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journal of environmental economics and management 35,164189 (1998)article no. EE981027

Optimal Exploration for and Exploitation of

Heterogeneous Mineral Deposits

Robert D. Cairns1

Department of Economics, McGill University, 855 Sherbrooke St. W.,

Montreal, Canada H3A 2T7

and

Nguyen Van Quyen1

Department of Economics, Faculty of Social Sciences, University of Ottawa,

Ottawa, Canada K1N 6N5

Received November 1993; revised February 1998

We model exploration for and production from a multiple-deposit, nonrenewable resource.Exploration is viewed as a Bayesian process involving learning about undiscovered reserves,

and is undertaken both to improve the quality of and to replenish known reserves. Known

reserves are not physically defined, but depend on the realizations of discrete episodes of

exploration elsewhere and on decisions at each instant about the margin between ore to be

mined and low-grade material to be rejected. The appropriate measure of resource scarcity,

the sustainable level of the contribution of the resource, and the treatment of depletion in the

national accounts are considered. 1998 Academic Press

Key Words: exploration; extraction; learning; microfoundations; scarcity.

1. INTRODUCTION

Optimal mineral exploration is a multifaceted problem, and economists havetaken a number of perspectives on it. For want of space, we cannot do justiceto a copious literature; Cairns [5]provides a detailed review. Among the informa-tive contributions are those of Peterson [17], Pindyck [18, 19], Devarajan and Fisher

[7],A rrow and Chang [2],Swierzbinski and Mendelsohn [24],and Quyen [20, 21].Optimal management of mineral wealth typically requires trade-offs among three

types of decision, namely, (1) when to explore for new deposits, (2) when to developpreviously known but inactive deposits, and (3) the margins of exploitation. Tomodel these trade-offs, we stress that deposits are distinct. Mine managers and anexploration unit are perceived as having distinct decision problems.

InSection 2,we characterize the equilibrium extraction profile when there is noexploration. We find that some deposits will not be brought onstream immediately,

1 We thank J on Conrad, Pierre Lasserre, Yang Zhao and three referees for helpful comments.

164

0095-0696/98 $25.00Copyright 1998 by Academic Press

All rights of reproduction in any form reserved.



mineral exploration and exploitation 165

but will be held on the shelf for more propitious conditions (in the model, ahigher price). Trocki [26]argues that this has been an important feature of ironand copper supply in this century. Also, we find that mines of varying quality willbe exploited simultaneously, clearly a feature of reality, and we revise notions about

the optimal order of exhaustion of existing deposits.Section 3 introduces exploration as a learning process. We provide three particu-

lar examples before solving the general problem of exploration and extraction. Wediscuss how the quality of existing deposits affects the optimal exploration program,and how the exploitation model ofSection 2is nested within the problem of op-timal exploration and extraction. Success and failure in exploration feed back intothe mining problem through the choice of cut-off grades.

Our results help to cast further light on some of the main issues of nonrenewable-

resource economics, including the role of the r-percent rule, the definition of mea-sures of scarcity and of depletion, and the role of human action through resourcemanagement. InSection 4 we interpret the conditions derived inSections 2 and3for (i) the choice of cut-off grade and quantity mined at a given time and (ii) thetiming of exploration. We find that price is the appropriate measure of resourcescarcity, and we eliminate other proposed measures. An expression identified asHotellings [12]rule for the rate of growth of aggregate rent is presented. But ag-gregate rent cannot be interpreted in terms of aggregate magnitudes. Moreover,there is no single shadow price which obeys Hotellings rule. Rather, there are r-percent rules explicit to each producing deposit, and to the unexplored reserves(which collectively form a nonrenewable resource) as well. An implication is thatexisting empirical work on Hotellings rule may be misspecified.

Additions to known reserves are often considered to be an appreciation of nat-ural capital, for the national accounts and other purposes. But the appreciation of

known reserves comes at the expense of depleting unexplored resources. We findthat an unbiased estimate of the gain in welfare is the cost of exploration itself.However, actual realizations of exploration programs will, with probability 1, lead

to changes in price, cut-off grades, the expected value of the program, and all futureanticipations, and necessitate revaluations as unexpected capital gains and losses.The ability to vary cut-off grades gives the decision maker an additional control inoptimizing resource use.

A detailed interpretation of the results of the model is presented in Section 4.

2. EXTRACTION

The most suggestive approach to our problem entails first discussing the sub-problem of exploiting mineral deposits after the region is fully explored. Thenwe discuss exploration. However, as the solution to the subproblem turns out tocharacterize optimal extraction between episodes of exploration, the approach hastwo benefits: (1) Our model can also be discussed in the context of the litera-ture on exploiting known deposits. (2) We illustrate how the extraction problemis nested within the exploration problem, which is itself a nonrenewable-resource

problem.Our argument, though stylized, is intricate. Many variables are introduced in this

and the next section. For the readers convenience, these variables are summarizedin a table in Appendix II.



166 cairns and quyen

2.1. Mineral Deposits

We assume that all mineral deposits have identical geometrical dimensions. Theirqualities, however, may be different. Following Cairns [4]and Krautkraemer[13],

we assume that a deposit takes the form of a vertical cylinder of fixed length andradius, both normalized to 1. The base of the cylinder is at depth 1, so that theorebody has no overburden. The grade-distribution function of a deposit of qualityx, as a function of the distance from the axis, is given by xg, 0 1.The functiong is assumed to be known and to be common to all deposits; it isstrictly decreasing and continuously differentiable, with g0 1 and g1 =0. Thegrade-distribution functions imply that deposits may differ through a multiplicativefactor representing their qualities. The quality, x 0 1, of a deposit is not known

before the deposit is found, but is revealed upon discovery.A deposit is exploited by excavating downward. At each instantta disc of oreof thicknesshtand radiust (and hence with cut-off gradexgt) is mined andprocessed. The volume of this disc is vt=

2tht, and its metal content is xmtht,

where mt =t

0 2g d.

We assume that, once excavation has proceeded beyond any given depth, it isnot possible to return to mine the remaining mineral at that depth. This mayappear to be a strong assumption; occasionally, mining firms can return to re-work mines which have already been exploited. Technological change usuallyoccurs, however, to enable this change (cf. Boldt[3],p. 137ff.). We offer the fol-lowing explanation for our simplification. First, it is assumed in this article thatthere is no technological progress. Second, when royalty rates in British Columbiawere raised substantially in the 1970s, base-metal mining firms responded withcomplaints that huge quantities of close-to-marginal ore would be lost foreverbecause it would later be uneconomic to return to extract ore rendered sub-marginal by the policy. This argument suggests that, most plausibly, the assumptionmay be viewed as the limiting case of a very high cost associated with return-

ing to exploit part of a mine which has already been worked. It is consistentwith our assumption, made in the following text, of a finite choke price andalso with the practice of backfilling previously worked parts of an undergroundmine.

The total cost of mining and processing a disc of ore of volume v at depth L willbe denoted bycv L. Because xg0 < 1 and cv L includes processing costs,cv 0> 0. We shall call cv L the extraction-cost function. L etDi be the differ-ential operator with respect to the ith argument of the function involved. We assume

that cv L has the following properties. (1) For all v 0 and for all L 0 1,cv Lis well defined and continuously differentiable. (2) For any given L, the func-tion cv Lis convex and strictly increasing in v, and satisfies c0 L =0 (costs arenil if extraction is not occurring) andD1c0 L>0. (3) For any v0,D1cv L,and for v >0,D2cv L are strictly increasing in L.

