- Email: [email protected]
Strategic resource extraction and substitute development
Resource and Energy Economics 36 (2014) 455–468
Contents lists available at ScienceDirect
Resource and Energy Economics journal homepage: www.elsevier.com/locate/ree
Strategic resource extraction and substitute development夽 Thomas O. Michielsen ∗ New College, University of Oxford, Oxford OX1 3BN, United Kingdom
a r t i c l e
i n f o
Article history: Received 6 February 2013 Received in revised form 10 January 2014 Accepted 8 February 2014 Available online 19 February 2014 JEL classification: O30 Q30 Keywords: Nonrenewable resource Substitute Innovation Closed-loop equilibrium
a b s t r a c t We analyze a dynamic game between a buyer and a seller of a nonrenewable resource. The seller chooses resource supply; the buyer can pay a fixed cost to invent a perfect substitute for the resource at any time. In closed-loop equilibrium, the buyer adopts the substitute when the resource is exhausted. Investing makes the buyer worse off because it decreases resource supply, destroys his ability to derive surplus from the resource through delaying the investment cost incurrence, and causes a larger share of the resource stock to be sold at his reservation price. From the seller’s perspective, the buyer’s ability to develop a substitute is equivalent to an already available substitute with a higher marginal cost. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The oil market has flavours of a bilateral monopoly. The largest exporters have united themselves in OPEC, a cartel that controls more than 75% of proven reserves1 and actively manages supply. Importing countries coordinate on energy policy and energy security issues through various international organizations such as the IEA, OECD and EU, and cooperate in the development of renewable alternatives.
夽 I would like to thank Reyer Gerlagh for helpful comments on earlier drafts of this article, as well as seminar participants at Tilburg University, SURED 2012 and the 19th Annual EAERE Conference. ∗ Tel.: +44 1865 279565.
E-mail addresses: [email protected], [email protected] Known reserves that with reasonable certainty can be recovered in the future under existing economic and operating conditions. Source: BP Statistical Review of World Energy 2012. 1
http://dx.doi.org/10.1016/j.reseneeco.2014.02.001 0928-7655/© 2014 Elsevier B.V. All rights reserved.
456
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
Consuming countries are vulnerable to monopoly power because of their heavy dependency on oil, but also have the means to end this dependency through developing a backstop: a substitute that can replace oil as the dominant energy source. To prolong consumers’ dependence on their resource, oil exporters have an incentive to prevent prices from becoming too high. Indeed, one of OPEC’s aims is to ‘secure an efficient, economic and regular supply of petroleum to consumers’. Conversely, importing countries realize that investment in alternative energy sources, such as shale gas or renewable energy, is not only a remedy to the physical scarcity of oil, but also affects exporters’ supply decisions. The resulting strategic interaction is the subject of this paper: we ask how a nonrenewable resource seller can adjust the supply path to preserve its monopoly, and how a buyer can optimally use the ability to develop a substitute. A key feature of nonrenewable resource markets is that expectations about future demand affect current supply. A binding promise by the buyer in the initial period about the arrival time of the substitute (Dasgupta et al., 1983; Gallini et al., 1983) will therefore not only change market conditions when the substitute comes on line, but also affect the supply path in all preceding periods (Karp and Newberry, 1993). As time passes, the effect on supply at the beginning of the time horizon becomes sunk, so the buyer faces a different trade-off than in the initial period. As a result, the buyer’s optimal open-loop strategy is not time-consistent (Olsen, 1986, 1993): the buyer has an incentive to commit to late development in order to depress current prices, but would like to renege on his promise when the resource becomes scarce (see Section 3). Sellers who are sufficiently rational to calculate dynamic equilibria are unlikely to be naive enough to believe announcements that are not credible. In order to present a realistic model of their supply decision, it is important to exclude non-credible promises by the buyer. The contribution of this paper is to derive the closed-loop solution to the investment and supply game. We use a simple model and a highly stylized representation of the innovation process. In each period, a monopolistic seller makes a supply offer to a buyer. After observing the seller’s offer, the buyer decides whether to pay a fixed investment cost and develop a perfect substitute. Upon investment, the substitute can immediately be produced at constant marginal cost and competes with the resource. We abstract from uncertainty, capacity constraints, R&D externalities and imperfect cartelization in order to focus on the strategic aspects of the resource supply and innovation decisions. In equilibrium, the seller induces the buyer to delay the adoption of the substitute until the resource is exhausted. When the buyer cannot commit to future actions, his only means to influence current supply is to invest immediately. Doing so adversely affects the buyer in three ways. Firstly, the seller immediately reduces supply following investment. Secondly, the buyer loses the ability to benefit from the resource through saving the interest on the investment cost. Thirdly, a larger share of the remaining resource stock is sold at the buyer’s reservation price. From the seller’s perspective, the buyer’s ability to develop a substitute is equivalent to an already available substitute with a higher marginal cost.2 Like in Hoel (1978), the seller limit-prices at the buyer’s reservation price, which is higher than the marginal cost of the substitute because of the fixed investment cost. The buyer’s indifference condition for investment only becomes binding when the remaining resource stock is sufficiently small. The paper highlights the importance of the timing of moves. When the buyer moves after the seller in each period, he is able to punish the seller for supplying a low quantity. This threat disciplines the seller and allows the buyer and seller to attain a higher surplus compared to when the buyer invests immediately. If the buyer must decide whether to invest before observing the seller’s quantity – as in Olsen (1993) – this disciplining device is unavailable, prompting the buyer to develop the substitute before the resource is exhausted.3 Our timing accords better with the sustained and mutually beneficial relationship between buyers and seller in the oil market, as well as the slow progress in renewable
2 The idea that potential competition resembles actual competition has a long tradition in industrial organization (Bain, 1956; Sylos Labini, 1957). The link between periods through the stock sets the resource market apart however: it creates a time-inconsistency for a consumer-surplus-maximizing entrant, and drives the immediate reduction in market supply when the buyer invests. 3 Fujiwara and Long (2011) and Fujiwara and Long (2012) explore the importance of timing when the buyer can levy a tariff on the resource, instead of being able to develop a substitute. The welfare ranking of different timing structures depends on whether the seller sets prices or quantities.
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
457
energy. Developing a substitute does not help industrialized countries to capture a larger share of the oil rent. So as long as oil prices do not become prohibitively high, there is little economic rationale for substantive efforts. Three recent papers consider similar questions. Liski and Montero (2011) study the optimal demand schedule of a nonrenewable resource monopsonist that has access to a substitute. As in our paper, the buyer would like to commit to postpone the switch to the substitute to depress current prices, and the buyer’s share of the resource rent increases in the cost of the substitute when the buyer cannot commit. The mechanism differs however: in Liski and Montero (2011), the resource is exhausted when its price reaches the substitute cost because the sellers are perfectly competitive. Postponing the switch to the substitute decreases the sellers’ reservation price and therefore the current price, but decreases current supply because the resource is used for a longer period. In our framework, the monopolistic seller can marginally undercut the substitute for a nondegenerate period (Olsen, 1993), to the detriment of the buyer’s and social welfare. Postponing the arrival of the substitute increases cumulative resource supply before invention. Current prices go down because current supply goes up, rather than because of an increase in monopsony power as in Liski and Montero (2011). As a result, initial resource use is higher under commitment than under discretion in our paper, but lower in Liski and Montero (2011).4 In Harris and Vickers (1995), the arrival of the substitute is a stochastic event and depends on the buyer’s R&D effort. The buyer’s effort increases when the stock dwindles, so the seller has an incentive to slow down depletion in closed-loop equilibrium. The model Gerlagh and Liski (2011) is the same as ours except that the substitute is not available immediately following investment, but after an exogenous transition period. By investing, the buyer can force the seller to sell the remaining stock during the transition period. The value of this option decreases as the resource stock dwindles. To compensate the buyer for this decrease, supply is increasing over time during an interval before investment. Although Gerlagh and Liski look for a Markov-perfect equilibrium, the time-to-build delay acts as a commitment device. We show that in the absence of this commitment possibility, the buyer only shares in the resource rent because of discounting, because he is no longer able to decrease the current resource price by making credible promises that the substitute’s arrival is imminent. 2. Model Consider a model with a buyer and a seller of a nonrenewable resource. We adopt the notation of Gerlagh and Liski (2011). The buyer derives net (flow) surplus u(q(t)) from consuming q(t) units of the resource. Let u(.) be increasing and continuously differentiable. The net surplus function is u(.) satisfying u(q(t)) = u(q(t)) − u (q(t))q(t). associated with a strictly concave gross surplus function A monopolistic seller is endowed with a finite stock s0 that is costless to extract. The seller aims to maximize the discounted stream of instantaneous profits (q(t)) = u (q(t))q(t). Utility and profits are discounted at rate r. The buyer can instantaneously develop a perfect substitute for the resource. By paying an upfront investment cost I, he gains the ability to produce the substitute at constant marginal cost c. The buyer is then guaranteed a flow net surplus u ≡ u(( u )
−1
(c)).
There are i = 1, . . ., N periods of time of length ε. Time is continuous but agents only act at the beginning of each period, i.e. at time points ti = ε(i − 1). They commit their actions for the entire period. We distinguish between a pre-investment phase A and a post-investment phase B. In phase A, the buyer has a binary decision variable k(ti ) ∈ {0, 1}, where k(ti ) = 1 indicates that the buyer develops the substitute. In phase A, each period has three stages:
4 A formal proof for the general case is beyond the scope of this paper. We provide an example with iso-elastic demand in Appendix D.
