Agricultural & Applied Economics Association

Bargaining, Strategic Reserves, and International Trade in Exhaustible ResourcesAuthor(s): Vincent P. Crawford, Joel Sobel and Ichiro TakahashiReviewed work(s):Source: American Journal of Agricultural Economics, Vol. 66, No. 4 (Nov., 1984), pp. 472-480Published by: Oxford University Press on behalf of the Agricultural & Applied Economics AssociationStable URL: http://www.jstor.org/stable/1240926 .Accessed: 26/09/2012 15:36

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

Agricultural & Applied Economics Association and Oxford University Press are collaborating with JSTOR todigitize, preserve and extend access to American Journal of Agricultural Economics.

http://www.jstor.org

Bargaining, Strategic Reserves, and

International Trade in Exhaustible Resources

Vincent P. Crawford, Joel Sobel, and Ichiro Takahashi

This paper proposes a framework for the analysis of optimal strategic reserve policy when ability to withstand embargoes influences terms of trade in an exhaustible resource like oil. The model embodies two stylized facts: competitive forces are not always strong enough to eliminate other influences on the negotiated price of oil, and long-term contracts for oil delivery cannot be enforced across national boundaries. Some modeling issues that arise in this context are discussed, and a model of optimal strategic reserve policies is proposed and solved in a leading special case.

Key words: exhaustible resources, international trade, strategic reserves.

Long-term international trading relationships in exhaustible resources play an important role in the world economy but have received little formal attention.' In particular, a country can benefit from holding strategic resource re- serves, but almost nothing in the voluminous literature on exhaustible resources indicates upon what the value of such reserves might depend. This paper studies one rationale for strategic reserves in a simple model that pro- vides a vehicle for discussing the modeling issues that arise in this area and allows a characterization of the optimal reserve policy in a leading special case.

To understand the aspect of the problem, consider, for definiteness, the determination of the best U.S. strategy for dealing with threats by the Organization of Petroleum Ex- porting Countries (OPEC) to interrupt oil supplies. Three separate, possibly conflicting, considerations are potentially relevant. The

first, reducing the frequency of embargoes, is of great interest and has not been considered in the literature. This paper also avoids this issue by assuming that embargoes never occur in equilibrium, although embargo threats might still influence the terms of trade. But recent developments in bargaining theory (see for example Crawford or Sobel and Takahashi and the references cited there) might have use- ful applications to this problem. The second consideration, reducing the cost of embargoes when they occur, has been discussed by Bhagwati and Srinivasan, Mayer, Sweeney, Tolley and Wilman, and others. These studies take the influence of the home country's ac- tions on the probability of an embargo as ex- ogenous and derive rationales for strategic re- serves from this specification.

The third consideration, reducing the cost of embargo threats in the negotiations that de- termine the terms of trade in oil, is the subject of this paper.2 Even though the U.S. has sub- stantial oil supplies of its own, it derives sig- nificant benefits over time from importing or exporting; and because the world oil market is

Vincent P. Crawford and Joel Sobel are associate professors of economics at the University of California, San Diego; Ichiro Takahaski is at Soka University, Tokyo, Japan.

Partial financial support for this study came from the National Science Foundation under grant SES7-05550.

The authors are grateful to Peter Berck, Avinash Dixit, James Mirrlees, Andrew Postlewaite, and to participants in seminar pre- sentations at the University of California, San Diego, and at the San Diego summer meetings of the Econometric Society for help- ful conversations.

Review was coordinated by Alan Randall, associate editor. I Notable partial exceptions are Bhagwati and Srinivasan,

Mayer, Nichols and Zeckhauser, Ryder, Sweeney, and Tolley and Wilman. None of these papers, however, are directly concerned with the issues discussed here.

2 There may be a tension between the second and third consid- erations because a change in policy that increases the cost of an embargo to the United States but causes a much greater increase in the cost to OPEC might reasonably be expected to enhance the U.S. negotiating position. While this tension is ignored by focus- ing on the third consideration in isolation, the analysis of this paper may prove helpful in future attempts to determine how sharp such trade-offs are in practice.