Sometimes, to derive sharper results, we shall assume the following special formof the extraction-cost function,

cv L =vbL b0> 0 (1)

In this specification, we interpretbL as the cost of mining and processing a unitvolume of ore at depth L.




2.2. Exploitation of Several Proven Reserves

L etpq be the inverse demand function for the metal in question. We assumethat pqis continuously differentiable and strictly decreasing until it reaches zero.

Furthermore, the choke pricep0 is positive and finite. L et uq =q

0 py dydenote the social welfare derived from consuming the quantity of metal q. Futuresocial welfare is discounted at rate r >0.

Suppose that at any time s 0 there are n proven reserves (deposits) havingqualities x1 xn, and that these reserves have already been exploited to depthsL1s Lns, respectively. A planner aims to exploit these deposits from the ini-tial condition Ls sx = L1s Lns s x1 xnto maximize discounted socialwelfare. (Note that we use bold type to represent a vector.) L etLit be the mining

depth at time t in the ith reserve.The problem to be solved can be formally stated as follows. GivenL s sx, ateach time ts and for each idetermine a disc of ore hit itto be extracted fromthe ith deposit so as to maximize the net present value of future welfare, i.e., to

max

sert

u

ni=1

ximithit

ni=1

cvit Lit

dt=nLs sx (2)

subject to dLit/dt= hit; hit 0; 0 it 1; 0 Lis Lit 1. We shall assumethat (2) has a piecewise- and a right-continuous solution, t hit it

ni=

1. For

each i, let Lit denote the depth at time t in the ith reserve under the solution, andlet vit=

2it

hit; qit=ximithit; and qt=n

i=1qit.For each i, let Ti be defined as follows. If the ith reserve is exploited indefinitely,

setTi = . Otherwise, letTi be the exact time mining is terminated at the ith re-serve. We shall call T =maxT1 T nthe time the exploitation of the nreservesis terminated.

To solve (2), we first define the (present-value) Hamiltonian,

HLh t =ert

u n

i=1ximihi

ni=1

cvi Li

+

ni=1

ihi

Next, let

it=DinL t tx i= 1 n

be the shadow price of mining depth in the ith reserve along the optimal trajectory.Because extraction costs increase with depth, and because of the assumption thatonce excavation has proceeded beyond a given depth it is not possible to return laterto that depth to mine the remaining mineral, we must have D

in 0: the shadow

price of depth is nonpositive. Furthermore, it evolves through time according tothe following adjoint equation:

ditdt

= DiHL thtt t t

= ertD2c

vit Lit

(3)

We remark that (3) is the r-percent rule at a deposit, modified for the effect on costof increasing depth. Although it will prove more direct to present our analysis inpresent-value form, the rule is often written in terms of thecurrent shadow value,

it=

itert. We can easily rewrite (3) as

ditdt

rit=D2c

vit Lit

(4)

to capture the intuition of the r-percent rule.




Because c0 L = 0 for all L, we have dit/dt= 0 when vit=0. Also, becauseD2cv Lis strictly increasing in Lfor any given v >0, we must have that dit/dt >0 when vit>0. In short, the shadow price of mining depth in any deposit is strictlyincreasing when it is in production and remains constant when it is not. Because

it0, it is clear that ifitreaches 0 at some time , then all mining from the ithdeposit will cease at.

At any time t when theith reserve is in production, we must have both thathit >0 and that it >0. Invoking the maximum principle, we obtain the followingfirst-order conditions,

xim

it

p

qt

2itD1c

vit Lit

+ itert =0 (5)

xigitpqt D1cvit Lit= 0 (6)If theith deposit is not in production at timet, then either hit = 0 or it = 0. Ifit < 0, then the Hamiltonian is maximized only with hit = 0. Ifit = 0, then asargued before, mining at this reserve will never be undertaken after time t. Hence,in this case, hit = 0 is also an optimal control. Therefore, when the ith provenreserve is not exploited, hit = 0. Now, when hit = 0, it is indeterminate and cantake on any value between 0 and 1. To remove the indeterminacy, recall condition(6) and consider the following function:it xigitpqt D1c0 Lit. It is

strictly decreasing and is equal to D1c0 Lit < 0 at it = 1. If xig0pqt D1c0 Lit > 0, then this function vanishes at a unique value ofit. In this case,choose itsuch that xigitpqt D1c0 Lit =0. Ifxig0pqt D1c0 Lit 0, then choose it=0.

Now, for each i= 1 n, we have

hit

xim

it

p

qt

2itD1c

vit Lit

+ ertit

= 0

Summing this expression over i= 1 n, we obtain

qtp

qt

n

i=1

vitD1c

vit Lit

+ ertn

i=1

ithit=0 (7)

Furthermore, the following result comes from the HamiltonJ acobiBellman equa-tion,

ertuqt n

i=1

cvit Lit +n

i=1

ithit= Dn+1n

L t tx=rertn

L t 0x

(8)

In (8), the differentiation of n in the middle expression is with respect to t,then+1th argument. The second equality comes from the obvious result thatnL t tx =e

rtnL t 0x. Equation (8) expresses the famous result (see Weitz-man [27]) that, along the optimal path, the Hamiltonian is equal to the interest onthe value of the objective function.

Using (7), we can rewrite (8) as

u

qt

qtp

qt

+n

i=1

vitD1c

vit Lit

c

vit Lit

= r n

L t 0x

(9)




Some observations on (9) are in order. First, note that uqt qtpqt 0, withequality if and only ifqt=0. Second, note that for each iwe havevitD1cvit Lit cvitLit 0 becausecv L is convex and strictly increasing in v andc0 L =0. Hence, on the left-hand side of (9), each expression inside a pair of square

brackets is nonnegative. Third, the right-hand side of (9) is strictly decreasing overthe interval s T, and nLT 0x = 0. Therefore, qT = 0; i.e., the pricepqtrises to the choke price p0 at the time all extraction is terminated.

Although the price must ultimately rise to the choke price, in a simpler modelL evhari and L iviatan[14]argued that the price path may not be monotonic. Theintuition is that, while the right-hand side of (9) is declining through time, the terminvolving the summation sign might fall sufficiently fast to make uqt qtpqtrise temporarily. Nevertheless, if the extraction-cost function assumes the form rep-

resented in Eq. (1), thenvD1cv L = cv L for all v 0 and for all L 0 1,and (9) reduces to

u

qt

qtp

qt

= r nL t 0x

(10)

In this case, it follows directly from (10) that qt is strictly decreasing to 0 as tincreases from s toT.

We summarize the results just derived in the following proposition.

Proposition 1. The price of metal rises to the choke price when excavation ac-tivities are winding down; i.e., pqt p0 when tT. Price need not be mono-tonically increasing. But the rise to p0 is monotonic if the extraction-cost function

assumes the special functional form given by (1).

Proposition 1 makes clear that Swierzbinski and Mendelsohn [24] and others findthat price is strictly increasing because they assume constant unit costs as in Eq. (1).

