including an altogether different alternative) is smooth (i .e., the cost growth rate, a, is constant over time) then both the growth rate of rent and the ratio of rent over price will, each, remain the same value (a and air, respectively) in every time period including both times of resource abundance and times of resource shortage. There will be nothing in market signals (prices, costs and rents) which will foretell impending coal shortage . A DYNAMIC OPTIMIZATION MODEL The results of the simple model are compared with results obtained using a large-scale detailed dynamic optimization model of the U.S. coal market. The optimization model is run over the time period 1985 to 2075. During this period, currently available coal resources in many regional coal basins are projected to be completely physically exhausted . If the growth rate of rent or the ratio of rent-over-price is a leading indicator of shortage , then there should be systematic evidence of the rent path or the rent-over-price path leading the coal production path . The simple model developed above, of course, provides a strong presumption that rents will not be systematically related to exhaustion . The optimization model allocates coal in the U.S . over the next 90 years (1985-2075), with the objective of minimizing total mining, transportation, and SOz scrubbing costs (in present value terms) while meeting projected coal demands and operating within allowable SOz emission standards, available supplies of mineable coal, and shipping capacities for locks and transmission lines . The optimization model has these main features ; - 60 coal supply regions each represented by a step function of up to 12 steps relating mineable tonnage to mining cost by sulfur and btu
content. - 243 demand regions (corresponding to the U.S. Environmental Protection Agency's (EPA) air quality control regions), each having one or more of eight possible types of coal demand, including demands by electric utility and industrial boilers, export demands and metallurgical demands. - 6 time periods of 15 years each, spanning the time period from 1985 to 2075 . (In the results reported below, each time period ~ represented by its central year - 1993, 2008, 2023, 2038, 2053 and 2068 . ) - Coal shipped by regular rail or unit train, both of which are unconstrained, or by barge , which is constrained by lock capacity . Coal (in effect) also shipped via transmission line, subject to engineering limits, from minemouth electric generating plants. - Coal economic depletion rents calculated as the present discounted value of the marginal increases in future coal mining, transportation, and scrubbing costs that result from current coal utilization (same as the rent calculation used in the simple model) . The dynamic optimization model has several advantages over the simple model . It does not impose the assumption of a smooth constant increase in extraction, transportation, and scrubbing costs . Most importantly, it incorporates a number of features designed to analyze coal allocation, coal rents, prices, and costs in a realistic framework . The solution to the model is a set of coal shipments from the 60 supply regions to the 243 demand regions for 6 time periods . Coal (in total) is not physically exhausted in the optimal solution, rather coal is used in order of increasing full marginal cost . Depletion rents provide economic linkage across time, in correspondence with dynamic market clearing processes . To provide insight on key relationships, here we provide a streamlined version of the model . The full programmed version is an exact duplicate except that cost functions are represented by step functions . Optimality conditions are derived using the Kuhn-Tucker mathematical theorems for optimization (Chiang, 1984) . To apply Kuhn-Tucker conditions, continuous differentiable functions are assumed for extraction, transportation, and scrubbing costs. Mining, transportation and SOX scrubbing costs are functions of the current level of coal production (x~) and the cumulative amount of past coal production (Zt). All subscripts except t are suppressed to keep derivations simple . The dynamic optimization model is represented by the following objective function (equation 6) and constraint inequalities (expressions 7 through 10) ; T t-l ~ [el ( XI,Zt ) 1-1
( l+r)
+ nl(xI , Zt) subject to ; ",X I ~ ~XI " xI ~ xI "
• bl (xt,zl) DI O.OO72D I
L
El
1
(6)
(PI ) (BI ) (C l) (Ht)
(7 ) (8 ) (9)
(10 )
Xl ;a 0
where K
r T
total cost of extracting, transporting, and scrubbing coal in present value terms over the planning horizon real rate of dis count fina l year of the planning horizon
Rising Economic Rent e, x,
z,
cost of extraction in time t (dollars/ton) tons of coal extracted in a time period t cumulative extraction up to t
,-,
(=
n, b,
~
q-'
xq )
cost of transportation in time t (dollars/ton) cost of scrubbing coal at a market in time t (dollars/ton)
a = factor for converting coal in actual tons to coal in normal tons, in terms of Btu ({Btu / ton.)/24,OOO,OOO) D_ coal demand at each destination in normal tons (tons containing 24,000,000 Btu/ton) B fraction of sui fur content in the coal by weight at the coal origin adjusted for sulfur removal by scrubbing 0 . 0072 EPA regulation (NSPS) for sulfur discharges at a consumption market (tons of sui fur per normal ton of coal) L capacity in tons at a waterway lock E~ electric transmission line capacity in tons of coal equivalent
The constraints (inequalities 7-10) and their dual variables or shadow values (symbols in parentheses on the right hand side of the inequalities) have the following interpretations: - Inequality 7. Coal shipments (transformed into Btu units by a) must be large enough to satisfy demand (in Btu units = D~) . P~ is the delivered price for a marginal ton of coal (dollars / ton at 24 million Btu per ton, in present value terms). If demand were relaxed (increased) by one ton (or 24 million Btu), P~ is the amount by which the objective function would decrease (increase ). In other words, p~ is the market clearing price for a delivered ton of coal containing 24 million Btu per ton . - Inequality 8 . Tons of sui fur discharged in the demand region cannot exceed SOX limits in the demand region . The factor, B, converts coal to tons of sulfur; it equals the sulfur fraction (by weight) of coal adjusted by sulfur scrubbing removal percentage . The factor, 0 .0072, converts demand (in Btu terms) into tons of sulfur allowed under a Federal New Source Performance Standard of 1.2 lbs of SOX per million Btu . B~ is the shadow value on sulfur discharges to the atmosphere (dollars / ton of sulfur , in present value terms). If the constraint on suI fur discharges were relaxed by one ton of sulfur, ~ is the amount of change in the objective function. B~ is like a market clearing price on sulfur discharges . If the atmosphere were owned and managed for waste disposal, B~ is the amount that could be charged for each ton of sulfur dumped. Since the atmosphere is not managed as a waste assimilating asset, B~ is a rent that can accrue to any of the participants in the coal market. Nonetheless, B~ is a marginal opportunity cost of limiting sulfur discharges to levels required by law . Other U.S. sulfur dioxide standards are implemented : (1) by shipping coal from certain coal supply regions, and only low-sui fur coal from those suppl y regions that can satisfy State Implementation Plan standards , and (2 ) by always scrubbing coal to satisfy sulfur dioxide standards for boilers under Revised New Source Performance Standards.