Copyright 1984 American Agricultural Economics Association

Crawford, Sobel, and Takahashi Strategic Reserves 473

far from competitive, the trading relationship with OPEC generates a large surplus relative to parties' next-best trading alternatives. Fur- ther, long-term contracts for the delivery of oil are generally acknowledged to be unenforce- able across national boundaries (Dasgupta and Heal and Maskin and Newbery), and might be uneconomic even if they could be enforced. The relationship is therefore naturally mod- eled as a dynamic bargaining problem in which a sequence of short-term negotiated agree- ments takes the place of competitive spot markets or a single long-term contract. Both the absence of long-term contracts and the fact that competitive forces are insufficient to de- termine completely the terms of trade are es- sential to the explanation of the role of strategic reserves presented here.

In the two-country model of this paper, the gains from trade stem from differences in re- source endowments and technologies. Domes- tic planners control their country's extraction path with the goal of maximizing its gains from the trading relationship over its entire life. Be- cause the ability to withstand embargoes is a natural benchmark from which to measure the gains from trade, it influences the negotiated terms of trade. Strategic reserves can yield real benefits, even when embargoes never oc- cur, by reducing the cost of threatened embar- goes in bargaining. If, for example, a relatively oil-poor country exhausts its oil supply too rapidly, its increased dependence on oil-rich countries in the later part of the trading rela- tionship can be expected to lead to unfavor- able terms of trade; this should create a motive to delay extraction. An oil-rich country faces the converse problem of how best to exploit its scarce bargaining advantage over the life of the relationship. These problems are inter- linked and are resolved simultaneously by countries' equilibrium extraction strategies. We show that, in a leading case, the equilib- rium involves the oil-rich country alone ex- tracting for the first part of the relationship, exporting to smooth production in both coun- tries until parity is reached. At that time au- tarky ensues, with both countries extracting until their stocks are exhausted. Total extrac- tion is typically inefficiently slow. The analy- sis of this leading case rationalizes intuition and indicates the likely qualitative features of the equilibrium for more general specifica- tions.

This paper is organized as follows. The next section introduces and motivates the model.

The following section shows how to charac- terize optimal extraction policies and con- sumption paths in general and solves for them in the leading case where countries have iden- tical technologies differing only in their initial resource endowments. The final section gives the conclusion.

The Model

This section presents the model. It could be extended in several directions, but a "stripped-down" version is presented for ease of exposition. Consider two countries, each with a given, known stock of the resource and perfect knowledge of all other aspects of the environment. Suppose each seeks to maxi- mize the undiscounted integral of con- sumption from time zero to time T, at which point a cheap backstop technology becomes available to both countries, making the re- source uneconomical. Discounting at equal rates could be allowed without significantly altering the results. Extraction is costless and storage above the ground is costly and there- fore uneconomical. The planners in each country set extraction rates unilaterally and negotiate an exchange of resource for the con- sumption good with the planners in the other country at each instant of time, taking extrac- tion rates as given. The amount of compensa- tion, which determines how countries share the gains from trade, is determined by the amount of resource exchanged and by coun- tries' relative "bargaining powers," which are in turn determined by how well each country could survive an embargo for the rest of the time horizon. These assumptions are stated more precisely and discussed further below.

The countries are denoted F and G. They have neoclassical instantaneous production functions fand g; each takes domestic extrac- tion plus or minus imports or exports of the resource as input and yields consumption as output. They should be regarded as indirect production functions that take into account domestic production possibilities as well as opportunities for trade in world markets; this could include trade with small competitive suppliers of the resource. The functions f and g are assumed to be twice continuously differ- entiable with strictly positive first derivatives and strictly negative second derivatives everywhere; in addition, they are assumed to satisfy an Inada condition at the origin, with

474 November 1984 Amer. J. Agr. Econ.

the limit of the marginal productivity of oil approaching infinity as total oil use ap- proaches zero. These restrictions could be de- rived from standard assumptions on the under- lying production technologies. Countries F and G begin with u0 and v0o units of the re- source at time zero. Extraction policies are described by specifying the time paths of re- maining reserves, denoted u([O, T]) and v([O, T]) for F and G, respectively. The physical constraints on these paths take the form u0 =

90, v0 = V0U, UT E 0, and VTr 0; they must, of course, also be nonincreasing.

At each moment in time, bargaining deter- mines how much trade in oil takes place and how the gains from trade are divided. Since only agreements about current trade are en- forceable, negotiations are restricted to cur- rent variables. Trade is assumed efficient at each instant, given extraction policies. This avoids begging the question of full intertem- poral efficiency that is a primary concern be- low.