Now suppose that theith deposit is in production at time t. If one maintainsthe thickness of the disc of ore excavated at hit, but increases the cut-off radius

slightly, from it to it+dit, then the increase in the amount of mineral pro-duced isdqit = xi2itgithitdit, and the increase in extraction cost is dcit =2ithitD1c

2it

hit Litdit. Thus, at the optimal cut-off radius, it, the marginalcost of metal is

dcit/dqit= D1cvit Lit

xigit =pqt (11)

The second equality is due to condition (6). On the other hand, if one maintains

the radius of the disc of ore at it but increases its thickness slightly from hit tohit+dhit, then the increase in its metal content is dqit = ximit dhit, and theincrease in discounted extraction cost isert dcit = e

rt2itD1c2it

hit Lit dhit.Furthermore, the variation in the user cost of mining depth is itdhit. Therefore,the full discounted marginal cost incurred in mining one more unit by increasingthe thickness of the disc of ore excavated is

ertdcitdqit

= ert2itD1cvit Lit it

ximit =ertpqt (12)

In (12), the second equality is due to condition (5).Therefore, there are two margins of exploitation, an intensive margin character-

ized by (6) and an extensive margin characterized by (5). At the extensive margin,




two kinds of cost are involved, extraction cost and user cost of mining depth. Thelatter captures the depletion of the reserve. As noted previously, condition (3) is amodified r-percent rule for mining depth, applying specifically at reserve i. To gainmore insight into that rule, we rewrite (12) as follows,

ert

p

qt

2itD1cvit Lit

ximit

=

itximit

(13)

In (13), the left-hand side represents the discounted net benefit contributed by aunit of metal at the extensive margin. The right-hand side, in general, differs amongthen reserves because of their differences in quality and depth. Indeed, the gapsbetween price and the marginal costs of producing metal at the extensive marginsare not in general equalized across the deposits that are in production at any instant:we can rewrite (13) using (11) to show that

itert

ximit=pqt

2itD1cvit Lit

ximit =p

qt

12itgit

mit

(14)

which depends on the cut-off grade it and hence the quality xi and the depth Lit.We have just proved the following.

Proposition 2. (i) At any instant, the costs of producing metal at the intensivemargins are identical for all deposits being exploited.(ii)Costs at the extensive marginsvary across producing deposits. A modified r-percent rule holds at the extensive margin,

but is particular to each deposit.(iii)Full marginal cost at each margin is equal to theprice of metal.

At any instant that the ith deposit is being exploited, (6) holds, and so

p

qt

= D

1cv

it L

it

xigit >

D1

c0 Lit

xig0

D1

c0 Lis

xig0 (15)

In (15), the strict inequality follows from the assumptions thatcv L is convex inv and that g is strictly decreasing in , and the second inequality follows fromthe assumption thatD1cv L is strictly increasing in L. Hence, for any t s T,

p0> p

qt

>

D1c0 Lis

xig0 (16)

if the ith proven reserve is exploited at time t. We can now prove the followingproposition. (Proofs can be found in Appendix I.)

Proposition 3. Consider a reserve, i, with initial mining depth satisfyingLis D1c0 Lis/p0g0.

For the remainder of this section, we assume that, given the choke pricep0,all n reserves are of at least the minimal quality that they will be brought into

production some time afters, i.e., that

xi > D1c0 Lis

p0g0 for all i (17)




We nowexploit Proposition 3 to demonstratehowthe qualityand the initial depthof a deposit affect its pattern of exploitation relative to the other proven reserves.First, letx and x be defined as follows,

x = D1c0 0p0g0

x = D1c0 1

p0g0

(18)

These are the minimal qualities for profitable exploitation of a reserve at depths 0and 1, respectively. It is clear that 0 < x < x. Furthermore, for any proven reserveof quality xwith initial mining depth equal to zero, the followingresults follow from

Proposition 3.

Corollary 1. Let x and x be defined by (18). Consider a deposit of quality xand zero initial mining depth. (i) If x x, then this deposit will never be exploited.(ii)Ifx > x, then the reserve will be mined all the way to depth1.(iii)Ifx < x x,then it will be mined down to the depth L defined implicitly by D1c0 L/xg0 =p0. This cut-off mining depth is a strictly increasing function ofxin x x.

Intuitively, we expect that reserves with lower extraction costs should be broughtinto production before those with higher extraction costs. It is well known thatwhen each proven reserve is of uniform quality and marginal extraction costs areconstant within each reserve, then a reserve with lower marginal extraction costshould beexploited and exhausted before a reserve with higher marginal extractioncost is brought into production. In our model, there is no single measure of marginalextraction cost for any deposit. Cost at the intensive margin depends only on thequality and the current mining depth. Cost at the extensive margin depends onthose same variablesas well as the qualities and current mining depths of the otherreserves. We provide the following partial answer to the question concerning theorder in which production in each reserve is started and ended.

Proposition 4. (i) Consider any two proven reserves, i and j, with qualities sat-isfyingx < xi < xj and initial depths satisfying (17). Ifxi < x

then it is optimal tomine the jth reserve to a greater depth than the ith reserve. If xj < x

, both reservesare exploited simultaneously, at least near the timeTwhen all mining is terminated. Ifxj> x

, then Tj< Ti.

(ii) Suppose that the extraction-cost functioncv L assumes the form given by(1). If the quality of a reserve is sufficiently high and its initial depth is shallow, thenit will be exploited immediately. Production from the other reserves may take place

only after the metal price has risen sufficiently. On the other hand, a reserve having acombination of low quality and high initial depth will be shelved temporarily.

Therefore, some mineralization (of lower quality than x) is never exploited at all.It maybe that some reserves (of lowqualitybut higher than x) areheld on the shelf.

Other reserves are exploited simultaneously. Deposits of high quality (exceedingx

)are exhausted (in the sense thatLT =1) in order of quality. But all deposits withquality between x and x are mined up to time Tand then closed with their depthsnot having reached L= 1.




3. EXPLORATION

Using geological knowledge of a given region, we partition it intoNcells in thebelief that in each cell there is a single potential mineral deposit.

3.1. The Exploration Process

The metal content of a mineral deposit of qualityx is 2x1

0g d, which

is clearly proportional tox. Hence, we can identify the content of deposit i withits (random) quality, xi. We assume that exploration at any cell can be carriedout instantaneously and yields perfect information about the value ofxi. Also, the

potentials of the N cells are identical in the following sense. First, it costs thesame amount, K >0, to explore each cell. Second, x1 xN are independent andidentically distributed random variables. Because all the potentials are identical, wecan assume that the cells are explored sequentially.

Consider a representative cell. We postulate a strictly increasing function 1x , with limx0

1x = 0 and limx1 1x = . The random variable

, which is an index of the quality, x, of the deposit in the cell, is taken to beexponentially distributed with parameter;2 its density function is then given by

f =e >0

To model learning in the exploration process, we apply Bayesian statistical de-cision theory by postulating a prior density for , then revising it repeatedly viaBayess formula, using the most recent data on discoveries as the exploration pro-gram unfolds. We assume that initially has a gamma density with parameters >0 and >0, and we write the conditional (on and ) density of as

g = e/1

>0

where =

0 u1eu du for >0, and = 1 1 for >1.