395
- Inequality 9. Tons of coal shipped cannot exceed river way lock capacities (L) . C~ is the shadow value on limited lock capacity (dollars/ton of coal, in present value terms). If the lock capacity were to be increased by one ton of coal, then C~ is the amount by which the objective function would decrease . C~ is the amount a lock owner can charge on each ton of coal passing through a capacity constrained lock. Since it reflects the cost of the next best alternative, it, like ~, is a marginal opportunity cost. - Inequality 10. Tons of coal shipped cannot exceed electric transmission capacities (E~). H_ is the shadow value on limited transmission capacity (dollars / ton of coal, in present value terms) . It is the marginal opportunity cost of transmitting coal through a capacity constrained transmission line. To develop necessary conditions for an optimal solution, the constraint inequalities are multiplied by their associated dual variables and added to or subtracted from the objective function (as appropriate) to obtain the following Lagrangian expression : T
K +
::E [1-,
P,{"x l
DI
-
)
BI {I3X I - O.0072D,) C,(x, - L) +H,(x,-E,)]
+ +
Necessary conditions for are obtained by applying to (11). The first step derivatives with respect
(11 )
a minimum cost solution the Kuhn-Tucker theorem is to take partial to x, :
2
0
( 12)
For the solution to be optimal, it is also necessary that Kuhn-Tucker complementary slackness conditions be satisfied, as follows : XI (
If
XI )
aZ/ ax , ) = 0
0 then , by complementary slackness,
aZ / ax I = 0
or from (12) (13a)
"PI (1 +r ) I - I
(1+r) 1-1[I3B, + C, + Ht) +
± (_1_ q_ITI
hr
)q-I . [ae q
a Zq
aZ q
aX t
(13b)
+
(13c)
The right-hand side of expression (13a) includes conventional marginal costs (the sum of marginal extraction, transportati on, and scrubbing cost ). Expression (1 3b) is the sum of marginal opportunity costs assoc iated with SOX discharge
396
W. D. Watson
li.its, river lock capacities, and transmission line capacities . Expression (13c) is the future .arainal opportunity cost from current extraction of nonrenewable coal . It is the dyna.ic Ricardian rent that gets added to the coal price in order to reflect future scarcity cost . The term (l+r)~-l in (13a) and (13b) moves the present values of price and shadow values forward from their discounted values in the initial period (time = 1) to non-discounted values in the current period (time = t) . The term a, dividing through on left- and right-hand sides, converts dollars per ton (by weight) for the right-hand side values into dollars per ton of coal containing 24 million Btu per ton. The term, P_(l+r)'-', is market price in undiscounted dollars per ton of coal containing 24 million Btu per ton . Thus, equation (13) says that a shipment can be in the optimal solution if its full marginal costs (normalized for Btu) are equal to (or are just covered by) the delivered price consu.ers are willing to pay at the demand center . That price equals the shadow value on the demand constraint. When delivered price does not cover full marginal cost (for all parts of the market transaction), the shipment will not be made since suppliers cannot cover all their costs . This logical outcome also is consistent with the Kuhn-Tucker complementary slackness condition . If XI = 0 then, by complementary slackness, aZ/ax I ~ 0 or from expression (12), full marginal cost 2 price . The Kuhn-Tucker conditions, also, provide insight on cost competition among different supply regions potentially shipping coal to a given demand center. By the Kuhn-Tucker conditions, competing coal supply regions would supply a given demand region if full marginal delivered costs from each supply region are covered by price . Finally , complementary slackness from the Kuhn-Tucker theorem also requires that the optimal solution satisfy : PI (az/aP I ) 0 BI (
RESULTS Coal allocati on patterns depend upon a complex interplay of factors that include (1) relative abundance of high quality coal, (2) growth in coal demands, (3) the changing spatial pattern of coal supply and demand, (4) transportation options, and (5) environmental restrictions . All of these factors are reflected in the solution provided by the dynamic OPtimization model . National coal demands are projected t o grow f rom 'The data used in the model, including coal quantity and coal quality data, are described in Watson and others (1989) .
800 million tons in 1985 to about 3.5 billion tons by 2075. Demand for boiler coal is projected to reach a saturation level by about 2040 . Thereafter, demand for coal by synfuels plants will keep total coal demand growing along an upward sloping linear path . The total U.S. supply of deliverable coal is estimated to be 338 billion tons. By the year 2075, 212 billion tons or 63~ of currently known U.S. deliverable coal resources are extracted in the solution. Many traditional coal supply basins would have their coal completely depleted in this time frame r Complete depletion is projected to occur , for example, in eastern and western Kentucky, in most of West Virginia , and in much of the Illinois basin. Remaining resources would be concentrated in high cost midwest districts and in the western U. S. Table 1 provides results for the total U. S. and for large geographic sections of the U. S. (The midwestern U.S., which is similar to the eastern U. S. , is not shown.) The absolute levels of delivered costs (sum of mining, transportation, and scrubbing costs ; and rents) differs by section of the U.S. The eastern U.S . , which has a high demand for low suI fur coal, has delivered costs and rents above the national average (see section B, Table 1), whereas the western U.S. with a relative abundance of low sulfur coal supplies - has costs and rents (in early years) below the national average (see section C, Table 1) .
