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Equilibrium path in oligopolistic market of nonrenewable resource
Nonlinear Analysis: Real World Applications 9 (2008) 1918 – 1927 www.elsevier.com/locate/na
Equilibrium path in oligopolistic market of nonrenewable resource Ding Zhanwen∗ , Tian Lixin, Yang Honglin Faculty of Science, Jiangsu University, Zhenjiang 212013, PR China Received 8 June 2005; accepted 5 June 2007
Abstract In this paper we study the oligopoly model of nonrenewable resource in which the unit production cost is variable and depends on the resource reserve level. We consider both the open-loop strategy and the closed-loop strategy of this dynamical differential game. For the case of linear cost function we have observed that the open-loop equilibrium and the self-feedback equilibrium satisfy the same equilibrium conditions, which can be described as a dynamical system. The analysis shows that the equilibrium path of the model is the stable orbit of this system, and this result leads to further studies of the properties of the total extraction and reserve and the individual ones of each producer. For the total extraction rate and reserve, some of the properties are similar to those of most oligopoly models with fixed unit production cost. For the individual behaviors, we have found out the solution expressions of the individual extraction rate and resource reserve and got the main result that the producer with larger initial stock has a larger but declining market share and the share of each producer converges toward the average one when time approaches to infinite. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Nonrenewable resource; Differential game; Dynamical system; Stable orbit; Equilibrium path
1. Introduction On the resources market there are often several oligopoly producers who compete with each other for the price and supply. Such competition is of course similar to that of any other goods production from the view points that each producer’s strategy has influences on the market price and demands thus their profits. But there are basic differences between nonrenewable resource and other ordinary goods. Because nonrenewable resource is exhaustible, the producer must allocate his extraction along the time horizon to maximize his aggregate profits. The extraction strategy of nonrenewable resource faces not only the constraint of cost, price and demand but also the constraint of resource reserve level. The oligopoly theory of nonrenewable source is to study the extraction path of each producer who is constrained by his initial stock. Since the game of competition is played in a finite or an infinite time interval, the mathematical model can be formulated as a dynamical differential game in which every producer’s extraction choice is an optimal control problem. If each producer has his own resource stock, for N players there are just N related maximization problems in all of which all the reserve constraints are included. This brings about the difficulty of general analysis of this model. But for the case of fixed unit cost, the problem becomes simpler and easier to deal with and many related works and results are given in the recent literature (e.g. [4–7]). However, it is more realistic that the unit cost of resource ∗ Corresponding author. Tel.: +86 0511 8791467; fax: +86 0511 8791467.
E-mail address: [email protected] (D. Zhanwen). 1468-1218/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2007.06.009
D. Zhanwen et al. / Nonlinear Analysis: Real World Applications 9 (2008) 1918 – 1927
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production depends on the reserve level. The hypothesis that cost increase with decreasing reserve can find its usage by researchers (e.g. [10]) and receives support in empirical studies [2,3,8]. For resource depletion, there are two concepts used by economists: Economic Depletion and Physical Depletion. For instance, Salo and Tahvonen [10] just based their work on Economic Depletion instead of Physical Depletion. The authors of this paper think that the essence of the question whether the depletion is physical or Economic arises from the question whether the unit cost is fixed or variable. If fixed, the net unit profits are, when resource is to be exhausted, the same as those of the initial stage, which means that the resource will be exhausted physically since the producer has positive profits all the time; if variable, in the stage of resource scarcity the production cost is too high for the producer to get positive net profits thus to continue his extraction, which means that he stops his extraction then and the resource is exhausted economically. The equilibrium of a dynamical game is mainly based on the open-loop or closed-loop strategy (e.g. [1]). The oligopoly theory of nonrenewable source has a lot of works in open-loop equilibrium study (including Cournot–Nash equilibrium and Von Stackelberg