The choice of a gamma density for is made to keep the exposition manageable.In the statistical literature, the gamma density is known as aconjugate priorof theexponential process. Usinga conjugate prior allows one to begin with a certain form

of the prior and to end up with a posterior of the same functional form, after theparameters have been updated by the sample information. Indeed, let

f =

0fg d

2The function 1 takes the actual distribution of x 0 1 onto the range of so that we can

apply statistical decision theory to . We assume that the distribution of involves a single parameter. It

would have been possible to use a less restrictive distribution, with more unknown parameters, such as

the lognormal which has two. However, the conjugate distribution would have a much more complicated

form. (See Raiffa and Schlaifer[22],p. 55.) Manipulations would be more difficult, but would not yield

greater economic insight.




be the marginal density of, given the parameters and that characterize theprior density of. The conditional density of, given , is

gf

f

= e/1e

f

=g+1 + /

f

+

+1

(19)

i.e., it is proportional to the gamma density with parameters 1= + 1 and1= + . Integrating (19) with respect to from 0 to, we obtain

/

f

+

+1

(20)

Using (20) in (19), we see that g11 is the posterior density of, given . It alsofollows from (20) that the marginal density of, given the parameters and ofthe prior gamma density of, is

f =

+

+1=1 (21)

Hence, can be characterized by the marginal probability mass function, f,in (21). Note that this marginal probability mass function is indexed by the doublesubscript,, that characterizes the prior density of.

3.2. Exploration and Learning

Now suppose that cell 1 is explored and a deposit with quality index 1is discov-ered. There remain N 1 unexplored cells. The uncertain qualityof a representative

unexplored cell is now believed to be exponential with parameter , which in turnhas a gamma density with updated parameters 1= + 1 and 1= + 1. Alter-natively, the uncertain quality of a representative potential mineral deposit can becharacterized using the marginal probability mass function,f11, whose explicitform is given by (21). This revision process is repeated each time a cell is exploredand comes to an end either when all the cells have been explored or when it isdecided that exploration will be discontinued. If we interpret as the cumulativemineral discovery in some former exploration region after cells have been ex-

plored, then at any point in time the cumulative discovery and the number of cellsalready explored embody all the learning from the exploration process. Indeed, ac-cording to this interpretation, immediately after the new cell has been explored, thenew cumulative discovery is 1and the number of cells explored is 1.

The distribution function associated with the density function f is

F =

0fy dy=1

+

(22)

For any given > 0, and for any > 0, the function F is strictlyincreasing, and F 1 as . Moreover, for any given >0, and for any >0, the function F is strictly decreasing, and F 1 as 0.

We summarize the results obtained in the following theorem.




Theorem 1. The family of distribution functionsF is stochastically decreasingin and is stochastically increasing in . More precisely, for each given >0,

(i) F is strictly increasing for each given and F 1 as

;(ii) F is strictly decreasing for each given and F 1as

0.

Theorem 1 plays a fundamental role in our analysis. Its economic content canbe explained as follows. Part (i) asserts that for any given,F increases as increases; i.e., as increases, the chance that the index of the uncertain quality ofa potential mineral deposit exceeds decreases. Ceteris paribus, the higher is pastexploratory effort (for a given level of discovery), the lower the expected quality of

the remaining potential deposits. In the extreme case that is large, Fwill beclose to 1 for any given; i.e., the quality of a potential mineral deposit will tendto be very low. Part (ii) asserts that for any given level of exploratory activity, ,the higher is , the cumulated discovery in the past, the richer will be the expectedquality of the remaining potential deposits. Furthermore, becauseF 1 as 0, the qualities of the remaining potential deposits will likely be poor if thegiven cumulative discovery has been low.3

To see how Theorem 1 is applied, suppose that we explore cell 1 and we find

a deposit with index of quality 1. At this point, we update the initial distributionfunction, F, to F11, where 1= + 1 and 1= + 1; F11 nowcapturesthe quality index of the N 1 remaining unexplored cells. Whether F11 is morefavorable thanF depends on the value of1. If1 0 (the exploration of cell 1resultsin a failure), then F11 is unambiguously less favorable than F by Theorem1(ii). On the other hand, if1 is large, then by Theorem 1(i), the discovery effectcan more than offset the negative effect resulting from the increase in the numberof cells explored, and renderF11 more favorable than F.

Before we attack the general case of exploration with extraction, let us considerthree special, illustrative situations.

3.3. One Unexplored Cell, No Proven Reserve

Suppose that we begin at time zero with no proven reserve, but there is theopportunity of exploring for a single potential reserve. L et be the index of qualityof the deposit. Once the deposit has been explored its quality is revealed to be x=. By Corollary 1, this newly discovered reserve will be exploited only ifx > x.

The exploitation of such a reserve yields a discounted net welfare of 10 0 x,where 10 0 x =0 ifx x. Before this deposit is explored, its expected value,in terms of discounted social welfare net of extraction (but not exploration) costs,

3At any time, the information about remainingunexplored cells is assumed to be identical; i.e., there is

no geological knowledge that allows distinguishingamongcells. Particularly within an oil play, exploration

may systematically turn up larger reserves first. See Cairns[5] for a discussion of the literature on

this problem. In a disaggregated model, modeling this feature would require modelinganother type of

learning, moving beyond thead hoc assumption hitherto made in the literature, that quality declines or

cost increases with cumulative exploration in some known, unchanging and exogenously given way. We

leave the modeling of this type of learning to future research.




is given by

V1=

010 0 f d (23)

In this notation,Vnis the discounted value ofn unexplored cells when there is no

proven reserve. We have the following result.

Proposition 5. For any given, V1 is strictly decreasing in and for any , V1

is strictly increasing in . That is, ceteris paribus, the higher is the past cumulativediscovery, the higher is the value of the potential mineral deposit. Also,ceteris paribus,the higher is the number of cells explored in the past, the lower is the value of the

potential deposit.

The decision whether or not to explore for the potential reserve depends on itsexpected value relative to exploration costs. We have the following obvious result.

Corollary 2. IfV1 > K, then one should search for the mineral deposit. Other-wise, no exploratory activity should be undertaken.

3.4. One Unexplored Cell, One Proven Reserve

Suppose that we begin at time s0 with a proven reserve of quality x1= 1

and that this reserve has already been exploited to depth Lis 0 1. Suppose alsothat there is a single unexplored cell. The index of quality of the contained depositis 2.

The problem we face can be stated formally as follows. Given Lis, choose anexploration date, , and an extraction program,h1t 1t s t < at the provenreserve, to

max

sertux1m1th1t c

21th1t L1tdt

erK+

02L1 0 x1 f d

=1 1L1s s x1 (24)

subject todL1t/dt=h1t;h1t0; 0 1t1; 0 L1s L1t1, for all t s .4

In (24), the first integral represents the net discounted social welfare derivedfrom exploitingthe proven reserve prior to exploring. Thesecond term, erK, isthediscounted exploration cost incurred at time . Immediately after time , two provenreserves are available for exploitation, the original reserve, which has been mined to

depth L1, and the newly discovered reserve, which is of revealed qualityx2= 2and has current mining depth L2 =0. The discounted social welfare derived fromexploiting these two proven reserves from the initial condition L1 L2 x1 x2is2L1 L2 x1 x2. The second integral is thus the present value of the expectedsocial welfare after time .

In the optimal-value function defined in (24), the first superscript refers to thenumber of proven reserves currently available, and the second superscript refersto the number of unexplored cells. The uncertainty concerning the quality of the

potential reserve is represented by the double subscript, . In general, if there

4Weitzman [28] provides a quite intuitive perspective on the types of condition governing the optimal

times of exploitation, exploration, etc.




aren proven reserves and n potential reserves, then the appropriate notation is

n n

Ls sx, withL s andx denoting vectors of the initial depths and qualities ofthe n proven reserves.

L et be the optimal exploration date and let h1t 1t s t < , be the optimal

program for exploiting the proven reserve prior to . Also, for eacht s, letL1tbe the mining depth at time t in the proven reserve under that program.