The aggregate rent - over-price ratios (R / P) in Table 1 from the optimization model (shown in column (6)) are almost identical to the rentover-price ratios calculated using the simple model (shown in column (7)) . The growth rate in delivered costs (column (4)) is estimated using the delivered cost estimates in column (3) . If the growth is not at a constant exponential rate or not smooth (indicated by "N Smooth" in column (2)), then the cost growth rate is calculated from year to year according to the entries in column (1). When costs grow smoothly, the cost
TABLE 1 Aggregate Coml2arisons (r
= 87.)
(1 )
(2)
(3)
(4)
Year
Cost Path
Cost'
a
2. 10 2 . 74 3 . 11 3.42 3.76 4.23
1.80 0 . 85 0. 64 0.71 0. 71 0 . 71
0 . 30 0 . 29 0 . 29 0.28 0 . 34 0. 36
0 . 14 0 .11 0 . 09 0 . 08 0 . 09 0. 09
0 . 23 0.11 0 . 08 0 . 09 0 .09 0 . 09
B. Eastern U.S . 1993 N Smooth 2.30 2008 N Smooth 3 . 00 2023 Smooth 3 . 29 2038 Smooth 3 . 56 2053 Smooth 3 . 92 2068 Smooth 4 . 50
1. 80 0 .62 0 . 70 0 . 70 0 . 70 0 . 70
0. 42 0. 36 0.30 0 . 32 0.35 0 .38
0.18 0 . 12 0 .09 0 . 09 0. 09 0 . 08
0.23 0 . 08 0.09 0 . 09 0 . 09 0 . 09
C. Western U. S. N Smooth 1. 97 1993 2008 N Smooth 2.55 2023 Smooth 2.97 2038 Smooth 3 . 33 Smooth 3.72 2053 2068 Smooth 4 . 15
1. 74 1. 00 0 . 75 0 . 75 0 .75 0 . 75
0 . 20 0.25 0. 28 0 . 30 0 .39 0. 36
0 . 10 0 . 10 0 . 09 0.09 0.10 0 . 09
0. 22 0.12 0 . 09 0 . 09 0. 09 0 . 09
A. Total U.S. 1993 N Smooth 2008 N Smooth 2023 N Smooth 2038 Smooth 2053 Smooth 2068 Smooth
'1985$/ 10"6 Btu
(5)
(6) Prog . Model Rent' RLP
(7) Simp . Model aLr
Rising Economic Rent
growth rate is calculated over the entire set of years displaying smooth growth . In the "N Smooth" periods, the simple model will tend to overestimate the actual rent-over-price ratio because the simulated growth is along an upwardly sloped linear path where cost growth rates are declining over time rather than staying constant as required by the simple model . As shown in Table 1, in the smooth growth periods there is very close agreement between the observed rentover-price ratios from the optimization model and the rent-over-price ratios from the simple model. Overall, the rent paths and the rent-over-price paths provided in Table 1 do not indicate impending coal shortage. The high rent-overprice ratios in the beginning periods are due primarily to relatively rapid increases in costs in early years incurred to meet EPA's S02 regulations. The rate of increase in costs and in rents remains constant in the smooth growth time periods even though by the end of the whole time period extending to the year 2075, U.S. coal, in total, is projected to be substantially depleted. And even though coal resources are completely depleted by the year 2075 in many eastern coal basins, the eastern U.S. paths for costs, rents, and rent-over-price ratios are similar to those for western U.S. coal resources where a large amount of resources is projected to be available for extraction in 2075. Table 2 provides rent-over-price ratios for selected coal supply regions that experience coal depletion by the year 2075. In the entire set of 60 regional coal supply basins, coal is depleted by 2075 in 28 basins. Among the 28 depleted basins, Region 16 (shown in Table 2) is the only basin where rent-over-price ratios are a leading indicator of coal depletion. Table 2 also shows that there is wide variation in rent-over-price ratios for any particular coal supply region within a single time period (see column (4». In a single period, each coal supply region ships coal to many different demand regions or markets. Each market has its own set of costs determined by transport distance, S02 constraints and rents. On the supply side, some coal supply regions ship more than one type of coal . Coal shipments to the various markets can be from various coal steps or blocks, each with different mining costs and coal quality . The diversity and complexity of the market process, as it gropes for a minimum cost response, necessarily means that a wide range of rents and costs will occur at any given time in a coal supply region, as is made evident by the data in Table 2. The seeming complexity, which is nothing more than a reflection of the market's success in balancing marginal opportunity costs, means that systematic trends in costs and rents are unlikely to exit in disaggregated data . CONCLUSION In a smoothly transitioning natural resource market, it appears that there will be little evidence in market signals of impending resource shortage . The rates of cost and rent increase can be high or they can be low and still nothing will be revealed about shortages as long as the rates of increase in costs are more or less constant. With a diverse natural resource base, such as we have with U.S. coal, one can presume that cost increases will be smooth (indeed, the simulation from the dynamic optimization model indicates such) . In this situation, the market is a perpetrator of a smooth transition . It will search out the lowest cost options at each point
397
TABLE 2
Regional Coal Rent and Production Patterns
(1)
(2)
Yr
Tons (MMI Yr)
(3)
(4)
RI P RI P Low to (%) Hi (%)
Pikevi lie KY (9)' 1993 2008 2023 2038 2053 2068
137 134 48 0 25 25
24 16 17 3 9 12
29 38 22 46 0 42
31 20 21 7 4 12
8-9 12-12
2-35 4-63 11-85 6- 7 11-12
Morganfield KY (3) 1993 2008 2023 2038 2053 2068
16 50 80 109 63
9 14 9 7 7 20
1-10 4 - 26 8-21 7-7 7-7 20-20
Moffat & Routt CO (93) 1993 2008 2023 2038 2053 2068
31 62 55 93 35
6 8 7 8 14 30
(2)
Yr
Tons (MMI Yr)
(3)
RIP (%)
(4)
RIP Low to Hi (%)
Pittsburgh PA (16) 4-48 6-63 7-68
Kittanning PA (17) 1993 2008 2023 2038 2053 2068
(1)
1-11 2-13 2-16 7-22 9-22 30-30
1993 2008 2023 2038 2053 2068
70 55 104 139 264 122
8 8 7 13 10
2-15 1-36 4-21 11-15 4-11 0-11
Charleston WV (21) 1993 2008 2023 2038 2053 2068
26 50 129 120 134 68
27 16 7 8 11 13
3-39 8-54 5-75 4-15 9-13 12-26
Price Utah
PO) 9 8 12 8 18 11
2-16 4-26 10- 52 8-10 5-24 4-15
