Article Contentsp. 277p. 278p. 279p. 280p. 281p. 282p. 283p. 284p. 285p. 286p. 287p. 288p. 289p. 290p. 291p. 292Issue Table of ContentsOperations Research, Vol. 33, No. 2 (Mar. - Apr., 1985), pp. 237-468Front MatterOR ForumOR and the Airline Overbooking Problem [pp. 237-248]Comparing Draws for Single Elimination Tournaments [pp. 249-262]Optimal Unemployment Insurance Policy [pp. 263-276]Optimal Supply of a Depletable Resource with a Backstop Technology: Heal's Theorem Revisited [pp. 277-292]Equilibrium Analysis of Disaggregate Facility Choice Systems Subject to Congestion-Elastic Demand [pp. 293-311]Optimal Ordering Strategies for Announced Price Increases [pp. 312-325]The Deterministic Dynamic Product Cycling Problem [pp. 326-345]A Generalized Uniform Processor System [pp. 346-362]A Branch and Bound Algorithm for the Total Weighted Tardiness Problem [pp. 363-377]The Transient Behavior of the M/M/s Queue, with Implications for Steady-State Simulation [pp. 378-396]The Push-Out-Priority Queue Discipline [pp. 397-403]Replacement of Systems and Components in Renewal Decision Problems [pp. 404-411]The Sequential Design of Bernoulli Experiments Including Switching Costs [pp. 412-426]Limits for the Precision and Value of Information from Dependent Sources [pp. 427-442]Assessment of Preferences by Conditions on Pricing-Out Amounts [pp. 443-454]Technical NotesThe Effect of Risk Aversion on the Expected Value of Perfect Information [pp. 455-458]Forest Outturn Optimization by Dantzig-Wolfe Decomposition and Dynamic Programming Column Generation [pp. 459-464]On Normalizing Constants in Queueing Networks [pp. 464-468]Back MatterOptimal Supply of a Depletable Resource with a Backstop Technology: Heal's Theorem Revisited SHMUEL S. OREN University of California, Berkeley, California STEPHEN G. POWELL Wang Laboratories, Inc., Lowell, Massachusetts (Received April 1983; revised January 1984; accepted April 1984) Heal's theorem states that if the extraction cost of a depletable resource increases with cumulative extraction, and if a backstop technology exists, the user cost of the depletable resource declines to zero at the date of exhaustion. In this paper, we first present a simple method for proving this proposition, using a social planning model that determines the optimal rates both of extrac- tion of the depletable resource and of production of the backstop technology. We then present two examples that show how this method can be used to solve more difficult problems in the theory of resource economics. The first example involves learning-by-doing in the backstop sector; that is, backstop costs decline with cumulative production. The second example involves uncer- tainty of backstop costs. JN A 1976 article in the Bell Journal, Geoffrey Heal established the following important proposition concerning the role of extraction costs in determining the socially optimal price of a depletable resource. Assume the resource is available in infinite amounts, but that its extrac- tion cost rises with cumulative extraction. The resource cost is bounded from above by a so-called "backstop" technology, which provides unlim- ited amounts of the resource at a constant cost. Heal claimed that in such a world the socially optimal price for the resource must start out well above the marginal extraction cost of the depeletable resource, and move toward it as cumulative extraction grows. Equivalently, the user cost of the resource is initially high, but declines to zero at the instant of transition to the backstop technology. This result contrasts sharply with the case in which extraction costs are constant and the total stock of the depletable resource is finite. With these alternative assumptions, the socially optimal price starts out close to the cost of extraction and rises steadily above it as the resource stock is consumed. The user cost of a constant-cost depletable resource reaches its maximum at the tran- sition to the backstop. Subject classification: 131 Heal's theorem revisited, 473 optimal supply of a depletable resource. 277 Operations Research 0030-364X/85/3302-0277 $01.25 Vol. 33, No. 2, March-April 1985 (? 1985 Operations Research Society of America278 Oren and Powell Heal's statement of these results has been sharpened in several impor- tant ways by subsequent authors. Clark [1978] pointed out that the user cost need not necessarily decline to zero monotonically, in particular if the change in the marginal extraction cost with cumulative extraction is initially small. Hanson [1980] then established an important relationship between the rate of change of user costs and the curvature of the resource price path. In particular, Hanson showed that the price path must be concave as it approaches the level of backstop costs, and the user cost must be decreasing as well. However, it is perfectly possible (again, depending on the time rate of change of marginal extraction costs) for the price path to be convex and user costs to be increasing over an earlier time interval. In this paper, we will denote by "Heal's theorem" the basic result that the user cost of depletable resources goes to zero at the transition to the backstop. In Figure 1 we illustrate the differences between the constant-cost and increasing-cost formulations. If the unit cost of the depletable resource is constant at value d, and the backstop is available at a cost c > d, the socially optimal price starts out close to d and rises until it reaches c at the transition date T (Figure la). The user cost component of the price, which measures the cost to society (in excess of the extraction cost) of consuming a unit of the finite stock, rises monotonically as long as the depletable resource is being consumed. Because the stock of the deplet- able resource is finite, consuming a unit today reduces the quantity available for later consumption. This hidden cost of consumption is reflected in the user cost, and it naturally rises as the remaining stock declines toward zero. In Figure lb we illustrate the alternative case, in which the resource stock is unlimited but the marginal cost of extraction rises with cumulative extraction. As in the previous case, the price of the P P t t c c USER It el) D(St I COST d D(O) USER I ~~~~~ ~~~~COSTI T t T t a. CONSTANT COST b. RISING COST Figure 1. User costs for depletable resources.Heal's Theorem Revisited 279 resource