First, we must establish the following result.

Lemma 1. IfK < V1, where V1 is defined by (23), then it is optimal to explore

for the potential reserve at some timets. IfKV1, then one should never explorebut should concentrate on producing from the proven reserve.

For the remainder of this section, we assume that V1> K: it is optimal to explore

for the potential reserve. We also assume that the extraction-cost function cv Lhas the form represented in (1).If the potential reserve is explored immediately, then = s, and L1 =L1s.If the proven reserve is exploited for some time before exploration, then > s

and L1 > L1s. For eacht s, letit=D11 1Lit t x1 be the shadow price

of mining depth at time t in the proven reserve. As before,1t0 and obeys thefollowing adjoint equation,

d1t

dt = D1HL1th1t 1t 1t t= ertD2cv1t L1t (25)

wherev1t=21th1t and H is the Hamiltonian defined by

HL1 h1 1 1 t =ert

u

x1m1h1

cv1 L1 + 1h1

Invoking the maximum principle, we can assert that for any t s, h1t and 1tsatisfy the following first-order conditions,

x1m

1t

p

q

21tD1c

v1t L1t

+ ert1t=0 (26)

x1g

1t

p

q

D1c

v1t L1t

= 0 (27)

where qt = q1t = x1m1th1t is metal output at timet. In the current context ofone proven reserve and one potential reserve, (26) and (27) are the counterpartsof (5) and (6) previously derived, while (25) is the counterpart of (3). The analysisof (26), (27), and (25) can, then, be conducted as in Section 2.This observationindicates that the extraction problem, the typical problem studied in nonrenewable-resource economics, is nested within the exploration problem. Also, because there isa limited number of cells, the exploration problem itself is a nonrenewable-resource

problem. In particular, the following version of (10) holds for any t s,

w

qt

u

qt

qtp

qt

= r 1 1

L1t 0 x1

(28)

Because 1 1L1t 0 x1 V

1 Kand because wqis strictly increasing in q 0,

it follows directly from (28) that

qtw1rV1 K> 0 s t < (29)

It then follows from (27) and (29) that

D1c0 L1t

x1g0 0 is defined in (29),then exploration takes place immediately. That is, if a combination of mining depth and

quality makes the initial marginal cost of production at the intensive marginwhen thismargin is close to the axis of the proven reservesufficiently high, then it is optimal to

search for the potential reserve immediately.

Proposition 6 provides a loose bound onx1 L1 for there to be immediate ex-ploration (= s); there may be values ofx1 L1for which D1c0 L1t/x1g0 s), as follows.

Corollary 3. It is not optimal to search immediately for the potential reserve if

the quality of the proven reserve is high and its current mining depth is shallow.

Now as t tends to from the left, let q =limt qt. Then by (28),

w

q

= r 1 1

L1 0 x1

= r

0

2

L1 0 0 x1

f d K

(30)

Immediately after exploration has taken place, there are two proven reserves toexploit and, given a newly discovered reserve of quality x2, the social welfare dis-counted to time is updated to2L

1 0 0 x

1 x

2. The solution to this problem

was discussed in Section 2,and a version of (10) holds. L et q+ =limt qt. Then

w

q+

= r 2

L1 0 0 x1 x2

(31)

It is evident fromthe second equality in (30) that 1 1L1 0 x1<

2L1 0 0

x1 x2 K/rfor at least one value ofx2. Indeed, 1 1L1s s x1is continuous at .

(See the definition of1 1 in (24).) Furthermore,

2L1 0 0 x1 x2 is increasing

and continuous in x2

. Thus, by (30) there is an x0

2

> 0 such that

2

L1 0 0 x1 x2

< 1 1

L1 0 x1

K

r ifx2< x

02

=1 1

L1 0 x1

K

r ifx2= x

02

1 1

L1 0 x1

K

r ifx2> x

02

(32)

By (30) and (31), we assert thatx02 is that value ofx2 which equates the expected

(before ) and the updated (after ) values of the objective function, so that

1 1

L1 0 x1

=

0

2

L1 0 0 x1

f d K

=2

L1 0 0 x1 x02

K

and, when x2= x02,

w

q

= w

q+

(33)

It follows from (33) that ifx2> x02 then q+ > q and ifx2< x

02 then q+ < q .

We have just demonstrated the following proposition.




Proposition 7. Suppose that > s; i.e., exploration is not carried out immediately.Then the expected value of the objective function, as updated, is continuous at the

optimal exploration date. Still, with probability 1, the metal price experiences a jumpat the exploration date. More precisely, if the quality of the newly discovered reserve

exceeds the critical value x02 defined in (32), then the price takes a downward jumpat time , and the larger the difference x2x

02, the larger the jump. If x2< x

02, then

the price takes an upward jump and the size of the jump increases with the differencex02 x2.

Applying Proposition 4(ii), we obtain the following result.

Corollary 4. (i)If the quality of the potential reserve is sufficiently high, then thisreserve will be brought into production immediately after its discovery. More precisely, if

p

w1

r2

L1 0 0 x1 x2

< D1c0 L1

x1g0

then production from the original proven reserve should be suspended and all produc-tion should be concentrated at the newly discovered reserve.

(ii) If x2 is low but still greater than the cut-off quality (x), then the newly

discovered reserve will be shelved temporarily. It will be brought into production afterthe price of metal has risen sufficiently. This will occur if

p

w1

r2

L1 0 0 x1 x2

< D1c0 0

x2g0 < p0

3.5. Two Unexplored Cells, No Proven Reserve

Suppose that we begin at time zero with no proven reserve, but have an explo-ration region of two cells. Cell 1 is explored first and a mineral deposit of quality x1is discovered. Immediately after the discovery of this reserve, we have one provenreserve, and one potential reserve in cell 2. Before any exploration is carried out,the density function ofi i = 1 2, is fi, as represented in (21). After cell 1has been explored, 1is known and 2is then believed to have the density functionf112, where 1= + 1 and1= +1. Givenx1, the expected discounted

utility is 1 111

0 0 x1, as discussed in the previous subsection. L et

V2 =

0

1 1+1+1

0 0 1f1 d1

As defined, V2 gives the expected discounted utilitygross of the cost of exploringcell 1under the decision to explore cell 1 immediately. The following result isevident.

Proposition 8. If V2 > K, then exploration of cell 1 should be undertaken im-mediately. Otherwise, the exploration region should be abandoned.

Now observe that the function1V1+1 +1

is strictly increasing by Proposi-tion 5. To avoid the degenerate case that the second cell will not be explored no

matter how largex1 is, we assume that lim1 V1+1 + > K, and we define1implicitly by

V+1 +1 =K (34)




Then,1is the critical index of quality such that if the first cell is explored and theindex of quality 1of the discovered reserve is found to be no greater than

1, then

cell 2 will be abandoned. It is clear that,ceteris paribus, 1 is decreasing in . Inparticular, ifV1+1 > K, then

1= 0; i.e., if , the past cumulative discovery, is

not too low and if the quality of the reserve found in cell 1 falls short of the criticalvaluex mentioned in Corollary 1, then cell 2 will be explored immediately aftercell 1.

Next, note that the argument used to prove Corollary 3 can be repeated verbatimto assert that ifx1is high enough, then exploration will be temporarily suspendedafter cell 1 has been explored. Metal production will come exclusively from thereserve discovered in cell 1, and cell 2 will be explored only after mining in thisreserve has reached a certain depth. Keeping these facts in mind, let us define

1=min

y

y1and if1y then cell 2 will not be

searched immediately after cell 1 has been explored.