1993 2008 2023 2038 2053 2068
8 8 14 15 3 7
Powder River WY (96) 1993 2008 2023 2038 2053 2068
199 221 217 266 322 369
7 11 10 11 11 11
1-16 2-23 3-25 6-21 4 - 21 0-18
'Coal supply region number used by the dynamic optimization model . in time and balance costs at the margin in order to achieve a smooth transition along the path of lowest cost . In other cases, if geology is unusually restrictive, or markets noncompetitive, or substitute resources high cost and in short supply, then the rate of cost increase can deviate from a smooth path and perhaps signal a resource shortage. It could be argued that if the transition is smooth, then there is no resource shortage . Still, in the simulations presented here, there are large and jarring changes in the U.S. coal industry. Regions of the country where coal mining has been a major industry for many decades will cease to produce coal . These large and dramatic changes are not revealed in the market signals provided by the programming model . At both national and regional levels, shortages are revealed in comparisons between estimates of extracted coal and the coal resource base . But the ability to make these comparisons and uncover the potential for shortage, requires detailed investigations of both geologic and economic conditions . It would appear that market signals alone are not likely to take us very far toward being able to make reliable forecasts of impending resource shortage . REFERENCES Brown, G. , and B. Field (1979) . The adequacy of measures for signaling the scarcity of natural resources . In V. K. Smith (Ed.) . Scarcity and Growth Reconsidered. Johns
W. D. Watson
398
Hopkins University Press, Baltimore. pp. 218- 245 . Chiang, A. C. (1984) . Fundamental Methods of Mathematical Economics . McGraw-Hill, New York. pp . 722-753. Dale, L. (1984) . The pace of mineral depletion in the United States. Land Econ .. ~ 255-267. Hotelling, H. (1931). The economics of exhaustible resources. J . Polit . Econ . ~ 137-175. Levhari, D. , and N. Liviatan (1977). Notes on Hotelling's economics of exhaustible resources . Can. J. Econ . , ~ 177-192. Slade, M. (1982) . Trends in natural resource commodity prices: an analysis of the time domain . J . Envir . Econ . and Manag . , ~ 122137 . Smith, V. K. (1978) . Measuring natural resource scarcity: theory and practice. J . Envir . Econ. and Manag .. ~ 150-171 . Watson, W. D. , A. L. Medlin, K. K. Krohn, D. S. Brookshire, and R. L. Bernknopf (1989) . Economic Effects of Western Federal Land Use Restrictions on U. S. Coal Markets. U. S. Geological Survey, Reston , Va. I
~lanagemenl
OIL AND GAS; PROSPECTS AND MANAGEMENT
RISING ECONOMIC RENT DOES NOT SIGNAL A COAL RESOURCE SHORTAGE W. D. Watson Alillis!r), of Fillallct alld Natiollal Economy, Kingdom of Saudi Arabia, Riya dh 11452, Saudi Arabia
Abstract. This paper analyzes the relationship between rent and resource depletion occurring as a result of increasing costs (i .e., economic exhaustion). Economic exhaustion is used as the framework because it is more representative of natural resources markets than a Hotelling-type physical exhaustion model . Rent is measured as the discounted present value of future marginal cost increases resulting from today's extraction. A simple model with exhaustion rents and smoothly increasing costs shows that there is no systematic relationship between rents and exhaustion . Empirical analysis, in a simulation mode, is then undertaken to examine rents, prices, and exhaustion in the U.S. coal market over the period 1985 to 2075 . These empirical results also show that rising rent is not a leading indicator of exhaustion . Thus, detailed investigations of both geologic and economic conditions are required for successful forecasts of an impending resource shortage. Keywords. Economics; time-varying systems; optimization; programmed control; mathematical programming. INTRODUCTION
basis of these results, Dale suggested that it may be possible to use rents as leading indicators of exhaustible resource shortage .
The economic theory of exhaustible resources distinguishes between two models that have somewhat different frameworks and different theoretical conclusions. The first model, originally presented by Hotelling (1931), is one of physical exhaustion. There is a large base of an exhaustible resource available at a constant extraction cost (an oft-cited example is Middle East oil). The resource is completely physically exhausted along an extraction path that maximizes consumer and producer net present value . User cost or in situ rent to the resource owner rises at the rate of interest. Using this model as a basis, some researchers (Smith (1978) and Brown and Field (1979» have examined market data for systematic evidence of increasing rents . In general, this research has not yielded any empirical findings that rents (or prices) for exhaustible resources are rising at rates equal to or close to real interest rates.
This paper analyzes the relationship between rent and resource depletion occurring as a result of increasing costs (i . e . , economic exhaustion) . An economic exhaustion model is used because it is more representative of natural resource markets than a Hotelling-type physical exhaustion model . A simple mathematical model with exhaustion rents is presented first, to show the basic relationships between rent and depletion. Empirical analysis, in a simulation mode, is then undertaken to examine rents and depletion in the U.S. coal market over the period 1985 to 2075 . In this time period, according to the simulation, many U.S. regional coal supply basins would have all of their currently reported coal resources extracted . The rent paths for these regions are examined to see if rising rents or rising rent proportions (rent over price) precede and, thereby, forecast the shortages that occur. The results presented in this paper extend those provided by Slade and Dale.
