Nội dung

     February  Accounting for Subsoil Mineral Resources  , A blue-ribbon panel of the National Academy of Sciences’ National Research Council completed a congressionally mandated review of the work that the Bureau of Economic Analysis () had published on integrated economic and environmental accounts. The panel’s final report commended  for its initial work in producing a set of sound and objective prototype accounts. The November  issue of the S  C B contained an article by William D. Nordhaus, the Chair of the Panel, that presented an overview of the major issues and findings and a reprint of chapter , “Overall Appraisal of Environmental Accounting in the United States,” from the final report. As part of its promise to inform users of the results of this evaluation,  is reprinting additional chapters from the panel’s report; below is a reprint of chapter , which reviews ’s development of a set of subsoil mineral accounts. This article is reprinted with permission from Nature’s Numbers: Expanding the National Economic Accounts to Include the Environment. Copyright of the National Academy Press, Washington, . This is a report of the National Research Council, prepared by the Panel on Integrated Environmental and Economic Accounting and edited by William D. Nordhaus and Edward C. Kokkenlenberg. INTRODUCTION  minerals—particularly petroleum, natural gas, and coal—have played a key role S in the American economy over the last century. They are important industries in themselves, but they also are crucial inputs into every sector of the economy, from the family automobile to military jets. In recent years, the energy sector has been an important contributor to many environmental problems, and the use of fossil fuels is high on the list of concerns about greenhouse warming. The National Income and Product Accounts () currently contain estimates of the production of mineral products and their flows through the economy. But the values of and changes in the stocks of subsoil assets are currently omitted from the . The current treatment of these resources leads to major anomalies and inaccuracies in the accounts. For example, both exploration and research and development generate new subsoil mineral assets just as investment creates new produced capital assets. Similarly, the extraction of mineral deposits results in the depletion of subsoil assets just as use and time cause produced capital assets to depreciate. The  include the accumulation and depreciation of capital assets, but they do not consider the generation and depletion of subsoil assets. The omission is troubling. Mineral resources, like labor, capital, and intermediate goods, are basic inputs in the production of many goods and services. The production of mineral resources is no different from the production of consumer goods and capital goods. Therefore, economic accounts that fail to include mineral assets may seriously misrepresent trends in national income and wealth over time. Omission of minerals is just one of the issues addressed in the construction of environmental accounts. Still, extending the  to include minerals is a natural starting point for the project of environmental accounting. These assets— which include notably petroleum, natural gas, coal, and nonfuel minerals—are already part of the market economy and have important links to environmental policy. Indeed, production from these assets is already included in the nation’s gross domestic product (). Mining is a significant segment of the nation’s output; gross output originating in mining totaled  billion, or . percent of , in . This figure masks the importance of production of subsoil minerals in certain respects, however, for they are intimately linked to many serious environmental problems. Much air pollution and the preponderance of emissions of greenhouse gases are derived directly or indirectly from the combustion of fossil fuels—a linkage that is explored further in the next chapter. Moreover, while the value of mineral assets may be a small fraction of the nation’s total assets, subsoil assets account for a large proportion of the assets of certain regions of the country. Current treatment of subsoil assets in the U.S. national economic accounts has three major limitations. First, there is no entry for additions to the stock of subsoil assets in the production or asset accounts. This omission is anomalous because businesses expend significant amounts of resources on discovering or proving reserves for future use. Second, there is no entry for the using up of the stock of subsoil assets in the production     or asset accounts. When the stock of a valuable resource declines over time through intensive exploitation, this trend should be recognized in the economic accounts: if it is becoming increasingly expensive to extract the subsoil minerals necessary for economic production, the nation’s sustainable production will be lowered. Third, there is no entry for the contribution of subsoil assets to current production in the production accounts. The contribution of subsoil assets is currently recorded as a return to other assets, primarily as a return to capital. There is a well-developed literature in economics and accounting with regard to the appropriate treatment