Because countries seek to maximize con- sumption, which can be transferred from one country to the other at a one-for-one rate sim- ply by agreeing on a different oil price, bar- gaining efficiency requires that the marginal productivity of the resource be equalized across countries at each instant. Thus, given our specifications off and g, a unique efficient level of imports for country F exists for any combination of extraction rates (-u', -v') where primes denote time derivatives; this level is denoted by m[u', v'] and characterized by the relationship

(1) f'[-u' + m(u', v')] = g'[-v' - m(u', v')];

m(-) is negative when efficiency requires that F export rather than import. Denoting partial derivatives by subscripts in the usual way,

ally enforce if an agreement is not reached; these fixed threats are commonly taken as the benchmark. A bargaining solution summarizes the influence of the benchmark (and the set of feasible agreements) on how the gains are shared. (See Roth for a good exposition and references to the literature.)

In world resource markets, each country has the right to impose an embargo, partial or total, for as long as it wishes. For this reason, assume that current compensation is deter- mined by countries' ability to withstand an embargo for the rest of the time horizon. This is not the only reasonable specification, but it leads to interesting results. It reflects the great importance of the future relative to the present in the continuous-time model and the fact that it may be significantly easier to commit to a long-term embargo, which can be enforced uni- laterally, than to a long-term contract, which cannot.

An alternative specification applies standard bargaining theory to the dynamic-program- ming value functions whose changes over time measure the long-run payoff gains from cur- rent agreements. As it happens, this appar- ently sensible approach does not yield a sensi- ble model of strategic reserve policy in this model. The reason, which is most conve- niently demonstrated in a discrete-time frame- work, is illuminating.

Let otr denote F's value function at time t, and let Ft similarly denote G's value function. If ut and vt denote stocks and xt and Yt denote extractions, ot + Ft = f[xt + m(xt, yt)] + g[y, - m(xt, Yt)] + Dt+ + rt+,. Consider the bar- gaining problem at time t, when these value functions are used to evaluate the future im- plications of the current bargain. This is a standard static bargaining problem, and the threat point is {(T - t)f[utl/(T - t)], (T - t)g[vt/(T - t)]}. Since the problem is symmet-

f(2) m"[-u' + m(u', v')] (2) m,(u', v') , and f"[-u' + m(u', v')] + g"[-v' - m(u', v')]A

-g"[-v' - m(u', v')] (3) m2(uv, v') m( ')] f"[-u' + m(u', v')] + g"[-v' - m(u', v')]

Taking extraction rates as given, an instan- taneously efficient agreement generates gains from trade that must be divided between coun- tries; this requires a benchmark from which to measure the gains from trade. In ordinary static bargaining theory, an outcome exists, by assumption, that each bargainer can unilater-

ric and has transferable utility, any of the classical bargaining solutions (see Roth) will yield a split-the-difference solution; this im- plies that Ot = ( ,:t + Ft + (T - t){f[ut/(T - t)] - g [vt/(T

- t)]})/2. By iteration, one can dis- T

cover that , + Ft = I {f[xt + m(xt, Yt)] + i=t

Crawford, Sobel, and Takahashi Strategic Reserves 475

g[yt - m(xt, Yt)]}. Substituting this expression into the above expression for (D and deriving the analogous expression for Ft reveals that F and G have a common interest in choosing intertemporally efficient extraction rates over the remaining horizon; but that their threats, and therefore how they divide the resulting gains from trade, are predetermined by past extraction decisions. This renders strategic re- serve policy ineffective, reducing it to plan- ners wishing that their ancestors had saved more oil.

To obtain a more sensible model, this paper proceeds as follows. In keeping with the above arguments that the appropriate benchmark from which to measure the gains from trade is the ability to withstand an embargo for the rest of the time horizon, assume that the outcome of a given period's bargaining is an instantane- ously efficient amount of trade in the resource and an implicit price of the resource, in terms of the consumption good, that depends on each country's remaining reserves and the amount of time remaining. These variables are sufficient to determine how well each country can withstand an embargo. This compensation function is assumed to be twice continuously differentiable and denoted h(ut, vt, T - t). (This function might, for example, be taken to be J {f'[ut/(T - t)] + g'[vt/(T - t)]}.) Total compensation paid by F to G is therefore given by m(-)h(ut, vt, T - t). Note that the price given by h(-) is not viewed as a market price; in particular, it does not imply a willingness to buy or sell, at the same price, quantities of the resource other than the one agreed upon. Compensation can depend nonlinearly on m[-], with minor changes in the conclusions, but this possibility is ignored because it does not seem to add insight.