A second model assumes that an exhaustible resource is heterogeneous, with a wide range of different qualities and increasing extraction costs. In this framework. rent is the present discounted value of future marginal cost increases that result from today ' s extraction (Levhari and Liviatan (1977». Exhaustion is economic exhaustion (not physical exhaustion), occurring as the market supplies the resource in order of increasing cost. Using this model as a basis, Slade (1982) presents empiri cal results that show a rising path for many natural resource prices . In particular, Slade finds that coal prices (f .o .b . per short ton) reached a minimum in 1910 and have been on an accelerating path since then . Also using an economic exhaustion model, Dale (1984) has examined the U.S. anthracite market and found that rents (as a percentage of price) for anthracite have an inverted V-shape (plotted against time) and that they peaked before anthracite production (also moving along an inverted V-shaped curve). On the
A SIMPLE MODEL Economic theory of increasing cost exhaustible resources states that the price of an exhaustible resource such as coal is equal to conventional marginal costs (for example, mining, transportation, and sulfur dioxide (SOz) scrubbing costs) plus static rents associated with transportation and S02 constraints plus opportunity costs of economic coal depletion. The latter term is the present value of future increases in extraction, transportation, and scrubbing costs that result as a consequence of choosing to extract certain tonnages at certain locations today. If increasing costs due to today ' s extraction can be approximated as
393
W. D. Watson
394
(1)
(where Co is current cost, t is time, e = 2.718+, and a is the rate of cost increase) then the current user cost or rent (R o )' which is the present value of the stream of increases in future marginal costs, is represented by T
Ro = a
f Coeate
-rtd t
(2)
o
where r is the real discount rate (r>a) and T is the terminal year of the discount period . Assuming T is very large then Rc
= (a/(r-a»C c
(3)
and (4)
From (4) it is seen that the growth rate for rent (= SRt/st + Rt = a) equals the growth rate of costs. Price (Pt) is the sum of rent and cost and , hence, Pt = Coe at
+
(a/(r-a»Coe
at
(5)
Using equations (4) and (5), the ratio of rent over price is air. The expression, air, is a constant that does not vary over time . From equation (4), it has already been noted that rent grows at a constant rate . These two results have an important implication . If the transition to a higher cost resource (including an altogether different alternative) is smooth (i .e., the cost growth rate, a, is constant over time) then both the growth rate of rent and the ratio of rent over price will, each, remain the same value (a and air, respectively) in every time period including both times of resource abundance and times of resource shortage. There will be nothing in market signals (prices, costs and rents) which will foretell impending coal shortage . A DYNAMIC OPTIMIZATION MODEL The results of the simple model are compared with results obtained using a large-scale detailed dynamic optimization model of the U.S. coal market. The optimization model is run over the time period 1985 to 2075. During this period, currently available coal resources in many regional coal basins are projected to be completely physically exhausted . If the growth rate of rent or the ratio of rent-over-price is a leading indicator of shortage , then there should be systematic evidence of the rent path or the rent-over-price path leading the coal production path . The simple model developed above, of course, provides a strong presumption that rents will not be systematically related to exhaustion . The optimization model allocates coal in the U.S . over the next 90 years (1985-2075), with the objective of minimizing total mining, transportation, and SOz scrubbing costs (in present value terms) while meeting projected coal demands and operating within allowable SOz emission standards, available supplies of mineable coal, and shipping capacities for locks and transmission lines . The optimization model has these main features ; - 60 coal supply regions each represented by a step function of up to 12 steps relating mineable tonnage to mining cost by sulfur and btu
content. - 243 demand regions (corresponding to the U.S. Environmental Protection Agency's (EPA) air quality control regions), each having one or more of eight possible types of coal demand, including demands by electric utility and industrial boilers, export demands and metallurgical demands. - 6 time periods of 15 years each, spanning the time period from 1985 to 2075 . (In the results reported below, each time period ~ represented by its central year - 1993, 2008, 2023, 2038, 2053 and 2068 . ) - Coal shipped by regular rail or unit train, both of which are unconstrained, or by barge , which is constrained by lock capacity . Coal (in effect) also shipped via transmission line, subject to engineering limits, from minemouth electric generating plants. - Coal economic depletion rents calculated as the present discounted value of the marginal increases in future coal mining, transportation, and scrubbing costs that result from current coal utilization (same as the rent calculation used in the simple model) . The dynamic optimization model has several advantages over the simple model . It does not impose the assumption of a smooth constant increase in extraction, transportation, and scrubbing costs . Most importantly, it incorporates a number of features designed to analyze coal allocation, coal rents, prices, and costs in a realistic framework . The solution to the model is a set of coal shipments from the 60 supply regions to the 243 demand regions for 6 time periods . Coal (in total) is not physically exhausted in the optimal solution, rather coal is used in order of increasing full marginal cost . Depletion rents provide economic linkage across time, in correspondence with dynamic market clearing processes . To provide insight on key relationships, here we provide a streamlined version of the model . The full programmed version is an exact duplicate except that cost functions are represented by step functions . Optimality conditions are derived using the Kuhn-Tucker mathematical theorems for optimization (Chiang, 1984) . To apply Kuhn-Tucker conditions, continuous differentiable functions are assumed for extraction, transportation, and scrubbing costs. Mining, transportation and SOX scrubbing costs are functions of the current level of coal production (x~) and the cumulative amount of past coal production (Zt). All subscripts except t are suppressed to keep derivations simple . The dynamic optimization model is represented by the following objective function (equation 6) and constraint inequalities (expressions 7 through 10) ; T t-l ~ [el ( XI,Zt ) 1-1
( l+r)
+ nl(xI , Zt) subject to ; ",X I ~ ~XI " xI ~ xI "
• bl (xt,zl) DI O.OO72D I
L
El
1
(6)
(PI ) (BI ) (C l) (Ht)
(7 ) (8 ) (9)
(10 )
Xl ;a 0
where K
r T
total cost of extracting, transporting, and scrubbing coal in present value terms over the planning horizon real rate of dis count fina l year of the planning horizon
Rising Economic Rent e, x,
z,