458
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
(1) the seller offers to supply a quantity q(ti ), (2) the buyer chooses k(ti ) ∈ {0, 1}, (3) when k(ti ) = 0, the market clears at price p(ti ) = u (q(ti )), or when k(ti ) = 1, the seller’s offer is u (q(ti )), c). rejected. The seller makes a revised offer q(ti ) and the market clears at price p(ti ) = min( At the end of the period, the economy moves to regime B.5 In phase B, the seller is the only mover and chooses quantity q(ti ). The market clears at price p(ti ) = min( u (q(ti )), c). Our timing reflects the mutual dependence of buyers and sellers in natural resource markets, and oil exporters’ concerns about security of supply: if the supply offer is sufficiently generous, the buyer will refrain from investing. Olsen (1993) uses the same setup as our paper, but the buyer moves before the seller in each period. This timing forces the buyer to invest very early: when he can only condition his action on the remaining stock, the buyer cannot credibly punish the seller for setting a high price.6,7,8 We will characterize the limit of the equilibrium when ε goes to zero.9,10 For convenience, the remainder of the analysis is therefore presented in continuous time with an infinite horizon. The only difference with Gerlagh and Liski (2011) is that in their paper, when the buyer rejects the offer and invests the seller makes a revised offer and the game enters a transition period of exogenous length h > 0 in which the market clears at price p(ti ) = u (q(ti )), after which the economy moves to regime B. 3. The time-consistency problem Olsen (1993) sketches why equilibria in which the buyer can commit the investment time are not consistent. According to the Hotelling rule, a monopolistic seller equates discounted marginal revenue −1
in all periods. When the arrival time of the substitute T is given, supplying q∗ ≡ ( u ) (c) in any period after T does not cause inframarginal losses because the resource price is fixed by the cost of the substitute (we assume the buyer cannot discriminate between the resource and the substitute). The seller does incur inframarginal losses before T , when the substitute is not present. Because marginal revenue is higher after the arrival of the substitute than before, the seller prefers to sell part of his stock after T . This is socially inefficient and reduces the buyer’s welfare, because resource units sold after T generate no consumer surplus in excess of what the substitute could provide. When the buyer commits to a late T , the seller’s post-investment revenues go down in present terms because of discounting. This makes it less attractive for the seller to marginally undercut the substitute after T , thereby reducing cumulative post-investment supply. A late T results in higher cumulative extraction in the pre-investment phase and increases current supply, but is costly for the buyer in later periods: when the resource becomes scarce, the buyer cannot dispose over the substitute as early as he would like. After having enjoyed the benefits of high supplies and low prices early on, the buyer has an incentive to avoid the costs of resource scarcity in subsequent periods by developing the substitute earlier than announced. We illustrate this incentive in Fig. 1. After some point, which we denote by T˜ , supply drops below the buyer’s reservation quantity q˘ – which is
5 In Gerlagh and Liski (2011), the economy moves into a transition regime in which the seller offers a quantity q(ti ) in each u (q(ti )). The terminal phase that follows this transition regime is the same as stage, with the market clearing at price p(ti ) = our phase B. 6 The exception is when the initial stock is small and within a sequence of equidistant ‘nontrigger intervals’ (Olsen, 1993). 7 The crucial difference between the timing in this paper and in Olsen (1993) is that the buyer moves in between the seller’s proposal and the market clearance. Reversing stages (2) and (3) would yield the same equilibrium as in Olsen (1993): after the seller’s offer in the initial period, the residual game is equivalent to Olsen (1993). 8 Enabling the buyer to condition his strategy on (s(ti ), q(ti )) also allows some less realistic equilibria, as we will see in Section 5. 9 The limit is taken analogously to Gerlagh and Liski (2011). The buyer’s ability to develop the substitute guarantees that the strategic interaction ends in finite time. When N is sufficiently large, changes in N do not affect the equilibrium. Moreover, when the number of remaining periods is large, the equilibrium strategies do not depend on the number of remaining periods. 10 In general, it is not clear whether the limit of the equilibrium is an equilibrium of the limit game; see Simon and Stinchcombe (1989) for an example.
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
459
Fig. 1. The buyer’s time-consistency problem.
defined such that u(˘q) equals the net surplus u of consuming the substitute minus the interest on the investment cost rI – and the buyer prefers to invest immediately instead of at the announced time T . 4. Post-investment phase When making decisions during phase A, agents take into account their payoffs in phase B. We therefore first turn to phase B. We assume that when the resource and substitute are available at the same price, consumers prefer the resource.11 In the limit when ε goes to zero, the seller’s problem in phase B is:
∞
max q(t)
min( u (q(t)), c)q(t)e−rt dt,
0
s.t. s˙ = −q(t),
(1)
q(t)≥0, s(t)≥0, s0 given.
The Hamiltonian for this problem and the necessary optimality conditions are H(q, s, ) = min( u (q(t)), c)q(t) − (t)q(t), (q(t)) − (t) ≤ 0, ˙ = r(t), (t)
q(t)≥q∗ complementary slackness,
(2)
lim (t)s(t) = 0,
t→∞
where is the scarcity value of the resource and
(q(t))
=
u (q(t)) + q(t) u (q(t)) for q(t)≥q∗ c
for q(t) < q∗
.
Because a perfect substitute is available at cost c, marginal revenue (q(t)) is equal to c for the first q∗ units and discontinuous at q∗ , as the seller starts incurring losses on inframarginal units.
11
Otherwise, the seller will offer the resource at an infinitesimally lower price, without changing the analysis.
460
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
Fig. 2. The seller’s optimal supply (left) and the present value of marginal revenue (right) during the post-investment phase.
Proposition 1 (Post-investment phase, Hoel (1978) and Salant (1979)). Let s∗ ≡ −
q∗ ln r
(q∗ ) c
.
When the remaining stock s < s∗ , q(t) = q∗
∀ t ∈ (0, T ), T =
s , q∗
where T denotes the exhaustion time of the resource. When s > s∗ , q(t) = ( )−1 (ce−r(T −t) ) ∀ t ∈ [0, T − ], q(t) = q∗
T −
s.t.
(3)
∀ t ∈ [T − , T ], ( )
−1
(ce−r(T −t) ) dt + q∗ = s,
(4)
0
where ≡
s∗ q∗
is the length of the limit-pricing phase. The above system defines the seller’s optimal post-
investment strategy I (s ; c). The seller compares whether it is more profitable to use a marginal resource unit to extend the limitpricing phase (in which q(t) = q∗ ) or to increase supply today. If the seller practices a limit-pricing strategy until exhaustion, the stock is exhausted at time s/q∗ and the present value of the marginal ∗ revenue of the last unit is ce−rs/q . The marginal revenue of supplying a unit in excess of q∗ today is −1
u (( u ) (c)) << c), but not subject to discounting. discretely lower in current value ( (q∗ ) = c + q∗ ∗ ∗ −s/q For small initial stocks, (q ) is smaller than ce and the seller supplies q∗ until exhaustion. When the remaining stock is large, limit pricing takes so long to exhaust the stock that (q∗ ) is ∗ higher than ce−rs(t)/q . It is then worthwhile to supply q(t) > q∗ today.12 In equilibrium, the seller is indifferent between marginally extending the limit-pricing phase and selling an extra unit in any period before the limit-pricing phase, i.e. between [0, T − ]. Fig. 2 illustrates the seller’s supply path during the post-investment phase. 5. Closed-loop equilibrium It will be useful to define the value function for the seller V(s) and buyer U(s) as a function of the remaining stock during phase A. Denote the buyer’s and seller’s value at the time of investment by 12 We implicitly assume that is positive for q ≥ q∗ , implying elastic demand, to rule out the case in which the seller always supplies q∗ until exhaustion regardless of the remaining stock (Andrade de Sà et al., 2012).
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
461
Fig. 3. Resource supply before and after investment as a function of time (left) and the remaining stock (right).
UI (s) and VI (s), respectively. Let k = (s, q) be the buyer’s investment strategy and q = (s) the seller’s supply strategy. The value for the seller is the payoff from choosing the optimal q for an arbitrarily small interval ε, given the strategic response of the buyer V (s) = max{[ε(q) + e−εr V (s − εq)](1 − (s, q)) + V I (s)(s, q)}. {q}
(5)
We define the value for the buyer analogously U(s) = max {[εu((s)) + e−εr U(s − ε(s))](1 − k) + U I (s)k}.
(6)
k∈{0,1}
The post-investment values UI (s) and VI (s) are determined by the dynamics described in Proposition 1. We look for a pair of equilibrium strategies such that (s) = argmax{q} V(s) and (s, q) = argmaxk U(s) for all s. Proposition 2 states the main result. Proposition 2. Let q˘ ≡ u−1 (u − rI). There is a continuum of subgame perfect equilibria in which the buyer’s strategy does not depend on the remaining stock that are indexed by the buyer’s threshold level q ∈ [˘q, q∗ ] and given by (s, q) =
1 ∀ q : q < q,
(7a)
(s, q) =
0 otherwise,
(7b)
(s) = (s; u ( q)), I
with
I (s ; c)
(7c)
as defined in Proposition 1.
Although the motivation differs in the pre- and post-investment phase, the seller always supplies at least the buyer’s reservation quantity. In the post-investment phase, the seller always supplies q ≥ q∗ because there is no loss on inframarginal units for smaller q. In the pre-investment phase, the seller’s best response to the buyer’s equilibrium strategy is to supply at least q≥ q, because he would trigger investment for smaller q. From the seller’s perspective, the threat of the substitute is therefore equivalent to an already available substitute with a higher marginal cost. The buyer does not invest before exhaustion because his benefit from the resource – the discounted surplus stream minus his reservation surplus stream from the substitute – decreases after investment. Adopting the substitute raises the buyer’s reservation surplus from u − rI to u, but not his actual surplus as the buyer pays interest rI on the sunk investment cost. The increase in the buyer’s reservation surplus has three q = q˘ equilibrium. effects as we shown in Fig. 3, which depicts the Firstly, the buyer’s benefit from the resource is lower for any given supply path: u(q) − u(˘q) > u(q) − u ∀ q. Secondly, the seller immediately decreases supply when the buyer invests: (s) > I (s) when s > s∗ (Hoel, 1978, 1983). To see this, imaginarily divide the seller’s remaining stock in two parts
462
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
and restate his problem as a free-terminal-time problem with a scrap value. Deciding when to exhaust the first stock and switch to using the second, the seller balances two concerns: by prolonging the use of the first stock, he can increase his profits from that stock by smoothing the sales over a longer period. On the other hand, he has to wait longer for the profits from the second stock, which go down in present terms because of discounting. When the buyer develops the substitute, the second concern becomes comparatively less important: because the maximum price of the resource goes down, so does the maximum profit from selling the second stock. As a consequence, the seller prolongs the use of the first stock and reduces current supply.13,14 Thirdly, a larger share of the resource stock is sold at the buyer’s reservation price: the area under the horizontal part of the dashed curve in the left panel of Fig. 3 is larger than the area under the horizontal part of the solid curve. Within this range of equilibria, the seller strictly prefers strategies with a low q. The buyer’s ranking depends on the utility function. When the buyer’s net flow surplus is convex in q, as with linear demand, the buyer would like to front-load supply with q = q˘ . When net flow surplus is concave, the buyer may q to increase his surplus in the limit-pricing phase.15 Strategies in which the buyer’s prefer a higher threshold q is above q∗ or below q˘ cannot be an equilibrium when (s) = I (s; u ( q)). For q > q∗ , the seller is insensitive to the buyer’s threat. Strategies with q < q˘ are time-inconsistent for the reason in Section 3: the buyer wants to renege and invest when supply drops below q˘ . Corollary 1.
limU(s) =
r→0
u r
− I.