equilibrium). Because closed-loop strategy takes the reserve variables of all the players into the maximization problem, each producer must take into consideration not only his own stock constraint but also his rivals’ stock constraint. Because of this complexity, there are less related works in the literature. Salo and Tahvonen [10] have studied a special case where the closed-loop strategy is a linear feedback with constant coefficients. In this paper we study the oligopoly model of nonrenewable resource in which the unit production cost is variable and depends on the resource reserve level. We consider both the open-loop and closed-loop strategies and find their equilibrium equations. We pay our attention to both the aggregate level and individual level of the economic variables such as stock, extraction and reserve, i.e. the total amount of these variables on the market and the individual amount of each producer. 2. The model Suppose that there are N ( 2) producers extracting some sort of exhaustible natural resource and supplying on the same market. On the demand side, the market demand schedule is assumed to be given and characterized by the time invariant inverse demand function p = p(q), p (q) < 0, where p denotes unit price of the resource, q represents the rate at which users of the resource desire to consume when the price is p. The inverse demand function p(q) is assumed to satisfy (D1) p(0) = p¯ = c(0); (D2) (p(q)q) < 0. The assumption (D1) states that the last unit resource gains the choke price p, ¯ or the resource consumer must pay the highest price for the last unit of the resource on the market. Assumption (D2) means that the industry marginal revenue is decreasing with increasing of the resource sale q. On the supply side, all producers have the same extraction cost function c(x), where x denotes the current resource reserve. And the cost function c(x) is assumed to be increasing with decreasing of the resource reserve, i.e. c (x) < 0. For simplicity, we study the linear form of the cost function1 c(x) = p¯ − cx. Now we consider the N producers. All players are endowed with a given initial stock of the resource and choose their extraction paths so as to maximize their discounted profits. We denote the initial stock of player i by Ri , player i’s extraction rate at time t by qit (simplified by qi if not confused), and the discount rate by . Then player i’s choosing of extraction strategy qit is to solve the following maximization problem: ∞ max e−t (p(q) − c(xi ))qi dt (1) qi
s.t.
0
x˙i = −qi , qi 0, xi 0, xi (0) = Ri ,
(2) (3) (4)
1 For general linear function c(X) = c − cX (X represents reserves), Salo and Tahvonen [10] define X ¯ by c0 − cX¯ = p¯ and let x = X − X, ¯ then 0 have c(X) = p¯ − cx. They call the case physical depletion when X = 0 and the case economic depletion when x = 0.
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where xi = xit is i’s the resource reserve at time t, c(xi ) = p¯ − cx i is his extraction cost, and q = qt is the total quantity of resource supply on the market, q = N i=1 qi . This is a dynamical differentiable game of N players. We seek below the equilibrium conditions of the open-loop and closed-loop equilibriums. 3. Equilibrium conditions 3.1. Open-loop equilibrium For the open-loop best strategy profile (q1 , . . . , qi , . . . , qN ), player i’s strategy qi is the solution of system (1)–(4) with his opponents’ strategies qj t (j = i) being given as known. We focus on the interior solution, qi > 0 and xi > 0, so that the constraints (3) and (4) become loose. Define the Hamilton function H i = (p(q) − (p¯ − cx i ))qi − i qi , then the interior solution is characterized by the necessary conditions (5) and the Euler equation (6) jH i = p (q)qi + p(q) − p¯ + cx i − i = 0, jqi
(5)
i
jH ˙ i = − + i = −cq i + i . jxi
(6)
Noting the state equation x˙i = −qi , from (5) and (6) we have p (q)qq ˙ i + p (q)q˙i + p (q)q˙ = (p (q)qi + p(q) − p¯ + cx i )
(i = 1, 2, . . . , N ).
(7)
The equations in (7) determine the open-loop equilibrium strategies profile (q1 , q2 , . . . , qN ). Sum them to get the aggregative equation on the market (p (q)q + (N + 1)p (q))q˙ = (p (q)q + Np(q) − N p¯ + cx), where x = xt = N i=1 xit is the total quantity of the resource reserve of the N players, obviously x˙ = −q. By the marginal revenue decreasing assumption (p(q)q) < 0, holds p (q)q + (N + 1)p (q) = (p(q)q) + (N − 1)p (q) < 0. Thus (8) can be written as a dynamical system of the state variable x and q, i.e. x˙ = −q, (p (q)q + Np(q) − N p¯ + cx) q˙ = . p (q)q + (N + 1)p (q)
(8)
(9)
(10,11)
3.2. Feedback equilibrium To search for the best closed-loop strategy, we view each producer’s extraction rate as a function of the states profile x = (x1 , x2 , . . . , xN ) and the time variable t, i.e. qi = qi (x, t) (i = 1, . . . , N ). Now producer i’s problem is to solve the following optimal control program: ∞ e−t (p(q(x, t)) − c(xi ))qi (x, t) dt (1a) max qi (x,t)
s.t.
