Exhaustible Resource Models with Uncertain Returns from Exploration Investment Department of John R. Birge Industrial & Operations Engineering The University of Michigan Ann Arbor, MI 48109 USA Technical Report 84-22

Exhaustible Resource Models with Uncertain Returns from Exploration Investment John R. Birge Department of Industrial and Operations Engineering The University of Michigan Ann Arbor, MI 48109 USA Abstract Exhaustible resource models that do not consider exploration investment typically have low values of perfect information and sometimes even optimal myopic policies. In this paper, we add exploration and capacity investment and allow the returns from exploration to be stochastic. We show that, in this model, the stochastic program solution may be quite valuable and that myopic policies are far from optimal. 1

1. Introduction Exhaustible resource models have been studied by a number of authors. Hotelling [3] initially formulated a model that demonstrated that the market price of an exhaustible resource grows exponentially as it is depleted. Nordhaus [7] introduced the idea of a "backstop" technology to this model. The result is the Hotelling-Nordhaus model in which a finite resource is used until its production cost exceeds that of the inexhaustible backstop technology. The backstop technology is then introduced and the two technologies are never used simultaneously. Manne [5] and Manne and Richels [6] use the Hotelling-Nordhaus model in their analysis of the effect of the uncertainty of the introduction date of the fast breeder reactor. They formulate a stochastic linear program and solve it to find the expected value of perfect information (EVPI). Their results indicate that the expected value of perfect information in this model is low and that, therefore, deterministic problem solutions provide close approximations to the solution of the stochastic problem. Chao [2] presents an analytical justification for the observations of Manne and Richels. He formulates a mathematical program for the Hotelling-Nordhaus model. Under certain assumptions that include a demand that is independent of price, Chao shows that a myopic policy of using the most inexpensive available technology first is optimal. He also introduces a price responsive demand function to his model and again shows that the EVPI is low. In this paper, we expand upon Chao's model by allowing exploration investment that could yield additional resource supplies. The amount of increase in the supply per unit of investment is however uncertain. We show that the EVPI and the value of the stochastic solution (VSS) (Birge [1]) can be large when this type of uncertainty is included. We give examples illustrating these observations. 2

2. The Basic Model Our results concern two measures of the effect of uncertainty in stochastic programs, the expected value of perfect information and the value of the stochastic solution. We present these measures in the context of two-stage stochastic programs with recourse. We first formulate the deterministic program Minimize 4(x,S) = cx + Min[qylWy = i + Tx, y > 0] (1) subject to Ax = b, x > 0 where the vectors c C Rn, q ~ Rn, and b ~ Rm are known, the m2-vector i is a random vector defined on the probability space (,F, F), and A, W, and T are correspondingly dimensioned known real-valued matrices. A decision vector x(i) obtained in Program 1 represents an optimal first period decision given a realization i of the random vector. If an optimal first period decision is taken for all possible realizations of the random vector, then we obtain in expected value the "wait-and-see" (WS) solution value (Madansky [4]), where WS = Eg [Min m (x,)]. x The stochastic program with recourse (Wets [8]) involves optimizing after taking the expected value. We write the value of this program as RP = Min Em [~(x,6)]. x For E(i) = i, we obtain a third value that is the expectation of the expected value (EEV) solution x(0) that is optimal in (1) for i = i. This quantity is EEV = E [c(x((),)]. The effects of uncertainty are measured by differences among WS, RP, and EEV. The expected value of perfect information represents the amount one is willing to spend in gaining information about the stochastic variables. It is calculated as EVPI = WS - RP. 3

The value of the stochastic solution, on the other hand, measures the additional value of solving the stochastic program over solving the deterministic expected value problem. We define VSS = EEV - RP. In the discussion below, we describe VSS and EVPI in the context of an exhaustible resource model originally due to Chao. Chao's basic model is a linear program to determine an optimal dynamic production schedule to minimize the present value of the cost of satisfying an increasing sequence of demand requirements over time. The demand may be satisfied by any of m-l substitutable technologies, each using one distinct finite resource, and by one backstop technology with no resource limit. The resulting linear program is m o m T min Z Z Bt ci Yit + Z Z t ki xit (2) i=l t=l i=l t=O oo00 s.t. Yit < Ri i=l,..., m; t=O m Z Yit = Dt, t=l,..., T; t=l 00 Yi,t+l = Yit + Z (6s - 6s-l) Xi,t-ss t=0,l,..., s=0 Yit ' 0, Xit > 0; t=0,l,...; i=l,...,m; where yit is the amount of period t demand, Dt, satisfied by resource i at time t, xit is the amount of resource i committed at t to be extracted later, ci is the current cost of technology i, ki is the capital cost of i, B is the discount factor, 6t is the extraction rate, and Ri is the initial availability of the resource used by technology i. It is assumed that yio and xit are known for i=l,..., n and 00 co for t=O, -1,..., an that yio = t 6 x It is also assumed that D1 < D <... t=0 < DT1 < DT Chao defines as the capital recovery factor for the standard time profile Chao defines Y as the capital recovery factor for the standard time profile 4