The buyer benefits from the resource when the seller is too impatient to sell the entire stock at the buyer’s reservation price. As r approaches zero, this consideration vanishes and the presence of the resource does not make the buyer better off than if he only had access to the substitute. The equilibria discussed above have the attractive property that they do not require the buyer to have detailed knowledge of remaining reserves, which are a potential source of asymmetric information,16 but they are not exhaustive. There are trivial equilibria with immediate investment, e.g. (s, q) = 1, (s) = 0 ∀ (s, q). If we consider equilibrium strategies where (s, q) depends on s as well as q, the range of equilibria becomes even larger. For example, the buyer may threaten to invest unless the seller implements a specific supply path, including the buyer’s first-best qB (s). Since lim V I (s) = 0, c→0
the seller’s indifference condition is always satisfied for small c, and (s) = qB (s), (s, qB (s)) = 0, (s, / qB (s) is formally a subgame-perfect equilibrium, even though the credibility of such q) = 1 ∀ q = ‘take-it-or-leave-it’ strategies in the real world is debatable.17 In the Appendix, we show that the result in Proposition 2 is not driven by the discrete nature of the buyer’s investment decision. We derive the same results when the buyer can invest a continuous amount in each period and the marginal cost of the substitute depends on cumulative investment. The buyer invests the amount that maximizes consumption utility minus the interest on the investment costs in one go when the resource is exhausted. Similarly, the seller supplies at least the buyer’s q until exhaustion. This extension is a limiting case of the model in Jaakkola (2012), threshold quantity which differs from Appendix C only in that it has convex instead of linear per-period investment costs. The convex costs in Jaakkola (2012) cause the buyer to invest in the substitute more gradually, but prohibit an analytical characterization of the closed-loop equilibrium. Jaakkola solves the closed-loop equilibrium numerically. The buyer develops the substitute more slowly than under commitment, for a similar reason as in the present paper: more rapacious investment leads to higher resource prices and lower supply in the short term. 13
This intuition is paraphrased from Hoel (1983).
Because the initial price is higher in equilibria with a high q, an increase in q lowers the seller’s total value of the stock V but increases the marginal value V when s > s∗ . If we were to endogenize s0 by allowing the seller to increase his resource base before the initial period at increasing marginal cost (as in van der Ploeg (2012)), exploration investments would be higher if q. the seller anticipates a high 15 There also exist mixed-strategy versions of the equilibria in Proposition 2 in which the buyer invests with probability ∈ (0, q. The seller’s gain from deviating for an arbitrarily small period is always smaller than (V(s) − VI (s)). 1) when q < 16 Sauré (2010) and Gerlagh and Liski (2012) argue that sellers have a strategic motive not to disclose the true value of these reserves. 17 Though not explicitly mentioned in that paper, these strategies are also equilibria in Gerlagh and Liski (2011). 14
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
463
6. Conclusion Binding promises about the adoption time of a substitute can give nonrenewable resource buyers a larger share of the resource rent but are not time-consistent. When the buyer cannot commit to the arrival time of the substitute, a monopolistic seller can induce the buyer to delay the invention of the substitute until the resource is depleted. In the absence of uncertainty or a fixed time-to-build delay, the threat of a substitute has a similar effect as an already available substitute in closed-loop equilibrium. We model the investment decision as binary for expositional clarity, but all results go through with continuous investment. The model is highly stylized – for example, OPEC and industrialized economies both face substantial coordination and free-riding problems that impede their ability to act as a monopolist or jointly develop a substitute. Nonetheless, the results accord with the slow technological progress in renewable energy and oil exporters’ efforts to stabilize prices, as unconventional hydrocarbons present an increasing threat to their revenues. Appendix A. Proof of Proposition 1 The seller supplies q∗ at time t when the MR of supplying an additional unit is lower than the discounted MR at the time of exhaustion, i.e. (q∗ ) ≤ (t) = (T )e−r(T −t) = ce−r(T −t) . Solving for T − t yields T −t ≤−
1 ln r
(q∗ ) c
.
(A.1)
The seller supplies q∗ in each period until exhaustion when this inequality holds for all t ∈ (0, (s/q∗ )). We can calculate the threshold stock s∗ such that the seller practices limit pricing until exhaustion for all s ≤ s∗ by substituting T = (s/q∗ ) and solving for the s such that (A.1) holds with equality for t = 0 s∗ = −
q∗ ln r
(q∗ ) c
.
When s > s∗ , the seller extracts the last s∗ units in a period of length = (s∗ )/(q∗ ). While extracting the first s − s∗ units, marginal revenue rises at the rate of interest and is equal the present value of marginal revenue at the time of exhaustion (q(t)) = ce−r(T −t) ⇒ q(t) = ( )
−1
(ce−r(T −t) ).
The stock constraint determines T
T −
( )
−1
(ce−r(T −t) ) dt + q∗ = s.
0
Appendix B. Proof of Proposition 2 We make use of a special case of Proposition 3 in Hoel (1983). Lemma 1 ((Post-investment phase, Hoel (1983))). When s > s∗ , I (s ; c) is increasing in c.
(A.2)
464
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
It will be useful to include c as a parameter argument to U(.) and u. We also write T(s ; c) and (c) to link the exhaustion time and the length of the limit-pricing phase to the cost of the substitute. The buyer’s strategy maximizes U(s ; c) with respect to k. His value from investing is
∞
T (s;c)−(c) ∞ =
T (s;c)−(c)
u(c)e−rt dt − I +
U I (s; c) =
(u(c) − rI)e
−rt
0 T (s;c)
dt +
0
u(I (s(t); c))e−rt dt
[u(I (s(t); c)) − u(c)]e−rt dt,
0
˙ s(t) = −I (s(t); c), s(0) = s.
(s) denote the buyer’s discounted surplus stream when he adopts the equilibrium strategy with Let U threshold q, and denote the corresponding exhaustion time and length of the limit-pricing phase by T (s; c) and (c), respectively. We have (s; c) = U =
∞
T (s;c) ∞ T (s;c) ∞
T (s;c) u(c)e−rt dt − e−r I+
(u(c) − rI)e−rt dt +
u(I (s(t); u ( q)))e−rt dt
0 T (s;c)
u(I (s(t); u ( q)))e−rt dt
0
(u(c) − rI)e−rt dt +
=
T (s;c)
T (s;c)
0
(u(I (s(t); u ( q))) − [u(c) − rI])e−rt dt,
0
˙ s(t) = −I (s(t); u ( q)), s(0) = s.
= (c). Comparing UI (s ; c) and U (s; c), we have u ( q)≥c, T (s; c) = T (s; c ) and (c) For cˆ ≡
∞
U I (s; c) =
(u(c) − rI)e
−rt
0
I
T (s;c)−(c)
I
dt +
[u( (s(t); c)) − u(c)]e
=0
[u( (s(t); c)) − u(c)]e
−rt
˙ dt, s(t) = −I (s(t); c), s(0) = s,
T (s;c)−(c)
(s; c) = U
∞
(u(c) − rI)e−rt dt +
0
≥0
T (s;c)−(c)
T (s;c)−(c)
(u(I (s(t); c )) − [u(c) − rI])e−rt dt+
0
(u(I (s(t); c )) − [u(c) − rI])e−rt dt+ ≥0
T (s;c)
II
T (s;c)−(c)
dt+
I
−rt
0
T (s;c)
˙ (u(I (s(t); c )) − [u(c) − rI])e−rt dt, s(t) = −I (s(t); c ), s(0) = s.
T (s;c)−(c)
By Lemma 1, I (s ; c) increases in c for s > s∗ , which implies II > I and, together with the increase in the limit quantity when c goes up, that the length of the non-limit-pricing phase T(s ; c) − (c) increases in c.18
18
Also note that
∂ ∂c
(q) (q) ≤ 0 when −q u˜u˜ (q) ≤ −q u˜u˜ (q) + 1.
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
465
q given that the buyer only invests when q < q. When the seller The seller always supplies at least q, he triggers immediate investment and is subject to the more stringent constraint chooses a q(t) < q(t) ≥ q∗ for the remainder of the game. Choosing q(t) < q therefore cannot be optimal. Appendix C. Continuous investment In this section, we generalize the buyer’s problem: rather than making a discrete investment decision, the buyer can invest a continuous amount in each period. The marginal cost of the substitute is a decreasing function of cumulative investment
t
1+
c=
c(X), c (X) ≤ 0, c (X)≥0, X(t) =
ε
εx(ti ).
i=1
We can write u(X) ≡ u(( u )
−1
(c(X))). The timing in each stage is
(1) the seller supplies a quantity q(ti ), (2) the buyer chooses x(ti ) ≥ 0, (3) the market clears at price p(ti ) = min( u (q(ti )), c(X(ti ))). Profits and utility depend on the stance of technology X in addition to the supply q: (X, q) = q min( u (q), c(X)) and u(X, q) = max(u(q), u(X)). We assume that investments at ti can immediately affect the market price during the interval (ti , ti+1 ), though this is inconsequential for the equilibria when the period length is small. Let (s, X) denote the seller’s supply strategy and (s, X, q) the buyer’s investment strategy. The value functions are given by V (s, X) = max{ε(X + ε (s, X, q), q) + e−εr V (s − εq, X + ε (s, X, q))},
(C.1)
U(s, X) = max{ε(u(X + εx, (s, X)) − x) + e−εr U(s − ε(s, X), X + εx)}.
(C.2)
{q}
{x}
Utility from the substitute is a concave function of cumulative investment. Assumption 1.