cost of extraction in time t (dollars/ton) tons of coal extracted in a time period t cumulative extraction up to t
,-,
(=
n, b,
~
q-'
xq )
cost of transportation in time t (dollars/ton) cost of scrubbing coal at a market in time t (dollars/ton)
a = factor for converting coal in actual tons to coal in normal tons, in terms of Btu ({Btu / ton.)/24,OOO,OOO) D_ coal demand at each destination in normal tons (tons containing 24,000,000 Btu/ton) B fraction of sui fur content in the coal by weight at the coal origin adjusted for sulfur removal by scrubbing 0 . 0072 EPA regulation (NSPS) for sulfur discharges at a consumption market (tons of sui fur per normal ton of coal) L capacity in tons at a waterway lock E~ electric transmission line capacity in tons of coal equivalent
The constraints (inequalities 7-10) and their dual variables or shadow values (symbols in parentheses on the right hand side of the inequalities) have the following interpretations: - Inequality 7. Coal shipments (transformed into Btu units by a) must be large enough to satisfy demand (in Btu units = D~) . P~ is the delivered price for a marginal ton of coal (dollars / ton at 24 million Btu per ton, in present value terms). If demand were relaxed (increased) by one ton (or 24 million Btu), P~ is the amount by which the objective function would decrease (increase ). In other words, p~ is the market clearing price for a delivered ton of coal containing 24 million Btu per ton . - Inequality 8 . Tons of sui fur discharged in the demand region cannot exceed SOX limits in the demand region . The factor, B, converts coal to tons of sulfur; it equals the sulfur fraction (by weight) of coal adjusted by sulfur scrubbing removal percentage . The factor, 0 .0072, converts demand (in Btu terms) into tons of sulfur allowed under a Federal New Source Performance Standard of 1.2 lbs of SOX per million Btu . B~ is the shadow value on sulfur discharges to the atmosphere (dollars / ton of sulfur , in present value terms). If the constraint on suI fur discharges were relaxed by one ton of sulfur, ~ is the amount of change in the objective function. B~ is like a market clearing price on sulfur discharges . If the atmosphere were owned and managed for waste disposal, B~ is the amount that could be charged for each ton of sulfur dumped. Since the atmosphere is not managed as a waste assimilating asset, B~ is a rent that can accrue to any of the participants in the coal market. Nonetheless, B~ is a marginal opportunity cost of limiting sulfur discharges to levels required by law . Other U.S. sulfur dioxide standards are implemented : (1) by shipping coal from certain coal supply regions, and only low-sui fur coal from those suppl y regions that can satisfy State Implementation Plan standards , and (2 ) by always scrubbing coal to satisfy sulfur dioxide standards for boilers under Revised New Source Performance Standards.
395
- Inequality 9. Tons of coal shipped cannot exceed river way lock capacities (L) . C~ is the shadow value on limited lock capacity (dollars/ton of coal, in present value terms). If the lock capacity were to be increased by one ton of coal, then C~ is the amount by which the objective function would decrease . C~ is the amount a lock owner can charge on each ton of coal passing through a capacity constrained lock. Since it reflects the cost of the next best alternative, it, like ~, is a marginal opportunity cost. - Inequality 10. Tons of coal shipped cannot exceed electric transmission capacities (E~). H_ is the shadow value on limited transmission capacity (dollars / ton of coal, in present value terms) . It is the marginal opportunity cost of transmitting coal through a capacity constrained transmission line. To develop necessary conditions for an optimal solution, the constraint inequalities are multiplied by their associated dual variables and added to or subtracted from the objective function (as appropriate) to obtain the following Lagrangian expression : T
K +
::E [1-,
P,{"x l
DI
-
)
BI {I3X I - O.0072D,) C,(x, - L) +H,(x,-E,)]
+ +
Necessary conditions for are obtained by applying to (11). The first step derivatives with respect
(11 )
a minimum cost solution the Kuhn-Tucker theorem is to take partial to x, :
2
0
( 12)
For the solution to be optimal, it is also necessary that Kuhn-Tucker complementary slackness conditions be satisfied, as follows : XI (
If
XI )
aZ/ ax , ) = 0
0 then , by complementary slackness,
aZ / ax I = 0
or from (12) (13a)
"PI (1 +r ) I - I
(1+r) 1-1[I3B, + C, + Ht) +
± (_1_ q_ITI
hr
)q-I . [ae q
a Zq
aZ q
aX t
(13b)
+
(13c)
The right-hand side of expression (13a) includes conventional marginal costs (the sum of marginal extraction, transportati on, and scrubbing cost ). Expression (1 3b) is the sum of marginal opportunity costs assoc iated with SOX discharge
396
W. D. Watson
li.its, river lock capacities, and transmission line capacities . Expression (13c) is the future .arainal opportunity cost from current extraction of nonrenewable coal . It is the dyna.ic Ricardian rent that gets added to the coal price in order to reflect future scarcity cost . The term (l+r)~-l in (13a) and (13b) moves the present values of price and shadow values forward from their discounted values in the initial period (time = 1) to non-discounted values in the current period (time = t) . The term a, dividing through on left- and right-hand sides, converts dollars per ton (by weight) for the right-hand side values into dollars per ton of coal containing 24 million Btu per ton. The term, P_(l+r)'-', is market price in undiscounted dollars per ton of coal containing 24 million Btu per ton . Thus, equation (13) says that a shipment can be in the optimal solution if its full marginal costs (normalized for Btu) are equal to (or are just covered by) the delivered price consu.ers are willing to pay at the demand center . That price equals the shadow value on the demand constraint. When delivered price does not cover full marginal cost (for all parts of the market transaction), the shipment will not be made since suppliers cannot cover all their costs . This logical outcome also is consistent with the Kuhn-Tucker complementary slackness condition . If XI = 0 then, by complementary slackness, aZ/ax I ~ 0 or from expression (12), full marginal cost 2 price . The Kuhn-Tucker conditions, also, provide insight on cost competition among different supply regions potentially shipping coal to a given demand center. By the Kuhn-Tucker conditions, competing coal supply regions would supply a given demand region if full marginal delivered costs from each supply region are covered by price . Finally , complementary slackness from the Kuhn-Tucker theorem also requires that the optimal solution satisfy : PI (az/aP I ) 0 BI (
RESULTS Coal allocati on patterns depend upon a complex interplay of factors that include (1) relative abundance of high quality coal, (2) growth in coal demands, (3) the changing spatial pattern of coal supply and demand, (4) transportation options, and (5) environmental restrictions . All of these factors are reflected in the solution provided by the dynamic OPtimization model . National coal demands are projected t o grow f rom 'The data used in the model, including coal quantity and coal quality data, are described in Watson and others (1989) .