of mineral resources. The major difficulty for the national accounts has been the lack of adequate data on the quantities and transaction prices of mineral resources. Unlike new capital goods such as houses or computers, additions to mineral reserves are not generally reflected in market transactions, but are determined from internal and often proprietary data on mineral resources. Moreover, there are insufficient data on the transactions of mineral resources, and because these resources are quite heterogenous, extrapolating from existing transactions to the universe of reserves or resources is questionable. Notwithstanding the difficulties that arise in constructing mineral accounts, the Bureau of Economic Analysis () decided this was the best place to begin development of its Integrated Environmental and Economic Satellite Accounts ().  in the United States and comparable agencies in other countries have in recent years developed satellite accounts that explicitly identify mineral assets, along with the changes in these assets over assets, along with the changes in these assets over time. This chapter analyzes general issues involved in minerals accounting and assesses the approach taken by  (as described in Bureau of Economic Analysis [b]). The first section provides an overview of the nature of subsoil mineral resources and describes the basic techniques for valuing subsoil assets. The second section describes ’s approach to valuation, including the five different methods it uses to value subsoil mineral assets. The third section highlights the specific strengths and weaknesses of ’s approach, while the fourth considers other possible approaches. The chapter ends with conclusions and recommendations regarding future efforts to incorporate subsoil mineral assets into the national economic accounts. GENERAL ISSUES IN ACCOUNTING FOR MINERAL RESOURCES Basics of Minerals Economics A mineral resource is “a concentration of naturally occurring solid, liquid, or gaseous material, in or on the earth’s crust, in such form and amount that economic extraction of a commodity from the concentration is currently or potentially feasible” (Craig et al., :). The size and nature of many mineral resources are well known, whereas others are undiscovered and totally unknown. Figure – shows a spectrum of resources that differ in their degree of certainty, commonly described as measured, indicated, inferred, hypothetical, and speculative. Another important dimension is the economic feasibility or cost of extracting and using the resources. Some resources are currently profitable to exploit; others may be economical in the future, but currently are not. Along this dimension, mineral resources are conventionally described as economic (profitable today), marginally economic, subeconomic, and other. Resources that are both currently profitable to exploit (economic) and known with considerable certainty (measured or indicated) are called reserves (or ores when referring to metal deposits). This means reserves are always resources, though not all resources are reserves. Over time, reserves may increase. Exploration may result in the discovery of previously unknown deposits or demonstrate that a known deposit is larger than formerly indicated. Research and development may produce new techniques that allow previously known but uneconomic resources to be profitably extracted. A rise in a mineral commodity’s price may also increase reserves by making previously unprofitable resources economic. The exploration required to convert resources into reserves entails a cost. As a result, companies have an incentive to invest in the generation of new reserves only up to the point at which reserves are adequate for current production plans. For many mineral commodities, therefore, reserves as a multiple of current extraction tend to remain fairly stable over time. . Two additional categories of mineral endowment are worth noting since they are commonly encountered. The reserve base encompasses the categories of reserves and marginal reserves, as well as part of the category of demonstrated subeconomic resources shown in Figure –. While reserves and the reserve base are typically a small subset of resources, resources in turn are a small subset of the resource base. The resource base, not illustrated in Figure –, encompasses all of a mineral commodity found in the earth’s crust. February  •   • February      While by definition all reserves can be exploited profitably, the costs of extraction, processing, and marketing, even for reserves of the same mineral commodity, may vary greatly as a result of the reserves’ heterogenous nature. Deposit depth, presence of valuable byproducts or costly impurities, mineralogical characteristics, and access to markets and infrastructure (such as deepwater ports) are some of the more important factors that give rise to cost differences among reserves. Figure – reflects the heterogenous nature of mineral resources by separating the reserves and other known resources for a particular mineral commodity according to their exploitation costs. The lowest-cost reserves are in class A ; their quantity is indicated in the figure as 0A and their exploitation costs as 0C 1 . The next least costly reserves are found in