2
∂u ≥0, ∂ u < 0. ∂X ∂X
The range of equilibria in which (s, X, q) does not depend on s is similar to Proposition 2. The buyer selects the cumulative investment level X∗ that yields the highest long-run utility u(X) − rX, and invest this amount in one go when the resource is exhausted. Proposition 3. Let X∗ be the (unique) solution to (∂u)/(∂X) = r and let ε be arbitrarily small. There is a continuum of subgame perfect equilibria in which the buyer’s strategy does not depend on the remaining q ∈ [u−1 (u(c(X ∗ )) − rX ∗ ), u−1 (u(c(X ∗ )))] and given by stock that are indexed by the buyer’s threshold level (s, X, q) =
X∗ − X ∀ q : q < q, ε
(C.3a)
(s, X, q) =
0 otherwise,
(C.3b)
(s, X) = max(I (s; u ( q)), ( u )
−1
(c(X))),
(C.3c)
with I (s ; c) as defined in Proposition 1. Proof. Any strategy with (s, X, q) > 0 for s > 0 cannot be an optimal response to (C.3c). By Proposition 2, when faced with a binary choice between either investing a positive amount now and zero afterward until exhaustion or investing the same positive amount at exhaustion and zero in all preceding periods,
466
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
the buyer prefers the latter.19 Because the buyer always wants to postpone his ‘last’ investment until exhaustion, by induction investment is zero when the resource stock is positive. When the resource is exhausted and X < X∗ , it is optimal for the buyer to reach the long-run optimal level X∗ as quickly as possible. Suppose that the buyer’s strategy entails x1 = (0, X1 , 0) > 0, x2 = (0, X2 , 0) > 0 for some 0 < X1 < X2 < X∗ . Then by Assumption 1, the buyer can marginally improve his welfare by choosing (0, X1 , 0) = (X2 − X1 )/(ε) + x2 . By induction it follows that the buyer invests (X∗ − X)/ε when the resource is exhausted. The proof that the seller’s strategy (C.3c) is optimal given buyer’s strategy (C.3a)–(C.3b) is analogous to the last part of the proof of Proposition 2. Appendix D. Commitment and subgame perfect equilibria In this section, we provide an example with an iso-elastic demand function in which a monopsonistic buyer that already has developed a substitute, as in Liski and Montero (2011), chooses a lower initial level of resource consumption under commitment than in closed-loop equilibrium. In the bilateral monopoly framework that we study in this paper, initial consumption is higher when the buyer can commit the invention time than in closed-loop equilibrium. √ Let u(q) = q, which corresponds to an iso-elastic demand with elasticity 2. We numerically compute the resource consumption paths in Liski and Montero’s framework when the buyer can fully commit q(t) for all t, in closed-loop equilibrium (where the buyer’s strategy q(t) = C(s(t)) is a function of the remaining stock) and in the social optimum. When the buyer has full commitment power, his problem is
Vt=0 ˙ s.t. s(t)
{q(t),T }
T
{
= max
q(t) − p(t)q(t)}e−rt dt +
0
= −q(t) s0 > 0,
1 1 −rT e r 4c
s(t) = 0,
˙ = rp(t), p(T ) = c, p(t) where T denotes the time at which the resource is exhausted and the buyer switches to the substitute. Liski and Montero (2011, p. 6) show that the buyer’s commitment optimum is fully characterized by 1 −1 1 ∂q 1 q 2 = √ + 2 q 4r ∂t 1 1 q(t) = − 4c 2
0,
(D.1a)
crs0 .
(D.1b)
In closed-loop equilibrium, the buyer’s value function is
∞
V (s(t)) =
[
C(s(t)) − P(s(t))C(s(t))]e−r( −t) d ,
t
where q(t) = C(s(t)) and p(t) = P(s(t)) are the equilibrium consumption and pricing rule, respectively. The authors demonstrate (p. 9) that the equilibrium is given by a first-order ODE in the pricing rule 1 2
−
P (S) − P(S) + P (S)S = 0, rP(S)
(D.2)
with boundary condition P(0) = c. Lastly, the social optimum is characterized by the Hotelling rule, p(t) = u (q(t)) and p(T) = c and has an analytical solution: T = 1/(2r) ln(1 + 8rs0 c2 ). Using MATLAB, we numerically solve (D.1) and (D.2) to determine the consumption paths under commitment and in
19
∗ −1 ∗ −1 ∗ The buyer strictly prefers )))) and if qˆ = u−1 (u(c(X ∗ )) − to delaythe investment if qˆ ∈ (u (u(c(X )) − rX ), u (u(c(X −1 (c(X ∗ ))
rX ∗ ), s > − (u˜ )
r
ln
((u˜ )−1 (c(X ∗ ))) c(X ∗ )
indifferent between investing zero and
−1 (c(X ∗ ))
. When qˆ = u−1 (u(c(X ∗ )) − rX ∗ ) and s ≤ − (u˜ )
X ∗ −X ε
.
r
ln
((u˜ )−1 (c(X ∗ ))) c(X ∗ )
, the buyer is
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
467
3
2.5
Closed−loop Commitment Social Optimum
Resource use
2
1.5
1
0.5
0
0
10
20
30
40
50
Time Fig. 4. Resource consumption in Liski and Montero (2011).
closed-loop equilibrium for s0 = 20, c = 0.5, r = 0.04.20 Fig. 4 depicts the results. In line with the intuition in the main text, resource use in Liski and Montero (2011) is lower under commitment than in closedloop equilibrium. When the buyer has full commitment power, p(t) = p(T)e−r(T−t) . By committing to a late switch to the substitute, the buyer reduces initial extraction but obtains a substantial price discount. Now return to the bilateral monopoly model in this paper. Dasgupta et al. (1983) and Olsen (1986) discuss the buyer’s optimal invention time under commitment at length. We demonstrate the following result through a series of Lemmas. Proposition 4. For iso-elastic demand with elasticity equal or larger than 2 and a sufficiently large initial stock, initial resource use is higher when the buyer commits the invention time at t = 0 than in closed-loop equilibrium. Lemma 2 ((Proposition 1, Olsen (1986))). For iso-elastic demand with elasticity equal or greater than 2 and s0 sufficiently large, the seller has zero stock remaining at the buyer’s optimal invention time. Lemma 3 ((Dasgupta et al. (1983), p. 1442)). Resource supply at t = 0 increases in the invention time between 0 and the buyer’s optimal invention time. Lemma 4 ((Dasgupta et al. (1983), p. 1442)). For iso-elastic demand with elasticity equal or greater than 2 and s0 sufficiently large, resource supply at t = 0 decreases in the buyer’s invention time when the invention time is greater than the buyer’s optimum. Lemma 1 implies that resource supply at t = 0 when the buyer commits to an infinitely large invention time (which is equivalent to c→ ∞) is larger than initial resource supply in closed-loop equilibrium. Then Lemmas 2–4 establish the Proposition. References Andrade de Sà, S., Chakravorty, U., Daubanes, J., 2012. Limit-pricing Oil When Demand is Inelastic, Resource Taxation and Substitutes Subsidies. lies http://www.cer.ethz.ch/sured 2012/programme/SURED-12 042 AndradeDeSa Chakravorty Daubanes.pdf
20
The code is available upon request.
MonopoMimeo
468
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
Bain, J.S., 1956. Barriers to New Competition. Harvard University Press, Cambridge, MA. Dasgupta, P., Gilbert, R.J., Stiglitz, J.E., 1983. Strategic considerations in invention and innovation: the case of natural resources. Econometrica 51 (5), 1439–1448. Fujiwara, K., Long, N.V., 2011. Welfare implications of leadership in a resource market under bilateral monopoly. Dynamic Games and Applications 1, 479–497. Fujiwara, K., Long, N.V., 2012. Optimal tariffs on exhaustible resources: the case of quantity setting. International Game Theory Review 14 (4), 1240004, http://dx.doi.org/10.1142/S021919891240004X. Gallini, N., Lewis, T., Ware, R., 1983. Strategic timing and pricing of a substitute in a cartelized resource market. Canadian Journal of Economics 16 (3), 429–446. Gerlagh, R., Liski, M., 2012. Resource Relationships. http://hse-econ.fi/liski/papers/Relationships.pdf Gerlagh, R., Liski, M., 2011. Strategic resource dependence. Journal of Economic Theory 146 (2), 699–727. Harris, C., Vickers, J., 1995. Innovation and natural resources: a dynamic game with uncertainty. RAND Journal of Economics 26 (3), 418–430. Hoel, M., 1978. Resource extraction, substitute production, and monopoly. Journal of Economic Theory 19, 28–37. Hoel, M., 1983. Future conditions and present extraction: a useful method in natural resource economics. Resources and Energy 5, 303–311. Jaakkola, N., 2012. Green technologies and the protracted end to the age of oil: a strategic analysis. OxCarre Research Paper 99. http://www.oxcarre.ox.ac.uk/files/OxCarreRP201299.pdf Karp, L., Newberry, D.M., 1993. Intertemporal consistency issues in depletable resources. In: Kneese, A.V., Sweeny, J.L. (Eds.), In: Handbook of Natural Resource and Energy Economics, vol. III. North Holland, Amsterdam. Liski, M., Montero, J.-P., 2011. On the Exhaustible-resource Monopsony. http://www.hse-econ.fi/liski/papers/HotCoase.pdf Olsen, T.E., 1993. Perfect equilibrium timing of a backstop technology. Journal of Economic Dynamics and Control 17, 123–151. Olsen, T.E., 1986. Strategic considerations in invention and innovation: the case of natural resources revisited. Econometrica 56 (4), 841–849. Salant, S.W., 1979. Staving off the backstop: dynamic limit pricing with a kinked demand curve. In: Pindyck, R.S. (Ed.), In: Advances in the Economics of Energy and Resources, vol. 2. JAI Press, Greenwich, CT, pp. 187–204. Sauré, P., 2010. Overreporting oil reserves. Swiss National Bank Working Paper 2010-7. http://www.relooney.info/SI Oil-Politics/Oil-Reserves 1.pdf Simon, L.K., Stinchcombe, M.B., 1989. Extensive form games in continuous time: pure strategies. Econometrica 57 (5), 1171–1214. Sylos Labini, P., 1957. Oligopolio e progresso tecnico. Milan, Giuffre. van der Ploeg, F., 2012. Breakthrough renewables and the green paradox. CESifo Working Paper 3986. http://www.econstor.eu/bitstream/10419/66869/1/730415155.pdf
Contents lists available at ScienceDirect
Resource and Energy Economics journal homepage: www.elsevier.com/locate/ree
Strategic resource extraction and substitute development夽 Thomas O. Michielsen ∗ New College, University of Oxford, Oxford OX1 3BN, United Kingdom
a r t i c l e
i n f o
Article history: Received 6 February 2013 Received in revised form 10 January 2014 Accepted 8 February 2014 Available online 19 February 2014 JEL classification: O30 Q30 Keywords: Nonrenewable resource Substitute Innovation Closed-loop equilibrium
a b s t r a c t We analyze a dynamic game between a buyer and a seller of a nonrenewable resource. The seller chooses resource supply; the buyer can pay a fixed cost to invent a perfect substitute for the resource at any time. In closed-loop equilibrium, the buyer adopts the substitute when the resource is exhausted. Investing makes the buyer worse off because it decreases resource supply, destroys his ability to derive surplus from the resource through delaying the investment cost incurrence, and causes a larger share of the resource stock to be sold at his reservation price. From the seller’s perspective, the buyer’s ability to develop a substitute is equivalent to an already available substitute with a higher marginal cost. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The oil market has flavours of a bilateral monopoly. The largest exporters have united themselves in OPEC, a cartel that controls more than 75% of proven reserves1 and actively manages supply. Importing countries coordinate on energy policy and energy security issues through various international organizations such as the IEA, OECD and EU, and cooperate in the development of renewable alternatives.