The assumption just adopted rules out any influence of current extraction rates on the outcome of the current bargain except as they influence m[- ]; this assumption turns out to be important for the analytical tractability of the model. The strongest justification for this as- sumption rests on the fact that in a continu- ous-time model, the present is insignificant when compared with the future, and on the argument that a planner's goal in this sort of situation ought to be the preservation of the trading relationship for the benefits it is ex- pected to yield over time. If this is correct, the current bargain ought to be evaluated as part of the sequence of bargains to which it be- longs, and planners should insist only that the

gains from trade throughout the sequence are shared appropriately. For example, a bargain need not be rejected because it violates static individual rationality-that is, if its instan- taneous current net benefits are negative- provided that it satisfies intertemporal indi- vidual rationality-that is, that it is part of a predictable sequence of bargains that, taken together, are beneficial to the country. While violations of static individual rationality may seem counterintuitive, they are not necessar- ily incompatible with the rationality of non- myopic planners who can predict the out- comes of future negotiations. Such planners will not wish to endanger a trading relationship that is beneficial in the long run to obtain addi- tional short-run gains that are of little sig- nificance over the long run.

These arguments are important in evaluat- ing the assumptions about compensation be- cause those assumptions are incompatible with static individual rationality in general. To see this, consider extraction rates (-u', -v') for which m[u', v'] = 0, so that the static individual rationality restrictions are just bind- ing. Differentiating the expressions for net in- stantaneous national income reveals that a necessary condition for the preservation of static individual rationality, following small changes in extraction rates away from such values of (-u', -v'), is that

(4) f'(-u') = g'(-v') = h(vt, ut, T- t).

Because this condition must hold for any such (-u', -v'), it is incompatible with the assump- tion thatf and g have negative second deriva- tives.

On the other hand, the assumptions about compensation should be generally compatible with intertemporal individual rationality. Sup- pose the equilibrium extraction paths charac- terized below are such that, with instanta- neously efficient trade, they generate more total consumption across countries and over time than would autarky throughout the time horizon with instantaneously efficient trade. Then the way h(.) enters planners' objective functions and the fact that u([0, T]) and v([O, T]) are sufficient to determine u'([0, T]) and v'([0, T]) suggest that compensation functions exist which satisfy intertemporal individual ra- tionality. If, on the other hand, no such func- tions exist because autarky generates more total surplus, at least one country would be better off under autarky, and this is incompat- ible with equilibrium. While trade is not con-

476 November 1984 Amer. J. Agr. Econ.

templated at period-by-period rates other than m[-], this is intended as an appropriate sum- mary of behavior when trade is intertempor- ally mutually beneficial, not a constraint that forces planners to trade against their will. So the analysis should be interpreted as an analy- sis of the case where optimal trade remains better than autarky throughout the time hori- zon. Simple sufficient conditions for this to hold in the model of this paper have not been found. But it is possible to show that, in the case that is solved, intertemporally individ- ually rational compensation functions always exist, and therefore trade remains potentially beneficial near the end of the trading relation- ship (which is just where possible failures of static individual rationality would lead one to expect trouble).

This completes the specification and jus- tification of the model except for behavioral assumptions about planners' best choices of extraction paths. Assume that the extraction paths are chosen as rational responses to ra- tional predictions about other planners' choices of future extraction; some form of Nash equilibrium is therefore the appropriate concept. The specification of this paper fol- lows what is known as the open-loop Nash equilibrium, which is simply a pair of extrac- tion paths that are best responses to each other. This solution concept has been widely used in the exhaustible resources literature because of its simplicity and because of its analytical tractability relative to more sophis- ticated solution concepts. It is fully appropri- ate (under the usual rationality assumptions) when planners must irrevocably commit themselves to extraction paths at the begin- ning of the time horizon, and cannot revise their plans after part of the equilibrium has been played out. Even when planners cannot or need not so commit themselves, an open- loop Nash equilibrium (unlike a Stackelberg equilibrium) is intertemporally consistent in the sense that continuations of equilibrium paths are also in equilibrium. However, when commitment is not possible or not required, it fails to be perfect: a planner whose opponent has deviated from an open-loop equilibrium path, if given an opportunity to deviate from his own planned path, will generally wish to do so. Thus, the conjectures about opponents' behavior implicit in an open-loop equilibrium are not fully rational in situations without commitment of strategy choices. Full ra- tionality in situations like the one considered