class B , with a quantity of A B and a cost of 0C 2 . The most expensive reserves are found in class M . These reserves are . Similar comparative cost curves are used to illustrate the relative costs of mineral production for major producing countries or companies. See, for example, Bureau of Mines () and Torries (, ). marginally profitable. The market price 0P just covers the extraction cost of class M (0C m ) plus the opportunity cost (C m P ) of using these reserves now rather than saving them for future use. This opportunity cost, which economists refer to as Hotelling rent (or sometimes scarcity rent or user cost) is the present value of the additional profit that would be earned by exploiting these reserves at the most profitable time in the future rather than now. Known resources in Figure – with costs above those of class M , such as those in classes N , O , and P , are by convention not reserves. In this case, mineral producers, like other competitive firms, will have an incentive to produce up to the point where the current production costs of the next unit of output, inclusive of rents, just equals the market price. When Hotelling rents exist, . Where the relevant market for a mineral commodity is global and transportation costs are negligible, Figure – reflects cost classes for reserves and other known resources throughout the world. Where a mineral commodity is sold in regional markets, a separate figure would be required for each regional market, and the cost classes shown in any particular figure are only for the reserves and other known resources in the regional market portrayed.     they are the same for all classes of reserves for a particular mineral commodity market. Thus, the total Hotelling rent shown in Figure – is simply the Hotelling rent earned on marginal reserves (C m P ) times total reserves (0M ). Those reserves whose marginal extraction costs are below those of the marginal reserves in class M are called inframarginal reserves. As a result of their relatively low costs, they yield additional profits when they are exploited. Mineral economists refer to these additional profits as Ricardian rents. In Figure –, the Ricardian rents per unit of output equal C 1 C m for reserves in class A , C 2 C m for reserves in class B , and so on. Unless technical or other considerations intervene, mineral producers will generally exploit first those reserves that have relatively low production costs and thus high Ricardian rents (like classes A and B ). This implies that the reserves currently being extracted have lower costs than the average of all reserves and that their Ricardian rents are likely to be above average. Since reserves by definition are known and profitable to exploit, they are assets in the sense that they have value in the marketplace. Although mineral resources other than those classified as reserves might have in-completely February  • defined characteristics (in terms of costs and quantities) or be currently unprofitable to exploit, they still may command a positive price in the marketplace. Petroleum companies, for example, pay millions of dollars for offshore leases to explore for oil deposits that are not yet proved reserves. Mining companies pay for and retain subeconomic deposits. The option of developing such deposits in the future has a positive value because the price may rise, or some other developments may make the deposits economic. Thus, a full accounting of subsoil assets should consider not only reserves, but also other mineral resources with positive market value. In the case of reserves, market value may reflect Hotelling rent, Ricardian rent, and option value. In the case of mineral resources other than reserves, a positive market value is due solely to their option value. Key Definitions in Mineral Accounting Changes in the value of the mineral stock come about through additions, depletions, and revaluations of reserves. P . The total value of reserves is V = i v i R i, where v i is the unit value of reserves in class i (i = A ,B,...,M ), and R i is the quantity of reserves of class i.   • February      • Additions are the increases in the value of reserves over time due to reserve augmentations. They are calculated as the sum of the price of new reserves times the quantity of new reserves for each reserve class. • Depletions are the decreases in the value of reserves over time due to extraction. They are similar to capital consumption (depreciation) and parallel the concept of additions. • Revaluations are changes in the value of reserves due to price changes. They measure the residual change in the value of reserves after correcting for additions and depletions. Techniques for Valuing Mineral Assets As noted in the last section, the major challenge in extending the national accounts to include subsoil minerals is to broaden the treatment of mineral assets to include additions and depletions and to incorporate depletion in the production accounts. This task involves estimating the value of the subsoil assets. A specific subsoil asset consists of a quantity of a mineral resource and the invested capital associated with finding and developing that resource. Invested capital includes physical structures such as roads and shafts, as well as capitalized