夽 I would like to thank Reyer Gerlagh for helpful comments on earlier drafts of this article, as well as seminar participants at Tilburg University, SURED 2012 and the 19th Annual EAERE Conference. ∗ Tel.: +44 1865 279565.
E-mail addresses: [email protected], [email protected] Known reserves that with reasonable certainty can be recovered in the future under existing economic and operating conditions. Source: BP Statistical Review of World Energy 2012. 1
http://dx.doi.org/10.1016/j.reseneeco.2014.02.001 0928-7655/© 2014 Elsevier B.V. All rights reserved.
456
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
Consuming countries are vulnerable to monopoly power because of their heavy dependency on oil, but also have the means to end this dependency through developing a backstop: a substitute that can replace oil as the dominant energy source. To prolong consumers’ dependence on their resource, oil exporters have an incentive to prevent prices from becoming too high. Indeed, one of OPEC’s aims is to ‘secure an efficient, economic and regular supply of petroleum to consumers’. Conversely, importing countries realize that investment in alternative energy sources, such as shale gas or renewable energy, is not only a remedy to the physical scarcity of oil, but also affects exporters’ supply decisions. The resulting strategic interaction is the subject of this paper: we ask how a nonrenewable resource seller can adjust the supply path to preserve its monopoly, and how a buyer can optimally use the ability to develop a substitute. A key feature of nonrenewable resource markets is that expectations about future demand affect current supply. A binding promise by the buyer in the initial period about the arrival time of the substitute (Dasgupta et al., 1983; Gallini et al., 1983) will therefore not only change market conditions when the substitute comes on line, but also affect the supply path in all preceding periods (Karp and Newberry, 1993). As time passes, the effect on supply at the beginning of the time horizon becomes sunk, so the buyer faces a different trade-off than in the initial period. As a result, the buyer’s optimal open-loop strategy is not time-consistent (Olsen, 1986, 1993): the buyer has an incentive to commit to late development in order to depress current prices, but would like to renege on his promise when the resource becomes scarce (see Section 3). Sellers who are sufficiently rational to calculate dynamic equilibria are unlikely to be naive enough to believe announcements that are not credible. In order to present a realistic model of their supply decision, it is important to exclude non-credible promises by the buyer. The contribution of this paper is to derive the closed-loop solution to the investment and supply game. We use a simple model and a highly stylized representation of the innovation process. In each period, a monopolistic seller makes a supply offer to a buyer. After observing the seller’s offer, the buyer decides whether to pay a fixed investment cost and develop a perfect substitute. Upon investment, the substitute can immediately be produced at constant marginal cost and competes with the resource. We abstract from uncertainty, capacity constraints, R&D externalities and imperfect cartelization in order to focus on the strategic aspects of the resource supply and innovation decisions. In equilibrium, the seller induces the buyer to delay the adoption of the substitute until the resource is exhausted. When the buyer cannot commit to future actions, his only means to influence current supply is to invest immediately. Doing so adversely affects the buyer in three ways. Firstly, the seller immediately reduces supply following investment. Secondly, the buyer loses the ability to benefit from the resource through saving the interest on the investment cost. Thirdly, a larger share of the remaining resource stock is sold at the buyer’s reservation price. From the seller’s perspective, the buyer’s ability to develop a substitute is equivalent to an already available substitute with a higher marginal cost.2 Like in Hoel (1978), the seller limit-prices at the buyer’s reservation price, which is higher than the marginal cost of the substitute because of the fixed investment cost. The buyer’s indifference condition for investment only becomes binding when the remaining resource stock is sufficiently small. The paper highlights the importance of the timing of moves. When the buyer moves after the seller in each period, he is able to punish the seller for supplying a low quantity. This threat disciplines the seller and allows the buyer and seller to attain a higher surplus compared to when the buyer invests immediately. If the buyer must decide whether to invest before observing the seller’s quantity – as in Olsen (1993) – this disciplining device is unavailable, prompting the buyer to develop the substitute before the resource is exhausted.3 Our timing accords better with the sustained and mutually beneficial relationship between buyers and seller in the oil market, as well as the slow progress in renewable
2 The idea that potential competition resembles actual competition has a long tradition in industrial organization (Bain, 1956; Sylos Labini, 1957). The link between periods through the stock sets the resource market apart however: it creates a time-inconsistency for a consumer-surplus-maximizing entrant, and drives the immediate reduction in market supply when the buyer invests. 3 Fujiwara and Long (2011) and Fujiwara and Long (2012) explore the importance of timing when the buyer can levy a tariff on the resource, instead of being able to develop a substitute. The welfare ranking of different timing structures depends on whether the seller sets prices or quantities.
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
457
energy. Developing a substitute does not help industrialized countries to capture a larger share of the oil rent. So as long as oil prices do not become prohibitively high, there is little economic rationale for substantive efforts. Three recent papers consider similar questions. Liski and Montero (2011) study the optimal demand schedule of a nonrenewable resource monopsonist that has access to a substitute. As in our paper, the buyer would like to commit to postpone the switch to the substitute to depress current prices, and the buyer’s share of the resource rent increases in the cost of the substitute when the buyer cannot commit. The mechanism differs however: in Liski and Montero (2011), the resource is exhausted when its price reaches the substitute cost because the sellers are perfectly competitive. Postponing the switch to the substitute decreases the sellers’ reservation price and therefore the current price, but decreases current supply because the resource is used for a longer period. In our framework, the monopolistic seller can marginally undercut the substitute for a nondegenerate period (Olsen, 1993), to the detriment of the buyer’s and social welfare. Postponing the arrival of the substitute increases cumulative resource supply before invention. Current prices go down because current supply goes up, rather than because of an increase in monopsony power as in Liski and Montero (2011). As a result, initial resource use is higher under commitment than under discretion in our paper, but lower in Liski and Montero (2011).4 In Harris and Vickers (1995), the arrival of the substitute is a stochastic event and depends on the buyer’s R&D effort. The buyer’s effort increases when the stock dwindles, so the seller has an incentive to slow down depletion in closed-loop equilibrium. The model Gerlagh and Liski (2011) is the same as ours except that the substitute is not available immediately following investment, but after an exogenous transition period. By investing, the buyer can force the seller to sell the remaining stock during the transition period. The value of this option decreases as the resource stock dwindles. To compensate the buyer for this decrease, supply is increasing over time during an interval before investment. Although Gerlagh and Liski look for a Markov-perfect equilibrium, the time-to-build delay acts as a commitment device. We show that in the absence of this commitment possibility, the buyer only shares in the resource rent because of discounting, because he is no longer able to decrease the current resource price by making credible promises that the substitute’s arrival is imminent. 2. Model Consider a model with a buyer and a seller of a nonrenewable resource. We adopt the notation of Gerlagh and Liski (2011). The buyer derives net (flow) surplus u(q(t)) from consuming q(t) units of the resource. Let u(.) be increasing and continuously differentiable. The net surplus function is u(.) satisfying u(q(t)) = u(q(t)) − u (q(t))q(t). associated with a strictly concave gross surplus function A monopolistic seller is endowed with a finite stock s0 that is costless to extract. The seller aims to maximize the discounted stream of instantaneous profits (q(t)) = u (q(t))q(t). Utility and profits are discounted at rate r. The buyer can instantaneously develop a perfect substitute for the resource. By paying an upfront investment cost I, he gains the ability to produce the substitute at constant marginal cost c. The buyer is then guaranteed a flow net surplus u ≡ u(( u )
−1
(c)).
There are i = 1, . . ., N periods of time of length ε. Time is continuous but agents only act at the beginning of each period, i.e. at time points ti = ε(i − 1). They commit their actions for the entire period. We distinguish between a pre-investment phase A and a post-investment phase B. In phase A, the buyer has a binary decision variable k(ti ) ∈ {0, 1}, where k(ti ) = 1 indicates that the buyer develops the substitute. In phase A, each period has three stages:
4 A formal proof for the general case is beyond the scope of this paper. We provide an example with iso-elastic demand in Appendix D.