0
x˙k = −qk (x, t),
qk 0,
k = 1, 2, . . . , N,
(2a)
k = 1, 2, . . . , N,
(3a)
xk (0) = Rk , k = 1, 2, . . . , N, where q(x, t) = N i=1 qi (x, t), c(xi ) = p¯ − cx i .
(4a)
xk 0,
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Let q(x, t) = (q1 (x, t), q2 (x, t), . . . , qN (x, t)), i = (i1 , i2 , . . . , iN ), and define the Hamilton function H (t, x, q(x, t), i ) = (p(q(x, t)) − p¯ + cx i )qi (x, t) − i
N k=1
ik qk (x, t).
In a similar way by Rincon-Zapatero et al. [9], the necessary conditions of the system’s closed-loop equilibrium strategy (q1 (x, t), q2 (x, t), . . . , qN (x, t)) are given by jH i (t, x, q(x, t), i ) = 0, jqi
i = 1, . . . , N,
jH i (t, x, q(x, t), i ) i + ij , ˙ j = − jxj
(12)
j = 1, . . . , N, i = 1, . . . , N.
(13)
Let qi = qi (x, t) and q = q(x, t), then we have its partial derivatives of the Hamilton H i jH i (t, x, q(x, t), i ) = p (q)qi + p(q) − p¯ + cx i − ii , jqi
(14)
N
N
k=1
k=1
jqk jqk jH i (t, x, q(x, t), i ) jqi − ik , + (p(q) − p¯ + cx i ) = qi p (q) jxj jxj jxj jxj N
N
k=1
k=1
j = i,
jqk jqi i jqk jH i (t, x, q(x, t), i ) + cq i + (p(q) − p¯ + cx i ) − k . = qi p (q) jxi jxi jxi jxi
(15)
(15a)
If each producer’s extraction depends only on his own resource reserve xi , his feedback strategy takes the selffeedback form2 qi = qi (xi , t). For this case, holds jqk /jxi = 0 (k = i). Now from (12) to (15a), we have d(p (q)qi + p(q) − p¯ + cx i ) jqi jqi = − qi p (q) − cq i − (p(q) − p¯ + cx i ) dt jxi jxi jq i + (p (q)qi + p(q) − p¯ + cx i ) + (p (q)qi + p(q) − p¯ + cx i ) jxi or ˙ i + p (q)q˙i + p (q)q˙ = (p (q)qi + p(q) − p¯ + cx i ). p (q)qq
(16)
This is just the same equilibrium condition (7) of the open-loop strategy, and its sums give the same dynamical system (10)–(11) of the extraction q and reserve x. Our work below is based on such system. 4. Equilibrium path and properties For system (10)–(11) we have first that Proposition 1. (0, 0) is its unique equilibrium point and just a saddle point. Proof. Since p(0) = p, ¯ (0, 0) is an equilibrium point and its uniqueness is obvious. The Jaccobian at (0, 0) is given by 0 −1 0 −1 c J= , m (N + 1)p (0) 2 In reality, a producer may consider the information of his opponents when make decision. But because of the asymmetry or lack of information, it may be not easy for one to adjust his extraction promptly to be accord with his opponents’ reserve. The self-feedback strategy considered here is to focus on the bounded rationality rather than complete rationality of the decision makers. This form of feedback can be done easily by the producers and simplify our mathematical analysis.
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D. Zhanwen et al. / Nonlinear Analysis: Real World Applications 9 (2008) 1918 – 1927 q
Q+
Γ (ϕ = 0) Q− L2
E
qT
L1
0
xT
x
Fig. 1. Phase diagram of equilibrium path.
where m = c/(N + 1)p (0) < 0. So the eigenvalue
( + 2 − 4m) ( − 2 − 4m) > 0, < 0 and 2 = 1 = 2 2 which implies (0, 0) a saddle point.