exploration and drilling expenses. The total value of the subsoil assets equals the sum of the value of the mineral and the value of the associated capital (see Figure – ). Currently, U.S. national economic accounts include the value of the associated capital, but exclude the value of the mineral resource. One of the goals of natural-resource accounting is to estimate the total value of subsoil assets and to separate this estimate into the value of the mineral and the value of the associated capital. An additional goal is to track over time changes in the value of the stock that result from additions, depletions, and revaluations. Three alternative methodologies are used in valuing mineral resources: () transaction prices, () replacement value, and () net present value. In developing its mineral accounts,  used one version of the first method and four versions of the third. This section explains the basic elements of each approach. Transaction Prices The most straightforward approach to valuing mineral resources relies on market transaction prices. This is the standard approach used across the national economic accounts for capital assets. When resources of petroleum, copper, gold, and other minerals are sold, the value of the transaction provides a basis for calculating the market value of the mineral component of the asset. A close look at the transaction-prices approach reveals, however, a number of difficulties that need to be resolved. The major difficulty is that a market transaction usually encompasses a number of assets and liabilities, such as the associated capital (e.g., surface roads, shafts, and refining operations), taxes, royalty obligations, and environmental liabilities. Because the transaction usually includes not only the mineral resources, but also associated capital, the value of the capital must be subtracted to obtain the mineral value. In addition, the property is usually encumbered with royalty obligations to prior owners or to owners of the land. Many mineral properties also have associated environmental problems, such as contaminated soils and water, and they may even be involved in complicated legal disputes, such as connection to a Superfund site with joint and several liability. Some of these associated assets and liabilities (such as mining structures) are true social costs or assets, while others (such as royalty obligations) are factor payments. Another difficulty with using transaction prices is the sporadic nature of the transactions. The infrequency of the transactions, coupled with the heterogeneity of the grade of the resource, makes it difficult to apply the transaction price for one grade or location of the resource to other grades in other locations. Because of the complex assortment of assets and liabilities associated with transactions of mineral resources, the price must be adjusted to obtain the value of a resource. As noted     above, the working capital and the associated capital must be subtracted from the transaction price, while any extrinsic environmental liabilities should be added, as should any factor payments, such as royalties or taxes, to obtain the value of the underlying resource. Box – provides an example of how to adjust the transaction price to obtain the market value of a mineral resource for a hypothetical sale involving the purchase of , barrels of oil. In this example, the buyer pays  million for a property containing , barrels of oil, and this is recorded as the transaction value. Attached to those reserves is a long-term debt of . million; this liability must be added to the purchase price. If the acquired reserves also include associated working capital of . million, this amount must be deducted from the purchase price. Correcting for these two items creates an effective purchase price or market value of the asset of . million. An additional issue arises because of payments such as future taxes and royalties. In acquiring the above property, the new owner must, for example, pay a  percent overriding royalty to the landowner. Such payments should be included in the value of the resource even though they do not accrue to the seller of the property. In the example shown in Box –, future royalties and taxes are assumed to have a present value of . million. These payments introduce a major new complication because taxes and royalties depend on future production. Not only are they uncertain, but they also cannot be easily estimated from market or transaction data. One approach is to adjust the transaction price by marking up the value of the transaction by a certain amount. Adelman and Watkins (:), for example, suggest that  percent be added to the “effective purchase price” to account for transfers. After adjusting for royalties, this yields a social asset value for the above property of . million. The final adjustment is for associated capital, which is assumed to have a value of . million. After this amount is subtracted, the estimated social value of the underlying petroleum reserve is calculated to be . million. Replacement Value A second approach uses the costs of replacing mineral assets to determine their value. Under this approach, it is assumed that firms have an incentive to undertake investments to find new resources up to the point where the additional cost of finding one more unit just equals the price February  •  Box –: Transaction Price Methoda Recorded Dollar Transaction (, barrels) ............................................... . million Adjustments Add: assumed liabilities .............. . million Subtract: working capital .............. million Effective Purchase Price of Asset ....................... . million Add: present value of taxes, royalty transfers ............................. million Value of Assets ......................................... . million Subtract: value of associated capital ...... million Value of Petroleum Reserve ............... ............ . million a This methodology is not followed in the conventional accounts. For instance, in valuing the stock of cars, we do not subtract tax credits, nor do we add in future liabilities such as property taxes. Similarly, to the extent that royalties are regarded as a sharing of profits (like dividends), they should not affect the value of an asset; to the extent that royalties are actually a deferred part of the purchase price, they can be capitalized to increase the value of an asset. Box –: Definitions of Symbols and Basic Concepts in Minerals Accounting For this discussion, assume that there is only one class of a mineral reserve, that extraction costs are constant, and that the unit value of the reserve rises at the social rate of discount. Variables are: R t = total quantity of reserves of the mineral commodity at year end H t = unit value of the reserves (say, petroleum reserves), which equals Hotelling rent under the above assumptions A t = quantity of new reserves discovered during the year q t = quantity of extraction or production during the year V t = total value of the reserves at year end In a given year, petroleum firms might discover new reserves totaling A t. Then the additions are given by: additionst = H tA t (.) During that year, petroleum production, and therefore depletion of existing reserves, is measured by q t. Depletion is, under the special assumptions listed above, quantity times the value of reserves: depletionst = H tq t (.) The total value of reserves at year end is: value of reserves = V t = H tR t (.) The change in the value from the end of year t − 1 to the end of year t is given by: change in value of reserves = V t − V t−1 = H tR t − H t−1 R t−1 (.) Revaluations are the change in the value corrected for the value of additions and depletions: revaluation = H tR t − H t−1 R t−1 − H tA t + H tq t (.)      • February  at which firms can buy that unit—that is, up to the market value. Therefore, the additional or marginal cost of finding a mineral resource should be close to its market price. Associated with this approach, however, are many of the same issues discussed above under transaction prices. For example, a particular replacement cost is relevant only for valuing deposits of comparable quality and cannot be used to value resources of another grade. This point can be illustrated using Figure –. Assume that exploration is resulting in the discovery of resources of class M . The market value of this class would be a function of the difference between 0P and production cost 0C M . It would be profitable for firms to continue exploring for such deposits until the finding costs (that is, the replacement costs) just reached the value of this class of resource. However, the replacement cost of class M cannot be used to value other classes, such as class A , which have a lower extraction cost and therefore a higher value. Because of cost differences, using class M to value classes A through L would yield an underestimate of the value of these reserves. Net Present Value A third valuation technique, the net present value or  method, entails forecasting the stream of future net revenues a mineral resource would generate if exploited optimally, and then discounting this revenue stream using an appropriate cost of capital. Under certain conditions—such as no taxes—the sum of the discounted revenue values from each time period will equal the market value of the resource. For example, assume that a  million-ounce gold asset generates a stream of net revenues (after accounting for all extraction and processing costs) that, when discounted at a rate of  percent per year, has a present value of . billion. According to this approach, the value of the asset is taken to be . billion. If the value of the plant, equipment, and other invested capital ultimately associated with the asset is estimated to be  million, the current value of the gold reserves is  billion, and their unit value is  per ounce. Again, as with the previous two methods, each class of resource should be separately valued, since the stream of revenues from a higher class of resource will be greater than that from a lower class. . The appropriate discount rate for energy and environmental resources is debatable. See Lind (, ) , Schelling () , and Portney and Weyant (). A special case of the  approach, known as the Hotelling valuation principle (see Miller and Upton,  ), avoids the difficulties of forecasting future net revenues and then discounting them back to the present. This approach makes the strong and generally unrealistic assumption that the unit value of a