here requires the adoption of a closed-loop solution concept in which planners' extrac- tions at each moment are functions of all the information available to them at that time. For the present model, conditioning on remaining reserves (and not past strategies) would likely be sufficient. An open-loop equilibrium is, triv- ially, a closed-loop equilibrium but generally not a perfect one. Unfortunately, the charac- terization of perfect closed-loop equilibria is usually quite difficult, particularly so in con- tinuous-time models (but see Levhari and Mirman and Reinganum and Stokey for exam- ple characterizations of closed-loop Nash equilibria in resource models). On the other hand, continuous time allows the great impor- tance of future over present bargains to be expressed most sharply and facilitates the analysis of the qualitative features of optimal extraction paths in the next section.

Characterization of Equilibrium and Best Responses

We now proceed to characterize the open-loop Nash equilibrium choice of extraction paths. Given the assumptions and definitions above, the problems that determine countries' equi- librium strategy choices can be written as

(F) max fI u([O, T]) {f [m(u't, V't) -

U't]

- m(u't, v't)h(ut, vt, T - t)}dt

subject to UT = 0, u't . 0 for all t E [0, T], and

(G) max T

' (G)

[o, rTl {g [-m(u't, V't) - V't]

+ m(u't, v't)h(ut, vt, T - t)}dt

subject to VT = 0, v't - 0 for all t E [0, T]. Problems (F) and (G) determine a standard differential game.

While explicit characterization of the equilibrium is difficult in general, as with most differential games, equilibrium is easily de- scribed in a leading, illuminating special case. Suppose that both countries have the same technology, so that f(-) - g(). This assump- tion is very strong, essentially implying that countries have identical capital stocks and non-resource market opportunities as well as identical available techniques of production. However, it leads to a solution not otherwise readily obtained, and it seems unlikely to give a distorted view of the qualitative form of

Crawford, Sobel, and Takahashi Strategic Reserves 477

equilibrium in general. With identical tech- nologies, (2) and (3) and the definition of m[u't, v't] imply that m,[u't, v't] - 1/2 and m2[u't, V' t] - 1/2, since assumptions aboutf and g imply that m[u't, v't] - u't = -m[u't, v't] - v't.

To aid further in the characterization, con- sider additional assumptions about the com- pensation function. In particular, suppose the integrands in (F) and (G) are concave in (ut, u't) and (vt, V't), respectively; these second- order conditions guarantee that the necessary conditions for optimality are sufficient. Also,

m['][hl(-) + h2(')] measures the change in

compensation associated with an equal in- crease in stocks. This change increases total supplies without arguably affecting relative bargaining strengths; at least in the symmetric case, such an increase can be expected to re- duce (in absolute value) the amount of com- pensation. Thus, assume that hi () + h2(0) < 0. This is satisfied, for example, when

h(ut, vt, T - t) + { ff'[ut/(T - t)]

+ g'[vt/(T - t)]}. The following result can now be established.

When countries have identical technologies, the differential game has an equilibrium with the following qualitative form. If stocks are equal, both countries extract simultaneously, at a constant rate; consequently no trade takes place. Because of symmetry, concavity, and stationarity, this extraction path is intertem- porally efficient. If initial stocks are different, say uo < io, only the country with greater reserves extracts until stocks are equalized. At that time autarky ensues, with both coun- tries extracting at the same, constant rate until stocks are exhausted at T. While equilibrium may not be unique without further restric- tions, all equilibria must be of this form, differ- ing only in the time when stocks are equalized, which always occurs before T if •o > 0.

To prove this result, note that if u([O, T]) and v([0, T]) are equilibrium extraction func- tions, then for their interior portions the Euler conditions for F and G must be satisfied,

(5) { f'[(-u't - v')/2] + h(ut, vt, T - t)}

- (U't - v't)hi(ut, vt, T - t)t 0, and

d (6) -~ f'[(-u't- v',)/2] + h(ut, vt, T- t)}

+ (u't

- v',)h2(ut, vt, T - t) O.

Given the assumption on hi(')

+ h2('), (5) and (6) can be satisfied simultaneously if and only if u't = v' constant. Clearly, u't v't. To see that they must be constant, note that when

u't = v't, (5) reduces to

(7) f'(-u't) + h(ut, vt,

T - t) constant.