Because x 0 and q 0, we need to pay attention only to the first quadrant of the x–q plane. Denote = p (q)q + Np(q) − N p¯ + cx. According to p (q)q + (N + 1)p (q) < 0 and by (11), q is increasing when < 0 and decreasing when > 0. We use the symbol to represent the curve = 0, Q+ to represent the area < 0 (in which q is increasing), Q− the area > 0 (in which q is decreasing), and E the stable orbit of the system. Apparently the stable orbit E is located in the area Q− (if not, the orbit E enters the area Q+ at some time, hence q increases thereafter and limt→∞ qt = 0, which is a contradiction to its stableness). And the norm vector of the curve defined by the function is given by (c, p (q)q + (N + 1)p (q)), where the x-coordinate is positive and the q-coordinate is negative, which means that the x-axis is located at the side of Q− (in which > 0) and the q-axis at the side of Q+ (in which < 0). In addition, there exits no more stable orbits other than E since (0, 0) is a saddle point. From these discussions, we get the phase diagram showed in Fig. 1. Theorem 1. The open-loop equilibrium path or the self-feedback equilibrium path is the stable orbit of the dynamical system (10)–(11). Proof. The system’s orbits except E must move along the path L1 or L2 showed in Fig. 1. For L1 , there exists some time T when L1 cuts x-axis at (xT , 0). With the constraint q 0, the optimal path of system (10)–(11) must end at this time T . Now the supply qT on the market is zero and the resource reserve xT is at a positive quantity. This means that there is at least a producer i whose reserve xiT > 0 and his unit profit p(qT ) − (p¯ − cx iT ) = p(0) − p¯ + cx iT = cx iT > 0, so it is profitable for him to extract his resource and supply on the market, which contradicts the zero supply when L1 cuts x-axis at (xT , 0). Therefore L1 is not the optimal path of the game. For L2 , it also cuts q-axis at (0, qT ) at some time T . By continuity, there exists a time such that when t ∈ (, T ), qt > qT /2 and cx t < p¯ − p(qT /2). In the period (, T ), each producer’s unit profit p(qt ) − (p¯ − cx it ) < p(qT /2) − p¯ + cx t < 0, which means it is not the best strategy to supply on the market in this period. But qt > 0, t ∈ (, T ), some producers have supplied. A contradiction. L2 is not the equilibrium path of the model, either. Proposition 2. On the equilibrium path, the price p and the resource extraction q satisfy (a) p˙ > 0 and q˙ < 0, i.e. p is increasing and q decreasing. (b) If x < (N − 1)p/c ¯ then p/p ˙ < .
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Proof. As stated above, E is located in the area Q− in which > 0, this means q˙ < 0, q is decreasing and hence p increasing. Noting that (p(q)q) < 0 (it means (p(q)q) = p (q)q + p(q) decreases with q), we have p (q)q + p(q) < p(0) = p¯ since q > 0 on the equilibrium path. Thus when x < (N − 1)p/c, ¯ holds − (N − 1)p(q) = p (q)q + p(q) + cx − N p¯ < p¯ + cx − N p¯ < 0 this means /(N − 1)p(q) < 1. And by the facts (p(q)q) < 0, p (q) < 0, q > 0 and > 0 on the equilibrium path, we have then p (q)q˙ p (q) p˙ = = p p(q) p(q)(p (q)q + (N + 1)p (q)) =
p (q) p(q)((p(q)q) + (N − 1)p (q))
<
p (q) = < . p(q)(N − 1)p (q) (N − 1)p(q)
Proof finished.