800 million tons in 1985 to about 3.5 billion tons by 2075. Demand for boiler coal is projected to reach a saturation level by about 2040 . Thereafter, demand for coal by synfuels plants will keep total coal demand growing along an upward sloping linear path . The total U.S. supply of deliverable coal is estimated to be 338 billion tons. By the year 2075, 212 billion tons or 63~ of currently known U.S. deliverable coal resources are extracted in the solution. Many traditional coal supply basins would have their coal completely depleted in this time frame r Complete depletion is projected to occur , for example, in eastern and western Kentucky, in most of West Virginia , and in much of the Illinois basin. Remaining resources would be concentrated in high cost midwest districts and in the western U. S. Table 1 provides results for the total U. S. and for large geographic sections of the U. S. (The midwestern U.S., which is similar to the eastern U. S. , is not shown.) The absolute levels of delivered costs (sum of mining, transportation, and scrubbing costs ; and rents) differs by section of the U.S. The eastern U.S . , which has a high demand for low suI fur coal, has delivered costs and rents above the national average (see section B, Table 1), whereas the western U.S. with a relative abundance of low sulfur coal supplies - has costs and rents (in early years) below the national average (see section C, Table 1) .
The aggregate rent - over-price ratios (R / P) in Table 1 from the optimization model (shown in column (6)) are almost identical to the rentover-price ratios calculated using the simple model (shown in column (7)) . The growth rate in delivered costs (column (4)) is estimated using the delivered cost estimates in column (3) . If the growth is not at a constant exponential rate or not smooth (indicated by "N Smooth" in column (2)), then the cost growth rate is calculated from year to year according to the entries in column (1). When costs grow smoothly, the cost
TABLE 1 Aggregate Coml2arisons (r
= 87.)
(1 )
(2)
(3)
(4)
Year
Cost Path
Cost'
a
2. 10 2 . 74 3 . 11 3.42 3.76 4.23
1.80 0 . 85 0. 64 0.71 0. 71 0 . 71
0 . 30 0 . 29 0 . 29 0.28 0 . 34 0. 36
0 . 14 0 .11 0 . 09 0 . 08 0 . 09 0. 09
0 . 23 0.11 0 . 08 0 . 09 0 .09 0 . 09
B. Eastern U.S . 1993 N Smooth 2.30 2008 N Smooth 3 . 00 2023 Smooth 3 . 29 2038 Smooth 3 . 56 2053 Smooth 3 . 92 2068 Smooth 4 . 50
1. 80 0 .62 0 . 70 0 . 70 0 . 70 0 . 70
0. 42 0. 36 0.30 0 . 32 0.35 0 .38
0.18 0 . 12 0 .09 0 . 09 0. 09 0 . 08
0.23 0 . 08 0.09 0 . 09 0 . 09 0 . 09
C. Western U. S. N Smooth 1. 97 1993 2008 N Smooth 2.55 2023 Smooth 2.97 2038 Smooth 3 . 33 Smooth 3.72 2053 2068 Smooth 4 . 15
1. 74 1. 00 0 . 75 0 . 75 0 .75 0 . 75
0 . 20 0.25 0. 28 0 . 30 0 .39 0. 36
0 . 10 0 . 10 0 . 09 0.09 0.10 0 . 09
0. 22 0.12 0 . 09 0 . 09 0. 09 0 . 09
A. Total U.S. 1993 N Smooth 2008 N Smooth 2023 N Smooth 2038 Smooth 2053 Smooth 2068 Smooth
'1985$/ 10"6 Btu
(5)
(6) Prog . Model Rent' RLP
(7) Simp . Model aLr
Rising Economic Rent
growth rate is calculated over the entire set of years displaying smooth growth . In the "N Smooth" periods, the simple model will tend to overestimate the actual rent-over-price ratio because the simulated growth is along an upwardly sloped linear path where cost growth rates are declining over time rather than staying constant as required by the simple model . As shown in Table 1, in the smooth growth periods there is very close agreement between the observed rentover-price ratios from the optimization model and the rent-over-price ratios from the simple model. Overall, the rent paths and the rent-over-price paths provided in Table 1 do not indicate impending coal shortage. The high rent-overprice ratios in the beginning periods are due primarily to relatively rapid increases in costs in early years incurred to meet EPA's S02 regulations. The rate of increase in costs and in rents remains constant in the smooth growth time periods even though by the end of the whole time period extending to the year 2075, U.S. coal, in total, is projected to be substantially depleted. And even though coal resources are completely depleted by the year 2075 in many eastern coal basins, the eastern U.S. paths for costs, rents, and rent-over-price ratios are similar to those for western U.S. coal resources where a large amount of resources is projected to be available for extraction in 2075. Table 2 provides rent-over-price ratios for selected coal supply regions that experience coal depletion by the year 2075. In the entire set of 60 regional coal supply basins, coal is depleted by 2075 in 28 basins. Among the 28 depleted basins, Region 16 (shown in Table 2) is the only basin where rent-over-price ratios are a leading indicator of coal depletion. Table 2 also shows that there is wide variation in rent-over-price ratios for any particular coal supply region within a single time period (see column (4». In a single period, each coal supply region ships coal to many