o00 where Y = 1/( Z Vs &s) and lets dt be the demand for new resource commitments 0 s=O where Dt = Z 6s dt_-s The result is that (1) can be rewritten as s=0 m T min ZE Z (ki + ci/ Y) t xt (3) i=l t=O s.t. Z Xit < Ri - ( s)xitx i=l=**m; t=O t=-l s=-t m Z xit = dt, t=0,...,T; i=l xit > 0, i=l,...,m; and all t. Chao uses Program 3 to derive his results on myopic solutions. He shows that the corresponding transportation problem can be solved optimally by the Northwest Corner Rule if the resource costs ki + ci/Y are arranged in increasing cost order within each period. The result leads to an expected value of perfect information of 0 because the WS solution is the same as the RP solution. It also yields a VSS of 0 because the EEV value is the same as RP when myopic solutions are optimal. Chao introduces price-responsive demands to the basic model in (3) and obtains a nonlinear programming model that does not have myopic optimal decisions. He computes an upper bound on the EVPI and shows that distant future uncertainties and low price elasticities lead to a small EVPI. In the next section, we introduce investment uncertainty into the basic model and show that this may lead to a significant EVPI and VSS. 3. A Model with Uncertain Exploration Returns We assume that Ri in Program 3 represents the amount of resource i that is known to be available at time 0. This amount can be increased by exploration investment, but the amount of the increase is uncertain. We also assume that there 5

is a capacity limit Li on the the amount of a resource which may be committed at time 0. This amount may also be increased by investment in new capacity and that return is assumed known with certainty. The stochastic linear program derived from (3) is then m m m T m K min Z (kl+ci/y)xio + Z duio + gv + Z pt(ki+ci/Y)xJt+diuJt+givJt} (4.0) i=l i=l i=l t=l i=l j=l subject to t-l t-1 xjt Ri + E ia(J a) _ Z xi(j) (4.1) it s is i s s=0 s=0 i=l,...,m; t=0,...,T; j=l,...,Kt; t-l xJ < Li + v is (4.2) s=0 i=l,...,m; t=0,...,T; j=l,...,Kt; m Z xt = dt; t=0,...,T; j=l,...,Kt; (4.3) i=l xit > 0, i=l....,m; t=0,...,T; j=l,...,Kt; (4.4) where di is the cost of one unit of exploration for resource i, uJt is the amount of exploration, gi is the cost of capital investment in resource i, vt is the amount of that investment, pJ is the probability of scenario j at time t, Kt is the number of scenarios at time t, and aJt is the return per unit of exploration for resource i under scenario j. Each scenario j is preceeded by ancestor scenarios in previous periods which are designated by a(j). The stochastic nature of Program 4 is contained only in the return on exploration investment, cJt. In general, these values may vary continuously, but the discrete formulation in (4) is used for simplicity. This program involves a stochastic technology matrix, but it may be formulated with stochastic right-hand sides by defining new variables wit, Z>O, such that 6

a(j1 - uit-l t Z=l and wit, at Li w -xa(,j) Z ait Wit - I19 t-1' P=l (5) xi < Ri - + it - j.,t-l where Ra(j) is the availability of resource i in period t-l, there are Li different Z Lj _ ~a (j) values of Ci t-l and wit < 0 for all L except for Z=qJ such that ait = it+l. Zj The upper bound on wit is sufficiently large to allow any investment level. The stochastic right-hand side problem is then formed by substituting (5), (6), and a constraint where RJ is set equal to the right-hand side of (6), for Constraint 4.1 in Program 4. In the deterministic version of (4), the investment decisions may skip from investment in one resource to another according to the values of act. This is due to the basic property of the linear program in which extreme point values correspond to investments in single resources. The solution of (4) allows for many more combinations of alternatives investment decisions and, hence, provides for hedging against other possibilities. This hedging characteristic yields a positive VSS for many cases and the value of knowing the investment return yields a positive EVPI. An example of these occurrences appear in the next section. 4. Example We consider a two period problem to demonstrate the potential effect of investment uncertainty. In this example, we consider three technologies. The first technology uses a resource in which investment return is highly variable. The second technology corresponds to a resource in which investment in additional capacity results in certain returns. The third technology is an infinitely available backstop. The data for the model is contained in Table 1. 7