(35)

The following proposition is now evident.

Proposition 9. Let 1and 1be defined by (34) and (35). Suppose that cell 1 is

explored and a reserve having index of quality 1 is discovered.

(i) If1< 1, then cell 2 will be abandoned.

(ii) If 1 1, then exploration will be temporarily suspended. The proven re-serve in cell 1 will be exploited for some time before exploration of cell 2 is carriedout.

(iii) If11< 1, then cell 2 will be explored immediately after cell 1, and in

a right neighborhood of time s, we have two reserves to exploit.

Now note that1 1

+1 +1x10 0 x1 is increasing inx1. There are two reasons

for this result. First, a higher value ofx1means a higher quality for the reserve just

found in cell 1. Second, the distribution function that characterizes the uncertainquality of the potential reserve in cell 2 is F+1 +12, which is stochasticallyincreasing in1by Theorem 1, and hence inx1. By the same proposition,F1is stochastically increasing in . The following two-cell version of Proposition 5 isevident.

Proposition 10. For any given, V2 is decreasing in and for any given , V2

is increasing in . That is, ceteris paribus, the higher is the past cumulative discovery,the higher is the potential of the two-cell region. Also, ceteris paribus, the higher is the

number of cells explored in the past, the lower is the potential of the two-cell explorationregion.

Proposition 10 deals with the influence of past exploration outcomes (from areference region, say) on the potential of a two-cell region. We now consider howthe net potential of an exploration region varies in terms of the number of cellsit contains. Suppose, then, that we begin at time zero with no proven reserve, buthave the opportunity of exploring for mineral reserves in an N-cell region. Then wehave the following.

Proposition 11. Suppose that we begin at time zero with an N-cell explorationregion but we have no proven reserve. For N = 1, the expected value in terms ofdiscounted welfare, net of exploration, and extraction costs, is V1K. For N = 2,




the corresponding value is V2KV1K. Furthermore, if we consider the value

of the first cell as given by V1 K, then the incremental contribution yielded by thesecond cell is less than the (incremental) contribution of the first cell.

3.6. The General Case

When there aren proven reserves andn unexplored cells at times, the valuefunction is

n n

L s sx =max

s

uqt

ni=1

cvit Lit

ert dt erK

+

0n+1 n

1+1 +L x

f d

(36)

where L = L 0 andx = x . The following theorem extends the analysis

to this general case.

Theorem 2. (i) The optimal value functionn n

, defined in (36), is decreasingin for given , and increasing in for given .

(ii) There are values, n+1 0 and n+1 >

n+1, for which the following

hold. (a) Exploration will not be undertaken again if the realized value of n+1

.

(b) Exploration will be undertaken immediately if n+1< n+1. (c) Exploration

will be suspended temporarily if > n+1.

(iii) The expected value of the objective function is continuous at an optimalexploration date. But, with probability one, there is a jump in price upon the realizationof exploration. There is a quality for the (n+ 1)th cell, x0n+1 (=

0n+1 for some

value 0n+1), which is realized with probability 0, and for which the jump is negative if

xn+1> x0n+1and for which the jump is positive ifxn+1< x

0n+1. Furthermore, the size

of the jump is increasing in the divergence ofxn+1from x0

n+1.(iv) The incremental value of an additional unexplored cell is positive but,ceterisparibus, is decreasing in the number of unexplored cells.

4. ECONOMIC IMPLICATIONS

Use of the exponential process and its conjugate, the gamma process, to modellearning yields an economically intuitive result, namely Theorem 1.Ceteris paribus,

a larger past cumulative discovery implies richer potential reserves while a greaternumber of cells already explored makes one more pessimistic about the qualitiesof the remaining potential reserves. We were able to model the multiple goals ofreplenishment of reserves, discovery of better quality reserves, and reduction ofuncertainty, and the trade-offs among them in an optimal program.

In reality, we do expect that some discoveries having only low-quality mineraliza-tion may never be mined. Medium-quality deposits are held on the shelf. Further-more, deposits of widely differing quality are mined simultaneously.

The model has implications for some of the important questions of resource eco-nomics. Tietenberg [25], Chap. 13, lists four possible measures of scarcity in perfectmarket: price, marginal discovery cost, marginal (extraction) cost, and user cost.Our examination eliminated the last three. Marginal discovery cost is not defined in




a model of discrete discoveries. Rather, expected incrementaldiscovery cost is equalto unity, in the following sense: exploration occurs when the optimal program valuehas fallen to the point that, when K is expended, benefits having expected value ofKwill be obtained. User cost varies at producing deposits: there is no global price

by which reserves in situcan be aggregated. Cost at the intensive margin is equal tothe price of the final product. Pindyck [19], Devarajan and Fisher[7], and Adelman[1]compared the resource rent per unit of reserves to the marginal discovery costof additional reserves. In our model there is no single measure of reserves, nor ofrent per unit of reserves.

The price of the final product aggregates present and anticipated future valuesof all relevant economic variables; it, not the shadow price of reserves in situ, isthe best indicator of scarcity. Using (4), (5), and (6) and recalling thatit =ite

rt

is the current rent at deposit i, we can express a modifiedr-percent rule (betweenepisodes of exploration) as follows,

1

it

ditdt

= r+

D2cvit Lit

D1cvit Lit

git

mit 2itgit

(37)

This rule applies at individual producing deposits. The current rent at depositi isithit. The total current rent for the sector, then, is Rt =

ni=1 ithit. In this

case, some manipulation using the same conditions yields

1

Rt

dRtdt

=r

1ni=1 ithit

n

i=1

ithit

D2cvit Lit

D1cvit Lit

git

mit 2itgit

+

hithit

(38)

We identify (38) asHotellings rule, a modified r-percent rule which holds for theentire sector. Themodification involves a complicated convex sum, with endogenous

weights, of the deposit-specific terms (and rules) represented in (37). There is norule for the rate of growth of aggregate rent in terms of aggregate magnitudes.

Rent is usually proposed as the national-accounts measure of depletion. Theeconomic definition of depletion is the reduction in the value of the optimalprogram due to current exploitation. Indeed, between episodes of exploration,Di

nn

L t 0x =it is the reduction in the expected current value at time t fromexploiting a unit length of deposit i. Therefore, in this model, (current) aggregatedepletion is given by Rt.

In connection with the measurement of depletion for the national accounts, someauthors distinguish the roles of average as opposed to marginal resource rents, andhence of total resource rent as opposed to Hotelling rent. Equations (5), (6), and(9) give a perspective on the attribution of rents. By (5), and (6),

p

qt

xi

m

it

2itg

it

hit= ithit

The right-hand side is the rent at depositi. The left-hand side is the current valueof metal mined less the value of metal at the marginal grade. Hotellings stance

on the issue of rent (as perhaps an oversimplification that, in a dynamic model,aggregate rent is a determinant of price) is evident in Eq. (7). But the physicallevels of cut-off grades are endogenous to the program. If the planner decentralizesdecisions to individual mine managers through quoting present and future prices,




then each manager could take Ricardos position to argue that price causes rent.AllRicardianrents (differential rents relating to the surplus above the cut-off grade,to qualityx, and to depthL) are incorporated into the rents which sum to theHotelling rent.