Proposition 2 means that on the oligopolistic market for nonrenewable resources the price of resource is increasing but will increase at a speed not more than the discount rate (noting that the condition of (c) will be satisfied at a moment since x approaches to zero). These results are similar to those of the models of constant production cost (cf. [6, Proposition 2, 7, Lemma 4]). For the fixed initial stock R = N i=1 Ri , we now use the variables with lower index N or in order to study the effects caused by the number of players or discount rate. The influences of the number of players N or discount rate on the equilibrium path are given with the following proposition. Proposition 3. For the equilibrium path E (a) qN (0)qM (0) and pN (0)pM (0), if N > M, (b) q (0)qϑ (0) and p (0)pϑ (0), if > ϑ. Proof. Let h=
(p (q)q + Np(q) − N p¯ + cx) = . p (q)q + (N + 1)p (q) p (q)q + (N + 1)p (q)
Obviously h is decreasing with and it can be verified easily that h is increasing with N since > 0 on the equilibrium path E. So q˙N > q ˙M when N > M. If qN (0) > qM (0), holds qN > qM . From the stable condition x(∞) = 0 we have ∞ ∞ R = 0 qN dt > 0 qM dt = R, a contradiction. Similarly p (0) > pϑ (0) is impossible. Here the result (a) of Proposition 3 differs from that of the fixed cost case (cf. [6, Proposition 3]). The difference is caused by the assumptions that the unit product cost is variable or fixed. For the fixed case, the increase of the numbers of players will increase competition and increase the current supply thus lower the current price since a change in the number of players will not alter the unit production cost. But for the variable case, when the total stock is fixed, the increase of the number of players will decrease their individual stock and hence increase their production cost, which means their extractions may decrease. However, this kind of result from the assumption of unchanged total stock is just a theoretic one. In the actual market, if all players have their own stocks, the total resource stock will be increased with his resource stock when a new player enters. 5. Individual behaviors The discussion above is about such aggregate variable as the total extraction rate q and the total resource reserve x. In this section, we pay our attention to the individual extraction rate qi and resource reserve xi . For calculating
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convenience, we consider the linear inverse demand function p(q) = p¯ − q. Taking such function into (7) or (16) and (10)–(11), we get the linear system as c cN c q˙i = qi + x1 + · · · − xi + · · · + xN , i = 1, . . . , N, (17,18) N +1 N +1 N +1 x˙i = −qi , i = 1, . . . , N of which the linear matrix is given by ⎡ (N + 1) c ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ c ⎢ ⎢ A= N +1⎢ ⎢ − (N + 1) ⎢ c ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
(N + 1) c
−
(N + 1) c
..
.
..
.
(N + 1) c
−N
1
...
1
−N
...
... 1
... 1
(N + 1) c By simple calculating we have two pairs of eigenvalues of A:
√ ∓ ( + 4c/(N + 1)) ∓ ( + 4c) r1,3 = and r2,4 = , 2 2
1
⎤
⎥ ⎥ 1 ⎥ ⎥ ⎥ ⎥ ... ... ⎥ ⎥ . . . −N ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
−
where r1 and r2 are negative, r3 and r4 are positive. The eigenvectors of i (i = 1, 3) are given by i = (−ri , −ri , . . . , −ri , 1, 1, . . . , 1) and the eigenvectors of i (i = 2, 4) are given by 1i = (ri , −ri , 0, 0, . . . , 0, −1, 1, 0, 0, . . . , 0) ,
2i = (ri , 0, −ri , 0, . . . , 0, −1, 0, 1, 0, . . . , 0) , .. . = (ri , 0, 0, 0, . . . , −ri , −1, 0, 0, 0, . . . , 1) N−1 i
thus there are just 2N eigenvectors with linear independence. The solution of the linear system (17) and (18) is determined by these eigenvalues and eigenvectors with 2N constants 1 , . . . , 2N . That is q N−1 1 r3 t r4 t = 1 er1 t 1 + er2 t 2 12 + · · · + N N−1 e + e + · · · + + . (19) 3 N+1 N+2 4 2N 4 2 x Now arises the question of how to decide the values of the 2N constants 1 , . . . , 2N . Of course, the initial stock conditions xi (0) = Ri (i = 1, . . . , N ) can be contributed to N of them. But we can not set the initial values of the extraction rate qi randomly since the extraction path should be interiorly determined by the model itself. However, we have got the results that the equilibrium path runs along the stable orbit on which both the variables, reserve variable x and the extraction rate variable q, approach to zero in the infinite horizon. Because r3 and r4 are the system’s positive eigenvalues, the coefficient constants 1 , . . . , 2N in (19) must be all zeroes (otherwise q or x cannot approach to zero). So the equilibrium solution of the linear takes the form q = 1 er1 t 1 + er2 t 2 12 + · · · + N N−1 . (20) 2 x
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By the initial reserve conditions xi (0) = Ri and from the last N equations of (20) ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ R1 −1 −1 1 ⎢1⎥ ⎢ 1 ⎥ ⎢ 0 ⎥ ⎢ R2 ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢R 1⎥ 1 ⎢ ⎢ . ⎥ + 2 ⎢ . ⎥ + · · · + N ⎢ . ⎥ = ⎢ .3 ⎣ .. ⎦ ⎣ .. ⎦ ⎣ .. ⎦ ⎣ .. 1