Resources Res 1 Res 2 Backstop Current Cost 5.0 10.0 16.7 Initial Availability 25.0 10.0 + 00 Investment Res 1 - Good Luck Bad Luck Res 2 Periods First Second Cost 1.0 1.0 1.0 Return 1.0 0.1 1.0 Demand 15.0 25.0 Scenarios Good Luck Bad Luck Discount Factor Probability 0.5 0.5 B = 0.6 Table 1. Model Input Data The only uncertainty in this model is in the return for Resource 1 exploration investment. Resource 2 investment can be interpreted as building additional capacity. This model can be formulated as a stochastic linear program with recourse and with uncertainty in the right-hand side by using constraints as in (5) and (6). In this case, we obtain the following two-stage stochastic linear program in which x represents first period decisions and y represents second period decisions. 8

min z = 5x1 + s.t. X1 10x2 + 16.7x3 + x4 + x5 + Eg[ 3y5 + 6Y6 + 10 X2 X1 + - X1 X2 + x3 + YI +.lY3 + Y4 - Y3 - Y4 x4 )y7] (7) < 25 10 > 15 = 0 = 0 = 0 < 25 < 10 Y7 > 25, - x2 x5 + Y2 Y4 + Y5 + Y6 Y5 + Y6 + Xl,...,x5 > 0, Yl 9 099Y7 > 0, where P{( = 0) = 0.5 and P{( = 10} = 0.5. In this program, xl, x2, and x3 represent commitments of the resources, x4 and x5 are investment variables, yl and Y2 represent the net changes in resource availabilities, Y3 and y4 represent the amount of new Resource 1 availability obtained through investment, and Y5, Y6, and Y7 represent commitments in the second period. The alternatives to Program 7 are to solve deterministic models that assume good luck, bad luck, a mean value with g = = 5, or a single myopic solution. For each of these solutions, we obtain the expectation of the two period costs after using the first period solution obtained by these deterministic problems (as in finding the EEV). These values are Scenario Deterministic Value Expectation Value Good Luck 175.0 196.5 Bad Luck 200.0 200.0 Mean Myopic 185.0 215.0 200.75 215.0 9

These values can be compared to the value of the stochastic program (7), which is 192.5. We can then obtain the information values, EVPI and VSS. The expected value of perfect information is EVPI = RP - WS = 192.5 - 187.5 = 5.0. The value of the stochastic solution is VSS = EEV - RP = 200.75 - 192.5 = 8.5. The value of the stochastic solution relative to the myopic, or no investment, solution is also of interest. It is 215.0 - 192.5 = 22.5. The difference between the EVPI and VSS values demonstrates how these quantities reflect different values of uncertainty. The EVPI is lower than the VSS because the RP solution can fairly adequately hedge against either of the future outcomes. In the RP solution, there is investment in both Resource 1 and Resource 2 capacity (X4 = 10 and x5 = 4) so that no backstop usage is necessary in either scenario. The mean value solution, however, only involves investment in Resource 1 so that the backstop must be used in the bad luck scenario. This leads to a higher VSS than EVPI and shows the merit of using the stochastic program solution. Investment in two resource is unique to the stochastic program solution. Any deterministic scenario only involves investment in one resource. This again shows the utility of the stochastic program. It is able to blend the deterministic solutions so that the decision maker does not have to decide among two completely different solutions. We also note that the addition of investment has a significant effect on the value relative to the myopic solution. If no investment is allowed then the myopic solution would be optimal, and the backstop would necessarily be used to satisfy five units of demand in the second period. An exhaustive resource model with investment therefore clearly must consider future scenarios, and the solution of an equivalent stochastic program can have significant advantages over the solution of a deterministic expected value problem. 10

References [1] J.R. Birge, "The value of the stochastic solution in stochastic linear programs with fixed recourse," Mathematical Programming 24(1982) 314-325. [2] H.P. Chao, "Exhaustible resource models: the value of information," Operations Research 29(1981) 903-923. [3] H. Hotelling, "The economics of exhaustible resources," J. Political Econ. 39(1931) 173-175. [4] A. Madansky, "Inequalities for stochastic linear programming problems," Management Science 6(1960) 197-204. [5] A.S. Manne, "Waiting for the breeder," in: Review of Economic Studies Symposium, (1974), pp. 47-65. [6] A.S. Manne and R.G. Richels, "A decision analysis of the U.S. breeder reactor program," Energy 3(1978) 747-767. [7] W.D. Nordhaus, "The allocation of energy resources," Brookings Papers on Economic Activity 3(1973) 529-576. [8] R.J-B Wets, "Stochastic programs with fixed recourse: the equivalent deterministic program," SIAM Review 16(1974) 309-339. 11