resource grows at exactly the same rate as the appropriate discount rate. In the above example, this would imply that the unit value of the gold resource would grow at the discount rate of  percent per year; that is, the unit value would be  in the first year,  in the next year, . in the following year, and so forth. Under this assumption, the present value of the resource would easily be calculated as the current period’s resource price multiplied by the current physical stock of the resource. Under a further set of assumptions, such as homogeneous resources and constant extraction costs, the current period resource price is simply the current net revenue (unit price less unit extraction cost). For example, assume that in a given year the United States has  million ounces of homogeneous gold reserves, that the price of gold in that year is  per ounce, and that the average extraction cost is  per ounce. Under the Hotelling valuation principle, the price of the gold reserves would be  per ounce, and the total value of the gold assets would be calculated as . billion. Note that it would still be necessary to deduct the value of capital from the . billion to obtain the value of the mineral reserve. Again, for this approach to be valid, the per unit price of gold reserves ( in this example) would need to grow at the discount rate appropriate for these assets. BEA’S VALUATION OF SUBSOIL MINERALS This section presents a more detailed description of ’s valuation methods (as set forth in Bureau of Economic Analysis, b). In the absence of observable market prices for reserves,  estimates mineral reserve and flow values using five valuation methods. These calculations are performed for reserves of fuel minerals (petroleum, natural gas, and coal) and other minerals (uranium, iron ore, copper, lead, zinc, gold, silver, molybdenum, phosphate rock, sulfur, boron, diatomite, gypsum, and potash) for each year from  through  (oil and gas figures are calculated from  to ). In addition, aggregate stock and flow values for five mineral categories (oil, gas, coal, metals, and other minerals) are en-     tered in the appropriate rows and columns of the  Asset Account for . This section first examines the five methods used by  in estimating mineral values, along with the data they require, and then describes ’s findings. Box – provides definitions of the symbols used in minerals accounting. February  •  Box –: Formulas for Current Rent Method I total mineral reserve valuet = V t = [p t − a t]R t − rR tK t/q t − R tD t/q t = [p t − a t − rK t/q t − D t/q t]×R t additionst = [p t − a t − rK t/q t − D t/q t]×A t depletionst = [p t − a t − rK t/q t − D t/q t]×q t BEA’s Five Basic Valuation Methods revaluationst = V t − V t−1 + depletionst − additionst Current Rent Method I Current rent methods I and II are  methods based on the Hotelling valuation principle. The attraction of the Hotelling valuation principle is the ease with which the calculation can be performed, avoiding the need to forecast mineral prices and to assume an explicit discount factor. In both methods, the value of the aggregate stock is calculated as the net price times the quantity of reserves, where the net price is as described below. Additions or depletions are similarly calculated as net price times the quantity of additions or depletions. One of the difficulties with this approach is that the Hotelling valuation principle tends to provide a systematic overvaluation of reserves, the reason for which is discussed in a later section. Current rent methods I and II are quite similar in construction. They differ primarily in the method of adjusting for the value of associated capital. (The algebra of the different formulas is shown in the boxes in this section.) Current rent method I (see Box – ) uses the normal rate of return on capital to determine the return on associated capital in the mining industry that should be subtracted from revenues. It then calculates the “resource rent per unit of reserve” by taking the net profits from mining, subtracting the return and depreciation on the associated capital, and dividing that sum (called “resource rent” by ) by the quantity of resource extracted during the year. The method thus yields an estimate of the unit value of the reserves currently extracted. To calculate the total value of the mineral reserve, the current resource rent per unit is multiplied by the total reserves, in the spirit of the Hotelling valuation principle. Additions and depletions are calculated as those quantities times the resource rent per unit. Revaluations are simply the residual of the change in the value of the stocks plus depletions minus additions. It has been observed that the value of the stock can be highly volatile; this volatility is due primarily to the revaluation effect. where V t = value of mineral reserves p t = price of commodity a t = average cost of current production R t = total quantity of reserves r = average rate of return on capital K t = value of associated capital, valued at current replacement cost q t = total quantity extracted D t = depreciation of associated capital A t = quantity of discoveries of new reserves additionst = value of discoveries of new reserves depletionst = value of depletions revaluationst = change