Arguments below (without using the con- stancy of u't and v't) imply that ut = vt as well. Further, with identical technologies, inter- temporal individual rationality clearly requires that

h(u, u, T - t) - f'[u/(T - t)] g'[u/(T - t)].

Otherwise, when stocks are nearly equal, one country could do better by switching to au- tarky. This reduces (7) to

(8) f'(-u't) + f'[ut/(T - t)] t constant.

A straightforward argument, using the feasibil- ity conditions on u([O, T]), shows that (8) has exactly one economically relevant solution, -u't = ut/(T - t), which establishes the claim.

The assumptions also ensure that an equilib- rium involves the country with more stock extracting until stocks are equalized. To see this, notice that the Inada conditions (and the fact that storage above the ground is uneco- nomical) guarantee that some country is ex- tracting at each instant. If both countries are extracting at time zero and initial stocks are unequal, then at some time strictly before T one country stops extracting. Otherwise, since u't = v't, both stocks could not be exhausted at time T. However, the country that first stops extracting can improve its welfare by delaying its extraction. Essentially, this happens be- cause

hi(') +

h2(') < 0 guarantees that the cost

of imports could be reduced by postponing some extraction until after an interval of equal depletion of resources. A similar argument shows that no country would choose to exhaust its stock before the other country. Hence, attention can be restricted to strategies that involve the resource-rich country extract- ing until its excess reserve is depleted and then both countries extracting at the same constant rate.

For each S E [0, T], denote by u(. ; S) the function defined by

T- for tE [0, Si],

(9) u(t; S) - oT

- t

uojTS for tE [S, J.

478 November 1984 Amer. J. Agr. Econ.

As already argued, this is the only possible kind of equilibrium strategy for the resource- poor country. The results below show that, for some S*, the best response to u([0, T]; S*) is an extraction path v([0, T]) that satisfies vt = u(t; S*) for t E [S*, T]. This merely requires showing that u(. ; S*) is a best response func- tion to this choice of v([0, T]).

For each S E (0, T), define the function v(. ; S) as the solution to the differential equa- tion

(10) 0 =

[f'(-v',/2) + h(ii0, vt, T - t)]

+ v'th2(UO, vt, T - t) that satisfies

(11) v(S; S) = 0, and

(12) v'(S; S) = -2t0/(T - S)

[denoting vu(t; S) by v'(t; S)]. For any given S, (10) is simply the Euler necessary condition for the optimality of the response function v(- ; S) to ut = uo on [0, S]. Equations (11) and (12) specify boundary values for v(S; S) and v'(S; S); (12) must be satisfied for any best response that satisfies v'(t; S) u'(t; S) for t _S.3

By the Cauchy-Peano theorem (10), (11), and (12) characterize a unique function v(. ; S) on [0, S] for any given S; v(t; S) is continu- ously differentiable in t and continuous in S. The Inada condition ensures that v'(t; S) < 0 for all t and S. If v(0; S*) = vo for some S*, then the function

v(t; S*) for t E [0, S*] (13)

vt- 0 ( T -t ) forte [S*, T] T ST- S*

is a best response to the resource-poor coun- try's extraction path u([0, T]; S*). This fol- lows since the v([0, T]) given by (13) satisfies all necessary conditions for a best response, and concavity assumptions make these condi- tions sufficient for maximization. The exis- tence of S* follows from the continuity of v(t; S) in S and the observation that

(14) /o

= lim v(0; S) < 3vo < lim v(0; S).

Next verify that u(. ; S*) is a best-response function to v(. ; S*). Let w be any other re- sponse function for the resource-poor country, and define zt =wt - u(t; S*). Notice that wt

= u(t; S*) = •o for t - S* so zt 5 0 for t 5 S*. Also, by concavity assumptions,4

(15) f{f[m(w', v') - w']

- m(w', v')h(w, v, T - t)}dt

- {f f[m(u', v' ) - v' ]

- m(u', v')h(u, v, T - t)}dt

+os -m(u', v')h(;o, v, T - t)

d 1 f'(m(u', v' )-V '] dt

- 2 ' ) -

+ h(u , v, T - t)) z(t)dt

+s dt If'[-uo/(T - S)]

+ h(u, u, T - t) z(t)dt - - z(S*) [f'(-v'L/2)

+ h(uo0, u, T - S*)] - {f'[-uo/(T - S*)]