458
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
(1) the seller offers to supply a quantity q(ti ), (2) the buyer chooses k(ti ) ∈ {0, 1}, (3) when k(ti ) = 0, the market clears at price p(ti ) = u (q(ti )), or when k(ti ) = 1, the seller’s offer is u (q(ti )), c). rejected. The seller makes a revised offer q(ti ) and the market clears at price p(ti ) = min( At the end of the period, the economy moves to regime B.5 In phase B, the seller is the only mover and chooses quantity q(ti ). The market clears at price p(ti ) = min( u (q(ti )), c). Our timing reflects the mutual dependence of buyers and sellers in natural resource markets, and oil exporters’ concerns about security of supply: if the supply offer is sufficiently generous, the buyer will refrain from investing. Olsen (1993) uses the same setup as our paper, but the buyer moves before the seller in each period. This timing forces the buyer to invest very early: when he can only condition his action on the remaining stock, the buyer cannot credibly punish the seller for setting a high price.6,7,8 We will characterize the limit of the equilibrium when ε goes to zero.9,10 For convenience, the remainder of the analysis is therefore presented in continuous time with an infinite horizon. The only difference with Gerlagh and Liski (2011) is that in their paper, when the buyer rejects the offer and invests the seller makes a revised offer and the game enters a transition period of exogenous length h > 0 in which the market clears at price p(ti ) = u (q(ti )), after which the economy moves to regime B. 3. The time-consistency problem Olsen (1993) sketches why equilibria in which the buyer can commit the investment time are not consistent. According to the Hotelling rule, a monopolistic seller equates discounted marginal revenue −1
in all periods. When the arrival time of the substitute T is given, supplying q∗ ≡ ( u ) (c) in any period after T does not cause inframarginal losses because the resource price is fixed by the cost of the substitute (we assume the buyer cannot discriminate between the resource and the substitute). The seller does incur inframarginal losses before T , when the substitute is not present. Because marginal revenue is higher after the arrival of the substitute than before, the seller prefers to sell part of his stock after T . This is socially inefficient and reduces the buyer’s welfare, because resource units sold after T generate no consumer surplus in excess of what the substitute could provide. When the buyer commits to a late T , the seller’s post-investment revenues go down in present terms because of discounting. This makes it less attractive for the seller to marginally undercut the substitute after T , thereby reducing cumulative post-investment supply. A late T results in higher cumulative extraction in the pre-investment phase and increases current supply, but is costly for the buyer in later periods: when the resource becomes scarce, the buyer cannot dispose over the substitute as early as he would like. After having enjoyed the benefits of high supplies and low prices early on, the buyer has an incentive to avoid the costs of resource scarcity in subsequent periods by developing the substitute earlier than announced. We illustrate this incentive in Fig. 1. After some point, which we denote by T˜ , supply drops below the buyer’s reservation quantity q˘ – which is
5 In Gerlagh and Liski (2011), the economy moves into a transition regime in which the seller offers a quantity q(ti ) in each u (q(ti )). The terminal phase that follows this transition regime is the same as stage, with the market clearing at price p(ti ) = our phase B. 6 The exception is when the initial stock is small and within a sequence of equidistant ‘nontrigger intervals’ (Olsen, 1993). 7 The crucial difference between the timing in this paper and in Olsen (1993) is that the buyer moves in between the seller’s proposal and the market clearance. Reversing stages (2) and (3) would yield the same equilibrium as in Olsen (1993): after the seller’s offer in the initial period, the residual game is equivalent to Olsen (1993). 8 Enabling the buyer to condition his strategy on (s(ti ), q(ti )) also allows some less realistic equilibria, as we will see in Section 5. 9 The limit is taken analogously to Gerlagh and Liski (2011). The buyer’s ability to develop the substitute guarantees that the strategic interaction ends in finite time. When N is sufficiently large, changes in N do not affect the equilibrium. Moreover, when the number of remaining periods is large, the equilibrium strategies do not depend on the number of remaining periods. 10 In general, it is not clear whether the limit of the equilibrium is an equilibrium of the limit game; see Simon and Stinchcombe (1989) for an example.
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
459
Fig. 1. The buyer’s time-consistency problem.
defined such that u(˘q) equals the net surplus u of consuming the substitute minus the interest on the investment cost rI – and the buyer prefers to invest immediately instead of at the announced time T . 4. Post-investment phase When making decisions during phase A, agents take into account their payoffs in phase B. We therefore first turn to phase B. We assume that when the resource and substitute are available at the same price, consumers prefer the resource.11 In the limit when ε goes to zero, the seller’s problem in phase B is:
∞
max q(t)
min( u (q(t)), c)q(t)e−rt dt,
0
s.t. s˙ = −q(t),
(1)
q(t)≥0, s(t)≥0, s0 given.
The Hamiltonian for this problem and the necessary optimality conditions are H(q, s, ) = min( u (q(t)), c)q(t) − (t)q(t), (q(t)) − (t) ≤ 0, ˙ = r(t), (t)
q(t)≥q∗ complementary slackness,
(2)
lim (t)s(t) = 0,
t→∞
where is the scarcity value of the resource and
(q(t))
=
u (q(t)) + q(t) u (q(t)) for q(t)≥q∗ c
for q(t) < q∗
.
Because a perfect substitute is available at cost c, marginal revenue (q(t)) is equal to c for the first q∗ units and discontinuous at q∗ , as the seller starts incurring losses on inframarginal units.
11
Otherwise, the seller will offer the resource at an infinitesimally lower price, without changing the analysis.
460
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
Fig. 2. The seller’s optimal supply (left) and the present value of marginal revenue (right) during the post-investment phase.
Proposition 1 (Post-investment phase, Hoel (1978) and Salant (1979)). Let s∗ ≡ −
q∗ ln r
(q∗ ) c
.
When the remaining stock s < s∗ , q(t) = q∗
∀ t ∈ (0, T ), T =
s , q∗
where T denotes the exhaustion time of the resource. When s > s∗ , q(t) = ( )−1 (ce−r(T −t) ) ∀ t ∈ [0, T − ], q(t) = q∗
T −
s.t.
(3)
∀ t ∈ [T − , T ], ( )
−1
(ce−r(T −t) ) dt + q∗ = s,
(4)
0
where ≡
s∗ q∗
is the length of the limit-pricing phase. The above system defines the seller’s optimal post-
investment strategy I (s ; c). The seller compares whether it is more profitable to use a marginal resource unit to extend the limitpricing phase (in which q(t) = q∗ ) or to increase supply today. If the seller practices a limit-pricing strategy until exhaustion, the stock is exhausted at time s/q∗ and the present value of the marginal ∗ revenue of the last unit is ce−rs/q . The marginal revenue of supplying a unit in excess of q∗ today is −1
u (( u ) (c)) << c), but not subject to discounting. discretely lower in current value ( (q∗ ) = c + q∗ ∗ ∗ −s/q For small initial stocks, (q ) is smaller than ce and the seller supplies q∗ until exhaustion. When the remaining stock is large, limit pricing takes so long to exhaust the stock that (q∗ ) is ∗ higher than ce−rs(t)/q . It is then worthwhile to supply q(t) > q∗ today.12 In equilibrium, the seller is indifferent between marginally extending the limit-pricing phase and selling an extra unit in any period before the limit-pricing phase, i.e. between [0, T − ]. Fig. 2 illustrates the seller’s supply path during the post-investment phase. 5. Closed-loop equilibrium It will be useful to define the value function for the seller V(s) and buyer U(s) as a function of the remaining stock during phase A. Denote the buyer’s and seller’s value at the time of investment by 12 We implicitly assume that is positive for q ≥ q∗ , implying elastic demand, to rule out the case in which the seller always supplies q∗ until exhaustion regardless of the remaining stock (Andrade de Sà et al., 2012).
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
461
Fig. 3. Resource supply before and after investment as a function of time (left) and the remaining stock (right).
UI (s) and VI (s), respectively. Let k = (s, q) be the buyer’s investment strategy and q = (s) the seller’s supply strategy. The value for the seller is the payoff from choosing the optimal q for an arbitrarily small interval ε, given the strategic response of the buyer V (s) = max{[ε(q) + e−εr V (s − εq)](1 − (s, q)) + V I (s)(s, q)}. {q}
(5)
We define the value for the buyer analogously U(s) = max {[εu((s)) + e−εr U(s − ε(s))](1 − k) + U I (s)k}.
(6)
k∈{0,1}
The post-investment values UI (s) and VI (s) are determined by the dynamics described in Proposition 1. We look for a pair of equilibrium strategies such that (s) = argmax{q} V(s) and (s, q) = argmaxk U(s) for all s. Proposition 2 states the main result. Proposition 2. Let q˘ ≡ u−1 (u − rI). There is a continuum of subgame perfect equilibria in which the buyer’s strategy does not depend on the remaining stock that are indexed by the buyer’s threshold level q ∈ [˘q, q∗ ] and given by (s, q) =
1 ∀ q : q < q,
(7a)
(s, q) =
0 otherwise,
(7b)
(s) = (s; u ( q)), I
with
I (s ; c)
(7c)
as defined in Proposition 1.
Although the motivation differs in the pre- and post-investment phase, the seller always supplies at least the buyer’s reservation quantity. In the post-investment phase, the seller always supplies q ≥ q∗ because there is no loss on inframarginal units for smaller q. In the pre-investment phase, the seller’s best response to the buyer’s equilibrium strategy is to supply at least q≥ q, because he would trigger investment for smaller q. From the seller’s perspective, the threat of the substitute is therefore equivalent to an already available substitute with a higher marginal cost. The buyer does not invest before exhaustion because his benefit from the resource – the discounted surplus stream minus his reservation surplus stream from the substitute – decreases after investment. Adopting the substitute raises the buyer’s reservation surplus from u − rI to u, but not his actual surplus as the buyer pays interest rI on the sunk investment cost. The increase in the buyer’s reservation surplus has three q = q˘ equilibrium. effects as we shown in Fig. 3, which depicts the Firstly, the buyer’s benefit from the resource is lower for any given supply path: u(q) − u(˘q) > u(q) − u ∀ q. Secondly, the seller immediately decreases supply when the buyer invests: (s) > I (s) when s > s∗ (Hoel, 1978, 1983). To see this, imaginarily divide the seller’s remaining stock in two parts
462
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
and restate his problem as a free-terminal-time problem with a scrap value. Deciding when to exhaust the first stock and switch to using the second, the seller balances two concerns: by prolonging the use of the first stock, he can increase his profits from that stock by smoothing the sales over a longer period. On the other hand, he has to wait longer for the profits from the second stock, which go down in present terms because of discounting. When the buyer develops the substitute, the second concern becomes comparatively less important: because the maximum price of the resource goes down, so does the maximum profit from selling the second stock. As a consequence, the seller prolongs the use of the first stock and reduces current supply.13,14 Thirdly, a larger share of the resource stock is sold at the buyer’s reservation price: the area under the horizontal part of the dashed curve in the left panel of Fig. 3 is larger than the area under the horizontal part of the solid curve. Within this range of equilibria, the seller strictly prefers strategies with a low q. The buyer’s ranking depends on the utility function. When the buyer’s net flow surplus is convex in q, as with linear demand, the buyer would like to front-load supply with q = q˘ . When net flow surplus is concave, the buyer may q to increase his surplus in the limit-pricing phase.15 Strategies in which the buyer’s prefer a higher threshold q is above q∗ or below q˘ cannot be an equilibrium when (s) = I (s; u ( q)). For q > q∗ , the seller is insensitive to the buyer’s threat. Strategies with q < q˘ are time-inconsistent for the reason in Section 3: the buyer wants to renege and invest when supply drops below q˘ . Corollary 1.
limU(s) =
r→0
u r
− I.