By the generalization of (9), however,

rn n

=u

qt

c

vit Lit

Rt=w

qt

+

vitD1c

vit Lit

c

vit Lit

Consumers surplus plus what Cairns [6]refers to as rent due to technological costconditions (reflecting decreasing returns to scale) is equal to interest on the valueof the objective. These rents can be consumed without affecting Hicksian income.Total surplus is equal to the sum of these surpluses and resource rent. This isconsistent with the identification of resource rent as the measure of depletion in

this model.Adelman[1]observes that depletion is not easily defined if, for whatever rea-

son (such as monopolistic behavior), an optimal program is not being followed. Theformal equivalence of the problems for a social planner and a monopolist allowsimmediate reflection on the actions of a monopolist. A monopolist would tend todiscourage consumption by charging a higher price than optimal early in the pro-gram, thereby appearing to have interests aligned with conservationists. Some havecalled this effectSolows paradox, as Solow [23]noted it in reference to Hotelling

[12].But a monopolist will have a nonoptimal pattern of exploration and level ofrecovery of the resource. Hotelling himself cautioned conservationists against fa-voring monopoly for its slower rate of exploitation.

In challenging economists approach going back to H otellings [12],Adelman [1]reasons that, at the margin, exploration cost, user cost, and price net of developmentcost are equated. Our model formalizes his intuition. By (36), at an exploration date, exploration cost is equal to the expected jump in the value of the objective,

K=

0

n+1 n1

+1 +L

s 0x

f d

n n

L

s 0x

By the definition ofRtand the generalization of (26), user cost is the rate of changeof the objective due to depletion of the mines,

Rt=

ithit=

Din n

L t 0xt

hit

Price obeys the relation

qt

pqt

vit

D1

cvit

Lit= D

i

n n

Lt

0xt

hit

Adelman calls existing oil deposits an inventory, to which additions through explo-ration are continually being made. It is possible to view both known deposits heldon the shelf for future production and unexplored cells as types of inventory.

These inventories are drawn down through time. Some authors, such as Hartwick[11],argued that depletion of the type previously mentioned should be a negativeentry in the national accounts, but that additions to reserves should be treated as a

positive entry. Suppose we abstract for the moment from the effect of exploration

on knowledge and on changes to values of known reserves (which should also betreated consistently). Then the optimal-value function in (36) stresses that the addi-tion to reserves comes at the expense of a reduction in the inventory of unexploredcells. These cells are also natural assets, and the exploration of a cell (depletion of




an unexplored cell) should be treated as anegative entry in the national accounts,to an extent offsetting the increase in reserves. As argued by Weitzman[27],thegoal of national income accounting would be, then, to account for changes in theoptimal-value function. Because there is no expected net gain to the program (in-

cluding information), the net contribution is random with mean zero. Thus, theexploration costK provides an unbiased estimator of the expected value of directcontributions of discoveries plus their informational value. In an expected-valuesense, then, treating exploration as an investment captures the full effect of addi-tions to reserves.

In a stochastic environment, there will be, with probability one, a jump in thevalue function at each episode of exploration. The interest on the optimal-valuefunction at time t has been called by (among others) Weitzman[27]and Hartwick

[10],the equivalent sustained level of the resources contribution to national prod-uct. With probability one, this level will change with exploration so that society willcontinually lurch away from what Weitzman calls theanticipated result.

The way that new knowledge is incorporated into the optimal trajectory helpsto resolve a controversy in the literature. According to Norgaard[16],theoreticalmodels involving Hotellings rule are valid only if resource exploiters are informed;if they are not, their calculations of rents will be faulty. Rather, the perceived rentand level of scarcity of resources reflect existing knowledge and the anticipations offuture knowledge. Expected values of unexplored cells are integrated directly intothe measures of scarcity, pqt, of the equivalent sustained contribution to nationalwelfare, rnn

, and of depletion, Rt. Resource managers utilize their information todirect their efforts, thereby determining paths of cost and price. But realizations ofevents affect their valuations and change the direction of their efforts. Prospectivemodels may be valid in incorporating what is known about resource scarcity but stillmay not readily explain economic history (ibid.: 24).

Even abstractingfrom exploration, Hotellingsrule would not be readily discernedin an aggregate empirical model, however. There is nostock effectbywhich costscan

be expressed as a function of remaining aggregate reserves. Halvorsen and Smith[9]were unable to confirm predictions of the theory of nonrenewable resourcesusing an econometric model with highly aggregated data. Compare also Livernoisand Uhler[15].

By the same token, L evhari and Liviatans [14]depletion effect, by which costs areheld to be a function of cumulative extraction, must be interpreted in a subtle way.In our model, what is depleted is a quantity,Lit, at each producing deposit,i, de-termined along an optimal path that accounts for qualities and depletion elsewhere.

Although defining Lit in the present model is a convenient simplification for math-ematical purposes, one would find it difficult to define an appropriate physical statevariable at a real deposit. Even in the model, what we called depth for purposes ofillustration is not cumulative production of ore or metal. These are economically,not physically, defined. In addition, cut-off grades, reserves, and expected rents arerevised upward or downward in response to exploration elsewhere. The model sug-gests, then, that empirical work on depletion and Hotellings rule using physicalmeasures may have been done using inappropriate data.

The ability to vary the cut-off grade gives the planner an additional control. Ifdiscoveries are not favorable, the consequent higher price implies a reduction of thecut-off grade and a higher ultimate level of recovery from known mineralization.Favorable discoveries allow for increased current production at lower cost. Choices




of cut-off grades and mine closings or openingsthe human contributionmitigateto an extent the effects of the physical attributes of deposits.

APPENDIX I: PROOFS

Proof of Proposition3. If theith reserve is exploited at some timet, then (16)must hold and we have D1c0 Lis/xig0 < p0. Conversely, suppose thatD1c0 Lis/xig0< p0; we want to prove that the ith reserve will be broughtinto production at some time t. Because pqt p0as tT, we can find a time s T such that pqt> D1c0 Lis/xig0. Now, if we excavate a disc of oreof thickness h >0and radius fromthe ith reserve at time , then the undiscountednet social welfare at isuj=i qj +ximh j=i cvj Lj c2h Lis.

The derivative of this expression with respect to is

2gxih

p

j=i

qj+ ximh

D1c2h Lis

xig

(39)

When 0, the expression inside the square brackets in (39) tends to the limitpqt D1c0 Lis/xig0 > 0. Hence, (39) is positive when is small; i.e., netsocial welfare can be increased by exploiting the ith reserve some time after.

Proof of Proposition4. (i) Consider two reserves, i and j, whose qualities satisfyxi < xj. There are several possibilities.

(a) If xi < x < xj, then by Corollary 1(iii), the optimal cut-off depth in

the ith reserve, LiTi , satisfies D1c0LiTi /xig0 =p0. Because Ti is the exact

time that all excavation in the ith reserve is terminated, by (5) we must have

p

qTi

=D1cviTi

LiTi

xigiTi

D1c0 LiTi

xig0 =p0

Therefore, pqTi =p0. Also, because qt>0 for allt < T, we can then conclude

thatTi =T. Because xi < xj, we can invoke Corollary 1 to assert that LjTj >LiTi .

(b) If xi > x, then by Corollary 1, LiTi =

LjTj =1.