+ h(uo, uo, T - S*)}) where v denotes the function v(- ; S*) and v'L

lim v'(t; S*). The last two terms on the t--*S right-hand side of (15) are zero because u"(t) =

0 for t E [S*, T] and by the definition of

v(. ; S*) [conditions (10), (11), and (12))]. In addition, for t E [0, S*],

d (16) -1 [f'(-v'/2) + h(k0, v, T- t)-

1 v'h2(uo, v, T - t) > -v'hi(fo, v, T- t)

by the Euler condition for G, equation (10), and because hi(-) + h2(') < 0; since z(t) - 0 for t E [0, S] the second term on the right- hand side of (15) is nonpositive. Thus,

3 Intuitively, (12) guarantees that the resource-rich country cannot increase its utility by shifting its stock from one side of S to the other. It can be derived formally from standard arguments if extraction paths with discontinuous derivatives at S are spe- cifically considered.

4 If

H(t, w, w') - f[m(w', v') - w'] - m(w', v')h(w, v, T - t)

for a fixed function v(t), then concavity implies

H(t, w, w') - H(t, u, u') + H2(t, u, u')z + H3(t, u, u')z'.

The inequality follows from integrating both sides from 0 to T, breaking the last two terms on the right into integrals from 0 to S and from S to T, and integrating the terms involving H3 by parts.

Crawford, Sobel, and Takahashi Strategic Reserves 479

u([O, T]; S*) is the best-response extraction path to v([O, T]; S*). This establishes the re- sult.

Several remarks are in order. First, from (12), aggregate extraction is continuous at S* and hence for all t. Second, when the two countries have identical, neoclassical tech- nologies, intertemporally efficient paths are characterized by a constant rate of aggregate production. Since the first part of the re- source-rich country's extraction path will not in general be flat in equilibrium, equilibrium production will be intertemporally inefficient. Plausible assumptions about the compensation function ensure that equilibrium extraction is less than the efficient rate until stocks are exhausted. Thus, countries delay extraction in order to obtain bargaining advantages in the future.

Note that the solution paths are intertem- porally consistent by construction. That is, a country whose expectations about the other country's actions are realized while carrying out the initial part of an equilibrium path will not wish to depart from the remainder of the path.

Finally, consider equilibrium and best-re- sponse extraction paths in more detail. Recall that intertemporal individual rationality im- plies that in the identical-technologies case, h(u, u, T - t) f' [u/(T - t)]; that is, when stocks are equal, the price of the resource is equal to its marginal productivity at u/(T - t), the rate at which the stock will be extracted under autarky. In general, the arguments above suggest that when technologies are identical, h(u, v, T - t) should be between f'[u/(T - t)] and f'[v/(T - t)]-exactly where depends upon a specification of bargain- ing power. An increase in a country's stock has two, possibly opposing, influences on compensation. First, because increased stock means increased total supply, the increase should tend to push prices down. On the other hand, the bargaining outcome should be influenced in favor of the country with in- creased stocks. If v < u, then an increase in v will lower prices because both effects move in the same direction. On the other hand, an in- crease in u has an ambiguous effect on

h(.). Similarly, since increasing t with u and v held constant both increases supply relative to the remaining time and increases the advantage of the resource-rich country, such a change has an ambiguous effect on h.

Now suppose that one country, say F, is

extracting at a constant rate, -u', over some interval. Consider the other country's best re- sponse. The Euler equation (6) for t in the interval becomes

(17) +-f"[(-u' - v')/2]v"

1 d - -h(u, v, T - t) 2 dt

1 2- (u' - v')h2(u, v, T- t) =_0.

This simplifies to

(18) f"[(-u' - v')/2]v"

1 1 t

+ ,h3)- u'[h&(.) + h2(.)] 0.

That is, if the net effect of not extracting is to increase the price paid to the exporting coun- try so that h3 < 0, G extracts at an increasing rate until stocks are equalized. Thus, the equilibrium of the differential game leads to an aggregate extraction path that is slower than the intertemporally efficient rate. (Recall that the aggregate extraction rate is continuous in equilibrium.) Alternatively, if the price reduc- tion caused by not extracting dominates the increased bargaining advantage, the equilib- rium extraction path, in aggregate, is faster than the efficient rate. In this unlikely case, the resource-rich country gets the most benefit from its greater stock by using it quickly.