in value of reserves corrected for depletions and additions The revaluation term is not directly calculated; it will include any errors in calculating additions, depletions, and opening and closing stock values. Current Rent Method II Current rent method II is virtually identical to current rent method I. The only difference is in the method of adjusting for associated capital. The value of the associated capital is subtracted from the total value of the mineral asset to obtain mineral-reserve values in current rent method II. Again employing the Hotelling valuation approach, the total value of the mineral asset (including the value of the associated capital) is calculated as the per unit net revenue times the total quantity of reserves. The total value of the mineral reserve is then calculated as the total value of the asset value minus the value of the associated capital. The unit resource value, which is used to price additions and depletions, is just this total reserve value divided by the total quantity of reserves. This approach is defined algebraically in Box – . As is discussed below, both current rent methods have major advantages in that they are easy to calculate on the basis of data  currently uses in its accounts (primarily profits and capital stock and consumption data). They both suffer from the serious disadvantage that they rely on      • February  Box –: Formulas for Current Rent Method II total mineral reserve valuet = V t = [p t − a t − K t/R t]R t additionst = [p t − a t − K t/R t]×A t depletionst = [p t − a t − K t/R t]×q t revaluationst = V t − V t−1 + depletionst − additionst where variables are as defined in Box .. the Hotelling valuation principle, thereby tending to overvalue reserves. Net Present Value Estimates If the basic assumptions of the Hotelling valuation principle do not hold—and there is strong evidence that they do not, as discussed below—life becomes much more complicated for national accountants. One approach that is sound from an economic point of view is to value reserves by estimating the present discounted value of net revenues. To render the present value approach workable,  makes three simplifying assumptions. First, it assumes that the quantity of extractions from an addition to proved reserves is the same in each year of a field’s life. The quantity of depletions in any year is assumed to result equally from all vintages (cohorts) still in the stock, i.e., all vintages whose current age is less than the assumed life. Second, the life for a new addition is assumed to be  years until  and  years thereafter. Third,  assumes that the discount rate applied to future revenues is constant at a rate of either  percent per year or  percent per year above the rate of growth of the net revenues (where the latter equals the rate of growth of the price of the resource). These assumptions lead to a tractable set of calculations. The present discounted value of the mineral stock as calculated using this present value method is simply the stock and flow values calculated with current rent method II, multiplied by a “discount factor” of between . and . for the  percent discount rate and between . and . for the  per cent discount rate. . According to , the rates were chosen to illustrate the effects of a broad range of approaches. The  percent per year discount rate has been used by some researchers to approximate the rate of time preference, while the  percent rate has been used by some researchers to approximate the long-term real rate of return to business investment. . At the  percent discount rate, the . discount factor holds for the years  through , with the rate edging upward thereafter as a result of commingling of reserves that were developed prior to  (which  assumes are extracted over  years) with those developed in  or later (for The calculated values are, then, lower than the values derived using current rent method II, with the difference depending on the discount rate employed. Additions and depletions are then calculated in a manner similar to that used with current rent method II. The average unit reserve value is calculated by dividing the total reserve value by the quantity of reserves, and then using this unit value to value additions and depletions. Additions would be calculated as  percent of the value of additions according to current rent method II if the discount rate is  percent per year, and  percent of the value of additions according to current rent method II if the discount rate is  percent. The calculated value of depletions would be  percent of the value of depletions under current rent method II at a  percent discount rate, and  percent at a  percent discount rate. In summary, the present value method as implemented by  takes the values of additions, depletions, and stocks calculated according to current rent method II and multiplies them by discount factors of between  and  percent. The reason for the discount is straightforward. Under current rent method II, which relies on the Hotelling valuation principle, it is assumed that net revenues rise at the discount rate. Under the present value approach, net revenues are assumed to rise at rates that are  or  percent slower than the discount rate applicable to mineral assets. The higher