The buyer benefits from the resource when the seller is too impatient to sell the entire stock at the buyer’s reservation price. As r approaches zero, this consideration vanishes and the presence of the resource does not make the buyer better off than if he only had access to the substitute. The equilibria discussed above have the attractive property that they do not require the buyer to have detailed knowledge of remaining reserves, which are a potential source of asymmetric information,16 but they are not exhaustive. There are trivial equilibria with immediate investment, e.g. (s, q) = 1, (s) = 0 ∀ (s, q). If we consider equilibrium strategies where (s, q) depends on s as well as q, the range of equilibria becomes even larger. For example, the buyer may threaten to invest unless the seller implements a specific supply path, including the buyer’s first-best qB (s). Since lim V I (s) = 0, c→0
the seller’s indifference condition is always satisfied for small c, and (s) = qB (s), (s, qB (s)) = 0, (s, / qB (s) is formally a subgame-perfect equilibrium, even though the credibility of such q) = 1 ∀ q = ‘take-it-or-leave-it’ strategies in the real world is debatable.17 In the Appendix, we show that the result in Proposition 2 is not driven by the discrete nature of the buyer’s investment decision. We derive the same results when the buyer can invest a continuous amount in each period and the marginal cost of the substitute depends on cumulative investment. The buyer invests the amount that maximizes consumption utility minus the interest on the investment costs in one go when the resource is exhausted. Similarly, the seller supplies at least the buyer’s q until exhaustion. This extension is a limiting case of the model in Jaakkola (2012), threshold quantity which differs from Appendix C only in that it has convex instead of linear per-period investment costs. The convex costs in Jaakkola (2012) cause the buyer to invest in the substitute more gradually, but prohibit an analytical characterization of the closed-loop equilibrium. Jaakkola solves the closed-loop equilibrium numerically. The buyer develops the substitute more slowly than under commitment, for a similar reason as in the present paper: more rapacious investment leads to higher resource prices and lower supply in the short term. 13
This intuition is paraphrased from Hoel (1983).
Because the initial price is higher in equilibria with a high q, an increase in q lowers the seller’s total value of the stock V but increases the marginal value V when s > s∗ . If we were to endogenize s0 by allowing the seller to increase his resource base before the initial period at increasing marginal cost (as in van der Ploeg (2012)), exploration investments would be higher if q. the seller anticipates a high 15 There also exist mixed-strategy versions of the equilibria in Proposition 2 in which the buyer invests with probability ∈ (0, q. The seller’s gain from deviating for an arbitrarily small period is always smaller than (V(s) − VI (s)). 1) when q < 16 Sauré (2010) and Gerlagh and Liski (2012) argue that sellers have a strategic motive not to disclose the true value of these reserves. 17 Though not explicitly mentioned in that paper, these strategies are also equilibria in Gerlagh and Liski (2011). 14
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
463
6. Conclusion Binding promises about the adoption time of a substitute can give nonrenewable resource buyers a larger share of the resource rent but are not time-consistent. When the buyer cannot commit to the arrival time of the substitute, a monopolistic seller can induce the buyer to delay the invention of the substitute until the resource is depleted. In the absence of uncertainty or a fixed time-to-build delay, the threat of a substitute has a similar effect as an already available substitute in closed-loop equilibrium. We model the investment decision as binary for expositional clarity, but all results go through with continuous investment. The model is highly stylized – for example, OPEC and industrialized economies both face substantial coordination and free-riding problems that impede their ability to act as a monopolist or jointly develop a substitute. Nonetheless, the results accord with the slow technological progress in renewable energy and oil exporters’ efforts to stabilize prices, as unconventional hydrocarbons present an increasing threat to their revenues. Appendix A. Proof of Proposition 1 The seller supplies q∗ at time t when the MR of supplying an additional unit is lower than the discounted MR at the time of exhaustion, i.e. (q∗ ) ≤ (t) = (T )e−r(T −t) = ce−r(T −t) . Solving for T − t yields T −t ≤−
1 ln r
(q∗ ) c
.
(A.1)
The seller supplies q∗ in each period until exhaustion when this inequality holds for all t ∈ (0, (s/q∗ )). We can calculate the threshold stock s∗ such that the seller practices limit pricing until exhaustion for all s ≤ s∗ by substituting T = (s/q∗ ) and solving for the s such that (A.1) holds with equality for t = 0 s∗ = −
q∗ ln r
(q∗ ) c
.
When s > s∗ , the seller extracts the last s∗ units in a period of length = (s∗ )/(q∗ ). While extracting the first s − s∗ units, marginal revenue rises at the rate of interest and is equal the present value of marginal revenue at the time of exhaustion (q(t)) = ce−r(T −t) ⇒ q(t) = ( )
−1
(ce−r(T −t) ).
The stock constraint determines T
T −
( )
−1
(ce−r(T −t) ) dt + q∗ = s.
0
Appendix B. Proof of Proposition 2 We make use of a special case of Proposition 3 in Hoel (1983). Lemma 1 ((Post-investment phase, Hoel (1983))). When s > s∗ , I (s ; c) is increasing in c.
(A.2)
464
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
It will be useful to include c as a parameter argument to U(.) and u. We also write T(s ; c) and (c) to link the exhaustion time and the length of the limit-pricing phase to the cost of the substitute. The buyer’s strategy maximizes U(s ; c) with respect to k. His value from investing is
∞
T (s;c)−(c) ∞ =
T (s;c)−(c)
u(c)e−rt dt − I +
U I (s; c) =
(u(c) − rI)e
−rt
0 T (s;c)
dt +
0
u(I (s(t); c))e−rt dt
[u(I (s(t); c)) − u(c)]e−rt dt,
0
˙ s(t) = −I (s(t); c), s(0) = s.
(s) denote the buyer’s discounted surplus stream when he adopts the equilibrium strategy with Let U threshold q, and denote the corresponding exhaustion time and length of the limit-pricing phase by T (s; c) and (c), respectively. We have (s; c) = U =
∞
T (s;c) ∞ T (s;c) ∞
T (s;c) u(c)e−rt dt − e−r I+
(u(c) − rI)e−rt dt +
u(I (s(t); u ( q)))e−rt dt
0 T (s;c)
u(I (s(t); u ( q)))e−rt dt
0
(u(c) − rI)e−rt dt +
=
T (s;c)
T (s;c)
0
(u(I (s(t); u ( q))) − [u(c) − rI])e−rt dt,
0
˙ s(t) = −I (s(t); u ( q)), s(0) = s.
= (c). Comparing UI (s ; c) and U (s; c), we have u ( q)≥c, T (s; c) = T (s; c ) and (c) For cˆ ≡
∞
U I (s; c) =
(u(c) − rI)e
−rt
0
I
T (s;c)−(c)
I
dt +
[u( (s(t); c)) − u(c)]e
=0
[u( (s(t); c)) − u(c)]e
−rt
˙ dt, s(t) = −I (s(t); c), s(0) = s,
T (s;c)−(c)
(s; c) = U
∞
(u(c) − rI)e−rt dt +
0
≥0
T (s;c)−(c)
T (s;c)−(c)
(u(I (s(t); c )) − [u(c) − rI])e−rt dt+
0
(u(I (s(t); c )) − [u(c) − rI])e−rt dt+ ≥0
T (s;c)
II
T (s;c)−(c)
dt+
I
−rt
0
T (s;c)
˙ (u(I (s(t); c )) − [u(c) − rI])e−rt dt, s(t) = −I (s(t); c ), s(0) = s.
T (s;c)−(c)
By Lemma 1, I (s ; c) increases in c for s > s∗ , which implies II > I and, together with the increase in the limit quantity when c goes up, that the length of the non-limit-pricing phase T(s ; c) − (c) increases in c.18
18
Also note that
∂ ∂c
(q) (q) ≤ 0 when −q u˜u˜ (q) ≤ −q u˜u˜ (q) + 1.
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
465
q given that the buyer only invests when q < q. When the seller The seller always supplies at least q, he triggers immediate investment and is subject to the more stringent constraint chooses a q(t) < q(t) ≥ q∗ for the remainder of the game. Choosing q(t) < q therefore cannot be optimal. Appendix C. Continuous investment In this section, we generalize the buyer’s problem: rather than making a discrete investment decision, the buyer can invest a continuous amount in each period. The marginal cost of the substitute is a decreasing function of cumulative investment
t
1+
c=
c(X), c (X) ≤ 0, c (X)≥0, X(t) =
ε
εx(ti ).
i=1
We can write u(X) ≡ u(( u )
−1
(c(X))). The timing in each stage is
(1) the seller supplies a quantity q(ti ), (2) the buyer chooses x(ti ) ≥ 0, (3) the market clears at price p(ti ) = min( u (q(ti )), c(X(ti ))). Profits and utility depend on the stance of technology X in addition to the supply q: (X, q) = q min( u (q), c(X)) and u(X, q) = max(u(q), u(X)). We assume that investments at ti can immediately affect the market price during the interval (ti , ti+1 ), though this is inconsequential for the equilibria when the period length is small. Let (s, X) denote the seller’s supply strategy and (s, X, q) the buyer’s investment strategy. The value functions are given by V (s, X) = max{ε(X + ε (s, X, q), q) + e−εr V (s − εq, X + ε (s, X, q))},
(C.1)
U(s, X) = max{ε(u(X + εx, (s, X)) − x) + e−εr U(s − ε(s, X), X + εx)}.
(C.2)
{q}
{x}
Utility from the substitute is a concave function of cumulative investment. Assumption 1.