To get a definite characterization of the best-response functions to a totally flat extrac- tion strategy, make the additional assumption that h(u, v, T - t) k[u/(T - t), v/(T - t)]. Then if u't is constant for all t E [0, T], (18) becomes

(19) f"[(-u - v')/2]v"

-k2(.)/2(T - t)2(v - u) = 0.

If v•

< i-0, earlier arguments suggest that k2(0) < 0 because increasing the resource-poor country's stock should reduce the price it must pay to import. Consequently, (19) im- plies that the best response of the resource- poor country to a constant extraction path for the resource-rich country is to extract at an increasing rate, equalizing stock only at T. Alternatively, if 3o > ui, (19) characterizes the resource-rich country's best response to a

480 November 1984 Amer. J. Agr. Econ.

constant rate of extraction by the resource- poor country. In this case, k2(-) > 0 implies v" < 0; this is possible if the increased bargaining strength derived from increasing the re- source-rich country's stock outweighs the ef- fect that increasing supply has on compensa- tion. However, a decreasing rate of extraction is possible here.

Conclusion

This paper has considered the determination of optimal strategic resource reserve policy in a simple model where delaying extraction en- hances a country's bargaining power by mak- ing it less vulnerable to embargo threats. The rationale for holding strategic reserves rests on two stylized facts of world markets for exhaustible resources, particularly oil: that competitive forces are typically too weak to eliminate all other influences on the negotiated price of oil and that trading relationships can- not be organized by long-term contracts be- cause they are unenforceable across national boundaries.

In a simple, two-country model reflecting these features, results show how to charac- terize countries' optimal extraction paths. In a leading special case, these paths take a strik- ingly simple, "bang-bang" form, in which only the resource-rich country extracts, ex- porting to smooth production in both countries until stocks are equalized, at which point au- tarky ensues. Total extraction is typically slower than the efficient path that would result if countries could organize their trading rela- tionship by a single long-term contract. Al- though the assumptions that underlie this characterization-identical technologies, in particular-are very strong, the qualitative features of equilibrium in more general spec- identified here. The robustness of the results is an interesting question, particularly with re- spect to the commitment assumption implicit in the open-loop equilibrium and to the as-

sumption that trade, but not extraction poli- cies, is negotiated.

[Received September 1983; final revision received May 1984.]

References

Bhagwati, Jagdish, and T. N. Srinivasan. "Optimal Trade Policy and Compensation under Endogenous Uncer- tainty: The Phenomenon of Market Disruption." J. Int. Econ. 6(1976):317-36.

Crawford, V. P. "A Theory of Disagreement in Bargain- ing." Econometrica 50(1982):607-37.

Dasgupta, P., and G. Heal. Economic Theory and EAhaus- tible Resources. Welwyn, Great Britain: Cambridge University Press, 1979.

Levhari, D., and L. Mirman. "The Great Fish War: An

Example Using a Dynamic Cournot-Nash Solution.' Bell J. Econ. 11(1980):322-34.

Maskin, E., and D. Newbery. Rational EApectations with Market Power-The ParadoA of the Disadvantage- ous Tariff on Oil. Massachusetts Institute of Tech-

nology Work. Pap. No. 227, Nov. 1978. Mayer, W. "The National Defense Tariff Argument Re-

considered." J. Int. Econ. 7(1977):363-77. Nash, J. "The Bargaining Problem." Econometrica

18(1950): 155-62. -- . "Two-Person Cooperative Games." Economet-

rica 21(1953):128-40. Nichols, A., and R. Zeckhauser. "Stockpiling Strategies

and Cartel Prices." Bell J. Econ. 8(1977):66-96. Reinganum, J., and N. Stokey. Oligopoly EAtraction of a

Nonrenewable, Common Property Resource: The

Importance of the Period of Commitment in Dynamic Games. MEDS Disc. Pap. No. 508, Northwestern

University, 1981. Roth, A. Axiomatic Models of Bargaining. New York:

Springer-Verlag, 1979. Ryder, H. Boycotts and Comparative Advantage: A Dif-

ferential Game Model. IMSSS Tech. Rep. No. 151, Stanford University, 1974.

Sobel, J., and I. Takahashi. "A Multi-Stage Model of

Bargaining." Rev. Econ. Stud. 50(1983):411-26. Sweeney, J. L. "Economics of Depletable Resources:

Market Forces and Intertemporal Bias." Rev. Econ. Stud. 44(1977): 125-41.

Tolley, G., and J. Wilman. "The Foreign Dependence Question." J. Polit. Econ. 85(1977):323-47.