percentage is the discrepancy between the rise in net revenues and the discount rate; the lower is the discount factor. The  approach is shown in Box – . Replacement Cost The fourth method of calculating the value of the mineral stock is used only for oil and gas reserves. Despite its name, this approach is similar to the  method, not to the replacement cost method described earlier. It adopts the approach of Adelman (), who calculates the present value of an oil field using special assumptions. It is assumed that the production from an oil or gas field declines exponentially over time. Under the assumption that the decline rate is constant and which a -year life is assumed). For the  percent discount rate, the . factor holds for the years  through . In , the year for which  calculates a more complete set of satellite accounts, the rate is . for the  percent discount rate and . for the  percent discount rate. . As with the calculation of mineral values, the factors shown in Box – vary depending on the year of the analysis. The factors reported are those for the  calculation. The factors differ in the various formulas because of the differing treatment of the timing of depletions and additions from reserves.     that the net revenue rises at a fixed constant rate that is less than the discount rate, a barrel factor is calculated. This barrel factor is multiplied times net revenue to obtain the present value of the reserves. Adelman estimates that the barrel factor is usually around ..  does not give the barrel factor used in its calculations, which should vary by deposit and depend on the rate at which future cash flows are discounted, but we estimate that it averages approximately .. The value of the asset—calculated with current rent method II using the Hotelling valuation principle—is then multiplied by the barrel factor. The justification is that this  approach, unlike the Hotelling approach, takes the physical specifics of oil and gas extraction into account and accordingly adjusts the unit value of reserves downward. As with the  approach discussed in the last section, this adjustment accounts for the overvaluation inherent in the Hotelling valuation principle. Once the value has been adjusted downward,  must again subtract the value of capital associated with the asset. With this method, the value of capital associated with each unit of existing reserves is assumed to be the current-year expenditure on exploration and development for oil and gas, divided by the quantity of oil and gas extracted during the year. This approach is loosely based on Adelman’s suggestion that the value of capital associated with a unit of production can be approximated by measuring the value of capital associated with finding new reserves. The replacement cost method is shown in Box – . February  • Box –: Formulas for Net Present Value Method total mineral reserve valuet@  percent discount rate = 0.88[p t − a t]R t − 0.88K t total mineral reserve valuet@  percent discount rate = 0.69[p t − a t]R t − 0.69K t additionst@  percent discount rate = 0.84[p t − a t − K t/R t]×A t additionst@  percent discount rate = 0.59[p t − a t − K t/R t]×A t depletionst@  percent discount rate = 0.83[p t − a t − K t/R t]×q t depletionst@  percent discount rate = 0.60[p t − a t − K t/R t]×q t revaluationst = V t − V t−1 + depletionst − additionst where variables are as defined in Box –. Note: The numerical values in this box apply to . As explained in the text, slightly different values will apply for different years. Box –: Formulas for Replacement Cost Method total mineral reserve valuet = V t = {0.375[p t − a t]−Z t/q t}R t additionst = {0.375[p t − a t]−Z t/q t} × A t depletionst = {0.375[p t − a t]−Z t/q t} × q t revaluationst = V t − V t−1 + depletionst − additionst where Z t = value of exploration and development expenditures in year t, and other variables are as defined in Box –. Transaction Price Method When oil and gas firms desire additional reserves, they can either buy them from other firms or find new ones through exploration and development. In the absence of risk, taxes, and other complications, the transaction price of purchasing new reserves should represent the market value of those reserves. For this reason, according to , “if available, transaction prices are ideal for valuing reserves” (Bureau of Economic Analysis, b:). In fact, transactions in reserves are few and far between outside of oil and gas, and even in oil and gas suffer from problems discussed above. To estimate transaction prices,  derived prices from publicly available data on the activities of large energy-producing firms for the period  to . The gross value of reserves was estimated by dividing expenditures for the purchase of the  Box –: Formulas for Transaction Price Method total mineral reserve valuet = V t = (T V t/T Q t − K t/R t)R t additionst = (T V t/T Q t − K t/R t)×A t depletionst = (T V t/T Q t − K t/R t)×q t revaluationst = V t − V t−1 + depletionst − additionst where T V t = value of reserve transactions, and T Q t = total quantity of reserves transacted, and other variables are as defined in Box –. rights to the proved reserves by the quantity of purchased reserves. The result was then adjusted for associated capital using the same method as