different demand regions or markets. Each market has its own set of costs determined by transport distance, S02 constraints and rents. On the supply side, some coal supply regions ship more than one type of coal . Coal shipments to the various markets can be from various coal steps or blocks, each with different mining costs and coal quality . The diversity and complexity of the market process, as it gropes for a minimum cost response, necessarily means that a wide range of rents and costs will occur at any given time in a coal supply region, as is made evident by the data in Table 2. The seeming complexity, which is nothing more than a reflection of the market's success in balancing marginal opportunity costs, means that systematic trends in costs and rents are unlikely to exit in disaggregated data . CONCLUSION In a smoothly transitioning natural resource market, it appears that there will be little evidence in market signals of impending resource shortage . The rates of cost and rent increase can be high or they can be low and still nothing will be revealed about shortages as long as the rates of increase in costs are more or less constant. With a diverse natural resource base, such as we have with U.S. coal, one can presume that cost increases will be smooth (indeed, the simulation from the dynamic optimization model indicates such) . In this situation, the market is a perpetrator of a smooth transition . It will search out the lowest cost options at each point
397
TABLE 2
Regional Coal Rent and Production Patterns
(1)
(2)
Yr
Tons (MMI Yr)
(3)
(4)
RI P RI P Low to (%) Hi (%)
Pikevi lie KY (9)' 1993 2008 2023 2038 2053 2068
137 134 48 0 25 25
24 16 17 3 9 12
29 38 22 46 0 42
31 20 21 7 4 12
8-9 12-12
2-35 4-63 11-85 6- 7 11-12
Morganfield KY (3) 1993 2008 2023 2038 2053 2068
16 50 80 109 63
9 14 9 7 7 20
1-10 4 - 26 8-21 7-7 7-7 20-20
Moffat & Routt CO (93) 1993 2008 2023 2038 2053 2068
31 62 55 93 35
6 8 7 8 14 30
(2)
Yr
Tons (MMI Yr)
(3)
RIP (%)
(4)
RIP Low to Hi (%)
Pittsburgh PA (16) 4-48 6-63 7-68
Kittanning PA (17) 1993 2008 2023 2038 2053 2068
(1)
1-11 2-13 2-16 7-22 9-22 30-30
1993 2008 2023 2038 2053 2068
70 55 104 139 264 122
8 8 7 13 10
2-15 1-36 4-21 11-15 4-11 0-11
Charleston WV (21) 1993 2008 2023 2038 2053 2068
26 50 129 120 134 68
27 16 7 8 11 13
3-39 8-54 5-75 4-15 9-13 12-26
Price Utah
PO) 9 8 12 8 18 11
2-16 4-26 10- 52 8-10 5-24 4-15
1993 2008 2023 2038 2053 2068
8 8 14 15 3 7
Powder River WY (96) 1993 2008 2023 2038 2053 2068
199 221 217 266 322 369
7 11 10 11 11 11
1-16 2-23 3-25 6-21 4 - 21 0-18
'Coal supply region number used by the dynamic optimization model . in time and balance costs at the margin in order to achieve a smooth transition along the path of lowest cost . In other cases, if geology is unusually restrictive, or markets noncompetitive, or substitute resources high cost and in short supply, then the rate of cost increase can deviate from a smooth path and perhaps signal a resource shortage. It could be argued that if the transition is smooth, then there is no resource shortage . Still, in the simulations presented here, there are large and jarring changes in the U.S. coal industry. Regions of the country where coal mining has been a major industry for many decades will cease to produce coal . These large and dramatic changes are not revealed in the market signals provided by the programming model . At both national and regional levels, shortages are revealed in comparisons between estimates of extracted coal and the coal resource base . But the ability to make these comparisons and uncover the potential for shortage, requires detailed investigations of both geologic and economic conditions . It would appear that market signals alone are not likely to take us very far toward being able to make reliable forecasts of impending resource shortage . REFERENCES Brown, G. , and B. Field (1979) . The adequacy of measures for signaling the scarcity of natural resources . In V. K. Smith (Ed.) . Scarcity and Growth Reconsidered. Johns
W. D. Watson
398
Hopkins University Press, Baltimore. pp. 218- 245 . Chiang, A. C. (1984) . Fundamental Methods of Mathematical Economics . McGraw-Hill, New York. pp . 722-753. Dale, L. (1984) . The pace of mineral depletion in the United States. Land Econ .. ~ 255-267. Hotelling, H. (1931). The economics of exhaustible resources. J . Polit . Econ . ~ 137-175. Levhari, D. , and N. Liviatan (1977). Notes on Hotelling's economics of exhaustible resources . Can. J. Econ . , ~ 177-192. Slade, M. (1982) . Trends in natural resource commodity prices: an analysis of the time domain . J . Envir . Econ . and Manag . , ~ 122137 . Smith, V. K. (1978) . Measuring natural resource scarcity: theory and practice. J . Envir . Econ. and Manag .. ~ 150-171 . Watson, W. D. , A. L. Medlin, K. K. Krohn, D. S. Brookshire, and R. L. Bernknopf (1989) . Economic Effects of Western Federal Land Use Restrictions on U. S. Coal Markets. U. S. Geological Survey, Reston , Va. I