2
∂u ≥0, ∂ u < 0. ∂X ∂X
The range of equilibria in which (s, X, q) does not depend on s is similar to Proposition 2. The buyer selects the cumulative investment level X∗ that yields the highest long-run utility u(X) − rX, and invest this amount in one go when the resource is exhausted. Proposition 3. Let X∗ be the (unique) solution to (∂u)/(∂X) = r and let ε be arbitrarily small. There is a continuum of subgame perfect equilibria in which the buyer’s strategy does not depend on the remaining q ∈ [u−1 (u(c(X ∗ )) − rX ∗ ), u−1 (u(c(X ∗ )))] and given by stock that are indexed by the buyer’s threshold level (s, X, q) =
X∗ − X ∀ q : q < q, ε
(C.3a)
(s, X, q) =
0 otherwise,
(C.3b)
(s, X) = max(I (s; u ( q)), ( u )
−1
(c(X))),
(C.3c)
with I (s ; c) as defined in Proposition 1. Proof. Any strategy with (s, X, q) > 0 for s > 0 cannot be an optimal response to (C.3c). By Proposition 2, when faced with a binary choice between either investing a positive amount now and zero afterward until exhaustion or investing the same positive amount at exhaustion and zero in all preceding periods,
466
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
the buyer prefers the latter.19 Because the buyer always wants to postpone his ‘last’ investment until exhaustion, by induction investment is zero when the resource stock is positive. When the resource is exhausted and X < X∗ , it is optimal for the buyer to reach the long-run optimal level X∗ as quickly as possible. Suppose that the buyer’s strategy entails x1 = (0, X1 , 0) > 0, x2 = (0, X2 , 0) > 0 for some 0 < X1 < X2 < X∗ . Then by Assumption 1, the buyer can marginally improve his welfare by choosing (0, X1 , 0) = (X2 − X1 )/(ε) + x2 . By induction it follows that the buyer invests (X∗ − X)/ε when the resource is exhausted. The proof that the seller’s strategy (C.3c) is optimal given buyer’s strategy (C.3a)–(C.3b) is analogous to the last part of the proof of Proposition 2. Appendix D. Commitment and subgame perfect equilibria In this section, we provide an example with an iso-elastic demand function in which a monopsonistic buyer that already has developed a substitute, as in Liski and Montero (2011), chooses a lower initial level of resource consumption under commitment than in closed-loop equilibrium. In the bilateral monopoly framework that we study in this paper, initial consumption is higher when the buyer can commit the invention time than in closed-loop equilibrium. √ Let u(q) = q, which corresponds to an iso-elastic demand with elasticity 2. We numerically compute the resource consumption paths in Liski and Montero’s framework when the buyer can fully commit q(t) for all t, in closed-loop equilibrium (where the buyer’s strategy q(t) = C(s(t)) is a function of the remaining stock) and in the social optimum. When the buyer has full commitment power, his problem is
Vt=0 ˙ s.t. s(t)
{q(t),T }
T
{
= max
q(t) − p(t)q(t)}e−rt dt +
0
= −q(t) s0 > 0,
1 1 −rT e r 4c
s(t) = 0,
˙ = rp(t), p(T ) = c, p(t) where T denotes the time at which the resource is exhausted and the buyer switches to the substitute. Liski and Montero (2011, p. 6) show that the buyer’s commitment optimum is fully characterized by 1 −1 1 ∂q 1 q 2 = √ + 2 q 4r ∂t 1 1 q(t) = − 4c 2
0,
(D.1a)
crs0 .
(D.1b)
In closed-loop equilibrium, the buyer’s value function is
∞
V (s(t)) =
[
C(s(t)) − P(s(t))C(s(t))]e−r( −t) d ,
t
where q(t) = C(s(t)) and p(t) = P(s(t)) are the equilibrium consumption and pricing rule, respectively. The authors demonstrate (p. 9) that the equilibrium is given by a first-order ODE in the pricing rule 1 2
−
P (S) − P(S) + P (S)S = 0, rP(S)
(D.2)
with boundary condition P(0) = c. Lastly, the social optimum is characterized by the Hotelling rule, p(t) = u (q(t)) and p(T) = c and has an analytical solution: T = 1/(2r) ln(1 + 8rs0 c2 ). Using MATLAB, we numerically solve (D.1) and (D.2) to determine the consumption paths under commitment and in
19
∗ −1 ∗ −1 ∗ The buyer strictly prefers )))) and if qˆ = u−1 (u(c(X ∗ )) − to delaythe investment if qˆ ∈ (u (u(c(X )) − rX ), u (u(c(X −1 (c(X ∗ ))
rX ∗ ), s > − (u˜ )
r
ln
((u˜ )−1 (c(X ∗ ))) c(X ∗ )
indifferent between investing zero and
−1 (c(X ∗ ))
. When qˆ = u−1 (u(c(X ∗ )) − rX ∗ ) and s ≤ − (u˜ )
X ∗ −X ε
.
r
ln
((u˜ )−1 (c(X ∗ ))) c(X ∗ )
, the buyer is
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
467
3
2.5
Closed−loop Commitment Social Optimum
Resource use
2
1.5
1
0.5
0
0
10
20
30
40
50
Time Fig. 4. Resource consumption in Liski and Montero (2011).
closed-loop equilibrium for s0 = 20, c = 0.5, r = 0.04.20 Fig. 4 depicts the results. In line with the intuition in the main text, resource use in Liski and Montero (2011) is lower under commitment than in closedloop equilibrium. When the buyer has full commitment power, p(t) = p(T)e−r(T−t) . By committing to a late switch to the substitute, the buyer reduces initial extraction but obtains a substantial price discount. Now return to the bilateral monopoly model in this paper. Dasgupta et al. (1983) and Olsen (1986) discuss the buyer’s optimal invention time under commitment at length. We demonstrate the following result through a series of Lemmas. Proposition 4. For iso-elastic demand with elasticity equal or larger than 2 and a sufficiently large initial stock, initial resource use is higher when the buyer commits the invention time at t = 0 than in closed-loop equilibrium. Lemma 2 ((Proposition 1, Olsen (1986))). For iso-elastic demand with elasticity equal or greater than 2 and s0 sufficiently large, the seller has zero stock remaining at the buyer’s optimal invention time. Lemma 3 ((Dasgupta et al. (1983), p. 1442)). Resource supply at t = 0 increases in the invention time between 0 and the buyer’s optimal invention time. Lemma 4 ((Dasgupta et al. (1983), p. 1442)). For iso-elastic demand with elasticity equal or greater than 2 and s0 sufficiently large, resource supply at t = 0 decreases in the buyer’s invention time when the invention time is greater than the buyer’s optimum. Lemma 1 implies that resource supply at t = 0 when the buyer commits to an infinitely large invention time (which is equivalent to c→ ∞) is larger than initial resource supply in closed-loop equilibrium. Then Lemmas 2–4 establish the Proposition. References Andrade de Sà, S., Chakravorty, U., Daubanes, J., 2012. Limit-pricing Oil When Demand is Inelastic, Resource Taxation and Substitutes Subsidies. lies http://www.cer.ethz.ch/sured 2012/programme/SURED-12 042 AndradeDeSa Chakravorty Daubanes.pdf
20
The code is available upon request.
MonopoMimeo
468
T.O. Michielsen / Resource and Energy Economics 36 (2014) 455–468
Bain, J.S., 1956. Barriers to New Competition. Harvard University Press, Cambridge, MA. Dasgupta, P., Gilbert, R.J., Stiglitz, J.E., 1983. Strategic considerations in invention and innovation: the case of natural resources. Econometrica 51 (5), 1439–1448. Fujiwara, K., Long, N.V., 2011. Welfare implications of leadership in a resource market under bilateral monopoly. Dynamic Games and Applications 1, 479–497. Fujiwara, K., Long, N.V., 2012. Optimal tariffs on exhaustible resources: the case of quantity setting. International Game Theory Review 14 (4), 1240004, http://dx.doi.org/10.1142/S021919891240004X. Gallini, N., Lewis, T., Ware, R., 1983. Strategic timing and pricing of a substitute in a cartelized resource market. Canadian Journal of Economics 16 (3), 429–446. Gerlagh, R., Liski, M., 2012. Resource Relationships. http://hse-econ.fi/liski/papers/Relationships.pdf Gerlagh, R., Liski, M., 2011. Strategic resource dependence. Journal of Economic Theory 146 (2), 699–727. Harris, C., Vickers, J., 1995. Innovation and natural resources: a dynamic game with uncertainty. RAND Journal of Economics 26 (3), 418–430. Hoel, M., 1978. Resource extraction, substitute production, and monopoly. Journal of Economic Theory 19, 28–37. Hoel, M., 1983. Future conditions and present extraction: a useful method in natural resource economics. Resources and Energy 5, 303–311. Jaakkola, N., 2012. Green technologies and the protracted end to the age of oil: a strategic analysis. OxCarre Research Paper 99. http://www.oxcarre.ox.ac.uk/files/OxCarreRP201299.pdf Karp, L., Newberry, D.M., 1993. Intertemporal consistency issues in depletable resources. In: Kneese, A.V., Sweeny, J.L. (Eds.), In: Handbook of Natural Resource and Energy Economics, vol. III. North Holland, Amsterdam. Liski, M., Montero, J.-P., 2011. On the Exhaustible-resource Monopsony. http://www.hse-econ.fi/liski/papers/HotCoase.pdf Olsen, T.E., 1993. Perfect equilibrium timing of a backstop technology. Journal of Economic Dynamics and Control 17, 123–151. Olsen, T.E., 1986. Strategic considerations in invention and innovation: the case of natural resources revisited. Econometrica 56 (4), 841–849. Salant, S.W., 1979. Staving off the backstop: dynamic limit pricing with a kinked demand curve. In: Pindyck, R.S. (Ed.), In: Advances in the Economics of Energy and Resources, vol. 2. JAI Press, Greenwich, CT, pp. 187–204. Sauré, P., 2010. Overreporting oil reserves. Swiss National Bank Working Paper 2010-7. http://www.relooney.info/SI Oil-Politics/Oil-Reserves 1.pdf Simon, L.K., Stinchcombe, M.B., 1989. Extensive form games in continuous time: pure strategies. Econometrica 57 (5), 1171–1214. Sylos Labini, P., 1957. Oligopolio e progresso tecnico. Milan, Giuffre. van der Ploeg, F., 2012. Breakthrough renewables and the green paradox. CESifo Working Paper 3986. http://www.econstor.eu/bitstream/10419/66869/1/730415155.pdf