(c) If xi < xj < x, then the argument used for the ith reserve in the

discussion of (a) can be applied to the jth reserve. In particular,Ti =T =Tj.Now, suppose that there are at least two deposits, i and j, with xi x

< xj.From (a),Ti = T. Thus, we need to prove that Tj < T. Definek1= D1c0 1 =

xg0p0. Then xjgjTp0 = k1. Therefore, gjT < g0, and hence, by(4) and (5), jT = p0xjmjT

2jTgjTe

rT < 0, so that jTj < 0 and

jTj > 0, as Tj T. Suppose that hjTj > 0. Then vjTj > 0 and hence, by (4),

pqTj+dTxjmjTj 2jTj

D1cvjTj 1 > 0. Clearly, it would be profitable to set

aside dh jTjerTj+dT dh

by the strict convexity ofc in v. Clearly,hjTj >0 is not optimal, and sovjTj =0.Now, by (4) and then the definition ofmand (5), at any time,t, for any deposit,

ert + r = pxm + pxm 2k1= pxm (40)




Also, by (3) and the fact thatvjTj =0, we can define k2=D2c0 1 = jTjerTj.

Suppose that Tj = T. For ease of exposition we drop subscripts, on the under-standing that, unless noted, variables refer to deposit j at time T. Then, by (5),pxg= k1. We solve (40) to obtain xm p + rp = k2 +

2k1r. Then,

k1p

1

g

= x =

2k1r k2m p + rp

< k1

p

2

m

This contradicts the fact thatm 2g. Therefore,Tj< T. A similar proof showsthat, ifxk > xj> x

then Tj> Tk, by substituting D1c0 1/pqTk xj fork1.(ii) L et us now assume that the extraction-cost function has the form given

by (1). Then Eq. (9) holds at each point in time before T, and, in particular, at theinitial time,s, we have

w

qs

= u

qs

qsp

qs

= r nLs sx (41)

Now observe that n increases if Lis decreases or if xi increases. Also observethat as Lis 0 andxi 1, the mineral in any disc of oreh h > 0 > 0,extracted from this deposit becomes very rich, and its extraction cost remains atc2h L at any given depth, L. Suppose that the richness of the other ore-bodies is such that wqs < r

nL/is 0 sx/i x, where, for any vector, u =

u1 un, u/i = u1 ui1 ui+1 un. By (41), if D1c0 Lis/xig0 pw

1rnL/is 1 sx/i x, then the ith

reserve will not be brought into production immediately.

Proof of Proposition5. By Theorem 1, the family of distribution functions asso-ciated with f, namely,F, is stochastically decreasing in and increasing

in . Hence, by Theorem 2.1 of Fishburn and Vickson[8], V1 is strictly

decreasing and V1 is strictly increasing.

Proof of Lemma 1. Suppose that at some time t s, when the proven re-serve was mined to depth L1t, one decides to explore the second cell, and onefinds a reserve of qualityx2. Then the discounted social welfare fromt onwardis2L1t 0 t x1 x2. Because of the concavity of the welfare function,uq, wemust have

2L1t 0 t x1 x2 1L1t t x1 +

10 t x2

Subtracting Kert from both sides and recalling the result following (7), we obtain

2L1t 0 t x1 x2 Kert 1L1t t x1 + e

rt10 0 x2 K (42)

Taking the expected value of (42) with respect to the distribution of2, we obtain 0

2

L1t 0 t x1 2

f2 d2 Kert

1L1t t x1 + ert

V1 K

(43)

The left-hand side of (43) is the expected discounted social welfare, net of explo-ration cost, from t onward if one decides to explore the potential reserve at timet. This expected discounted social welfare will be no greater than1L1t t x1 ifV1 K. Thus, when V

1K, one should never explore the cell.




Next consider the case V1> K, and suppose that one decides never to explore

the cell. The optimal discounted welfare is 1L1t t x1. Since metal output willbe near zero if t is in a left neighborhood of T1, social welfare discounted to twill be strictly less thanV1 K. Att, if one abandons the proven reserve and if

one explores the remaining cell, then the expected net discounted social welfareisV1K. The decision never to explore for the potential reserve is clearly notoptimal.

Proof of Corollary 3. Assuming that the first cell is in production at s, (26)givespqs = D1cv1s L1s/x1g1s. Let q1s be the quantity produced at s ifthere is only a reserve of qualityx1 at depthL1s and no unexplored cell. Thenq1s q

1s =w

1r1L1s 0 x1, and pq1s pq1s. Let x2 be the expected value

ofx2 under f; i.e., x2= 0 fd. Ifx2< D1c0 0/g0pq1s D1c0 0/g0pq1s, then it is expected that a newly discovered deposit will notcontribute to production at s. Therefore, K will not be incurred until some time > s.

Proof of Proposition11. Suppose that V1 > K. If N = 1, then the expected

value net of exploration cost is V1 K. IfN = 2 and only one cell is explored,

then the net expected discounted social welfare isV1K. However, dependingon the quality of the reserve found in this cell, one might find it advantageous to

explore the second cell as well. Hence, the expected net discounted value of a two-cell region,V2K, is strictly greater than V1K. Furthermore, we can show

that V2 K




Proof of Theorem 2. (i) For n given n, a simple induction argument serves.For n givenn, use induction backward from n+n proven cells (or n+k provencells if exploration is abandoned after the kth attempt), noting that

n+n1 1 Ls sxs =max

n

s

u

qt

n+n1

i=1

c

vit Lit

dt ern K

+

0n+n

1Ls 0 1xs

f d

and an analogous formula holds ifk < n.(ii) We define n+1 implicitly by V

n1+1 +n+1

= K if there is a for which

Vn1

+1 + < K, and 0 otherwise. Clearly, ifpqg0xn+1 D1c0 0, then we have

a loose upper bound for n+1, namely,1xn+1. The argument basically follows

that of Proposition 9.(iii) Since , and hence x, is continuously distributed, the probability of realiz-

ingany particular value is zero. The rest of the argument follows that of Proposition7 almost verbatim.

(iv) Fixn and the boundary conditions at time s. Let n+1 be the time at whichthe (n + 1)th cell is explored. As argued in the proof of Proposition 11, by the strict

concavity ofuq, at time s,n n+1 is bounded above by

n n

+ ern +1V1 K.

But, n+1 n . Hence, the bounds become progressively tighter as the numberof cells increases. Indeed,

n n+1

n n

n n

n n1 ; otherwise, noting the

strict concavity ofuq, expected social welfare under n n

could be increased byrearranging anticipated production from then+nth cell to coincide with what

would be anticipated from then+ n +1th cell under n n+1 , contrary to the

optimality ofn n

.

APPENDIX II: TABLE OF VARIABLES

t timex quality of a deposit distance of mining from axis; 0 1g grade at distance m metal content out toh thickness of disc of ore mined

v volume of ore mined,2hL depth of mining from surface; 0 L 1cv L mining and processing costDi differential operator wrt ith argument of functionbL unit cost at depth L in proportional cost functionqi quantity mined at deposit i,ximihiq total quantity mined in sector,

qi

pq inverse-demand function

uq utility functionr discount raten objective function when there are n depositsTi time at which deposit i is closed forever




T time at which mining ceases, maxi TiHLht Hamiltoniani discounted shadow price of mining depth at deposit ii current shadow price of mining depth at deposit i

x quality below which mineralization is never minedx quality above which mine is exploited to depth L= 1K cost of exploring a cell random variable representing an index of deposit quality function relating and x:x = parameter of exponential distribution of quality parameters of gamma distribution used in updatingVn discounted value ofn unexplored cells when there is no

proven reservewq consumers surplus,uq qpq

ij objective function;i proven reserves,junexplored cells

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