0
1
⎤ ⎥ ⎥ ⎥. ⎥ ⎦
RN
Thus we have 1 = R/N, 2 = R2 − R/N, . . . , N = RN − R/N , where R = Ri is the total initial reserve. Taking these results into (20), we can easily get the solution of the system (17) and (18) as follows: ⎧ R r1 t R ⎪ = − e − R − r r2 er2 t , i = 1, . . . , N, q ⎪ 1 i ⎨ i N N (21,22) ⎪ ⎪ ⎩ xi = R er1 t + Ri − R er2 t , i = 1, . . . , N. N N Proposition 4. If Ri > Rj holds for the initial stocks of player i and j , then their extraction rates satisfy (a) qi > qj , limt→∞ qi /qj = 1; (b) xi > xj , limt→∞ xi /xj = 1; (c) qi /xi > qj /xj , limt→∞ qi /xi = limt→∞ qj /xj . Proof. According to r2 < r1 < 0 and from (21) and (22), all these inequalities and equalities in (a) and (b) apparently hold and the ones in (c) can be easily and straightly verified. These properties show that the producer with a larger initial resource stock will supply at a larger rate and have a larger production/reserve ratio, but when time approaches to infinity all the players will play the same role on the market. In other words, the producer with larger initial stock has a larger but decreasing market share and in the infinite time horizon his advantage declines. Summing (21) and (22) from 1 to N , we have then the solution of the total variable in this situation: q = −Rr 1 er1 t and x = Rer1 t . From their expressions of the variables qi , xi , q and x we get the expressions of qi /q and xi /x as follows: R r2 (r2 −r)t1 xi R qi 1 1 1 1 e , = + Ri − = + Ri − e(r2 −r1 )t . q N R N r1 x N R N Noting that r2 < r1 < 0, we have the following further proposition. Proposition 5. If Ri =R/N or Ri > R/N or Ri < R/N , then producer i’s market share qi /q is constant or decreasing or increasing, and remains equal to or higher than or lower than 1/N , respectively, but converges toward 1/N in each situation; and all these hold for his resource reserve share xi /x. Proposition 5 means that if the stock of player i is greater (smaller) than the average, both his market share and his reserve share are decreasing (increasing), but when t → ∞ the shares of all producers converge toward the same one, i.e. the average share 1/N. Proposition 5 together with Proposition 4 characterize the individuals’ economic roles and the relationships between each other. Using constant coefficients linear feedback strategy, Salo and Tahvonen [10] have got some similar results of this game (cf. Propositions 4 and 5 in their work). Because we have got the stable equilibrium path, we have here found out the system’s solution expressions and the results are more obvious and straight. For the case N = 2, = 0.02 and c = 0.15, the phase picture of the resource reserve x1 and x2 is referred to Fig. 2. In Fig. 2 the two dotted lines isolate two areas (q1 = 0 and q2 = 0). In the area q1 = 0, for example, because of his resource stock shortage, producer 1’s production cost is so high that his net profit is less than zero and hence gives zero
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Producer 2's resource reserve level x2
100 90
q1=0
80 70 60 50 40 30 20 10
q2=0 10
20 30 40 50 60 70 80 90 100 Producer 1's resource reserve level x1
Fig. 2. Phase diagram for two oligopolies.
supply until the price grows up to the point where he begins to have positive profits. After entering the area between the two dotted lines, the two producers play on the equilibrium path. When their initial stock equals, their reserve levels are on the symmetry turnpike (the 45◦ bold line). If not, one with the higher stock keeps the higher market share. But all the equilibrium paths are tangent to the symmetry turnpike at the origin point (0,0), which means the differences between the two producers disappear gradually in an infinite time. 6. Concluding remarks In this paper we have reconsidered the game model of nonrenewable resource in which the unit production cost is variable and depends on the resource reserve level. For the case of linear cost, we find that the equilibrium conditions of open-loop and self-feedback strategies can be described as the same dynamical system for the case of linear cost. Our main mathematical result of this system is that its unique stable orbit is just the equilibrium path of the game. Based on this result, we have got the equilibrium properties of the main variables such as price, the total reserve and the total supply of resource. Most properties we give here are very similar to those of other oligopoly models with fixed unit production cost. It is also from this mathematical result that we have got the solution expressions of each producer’s maximization problem and some concrete and detailed properties are given. Although the method of this paper is different to that of Salo and Tahvonen [10], the main equilibrium properties of the individual behaviors are very similar to their works. Acknowledgements This work is financially supported by the National Natural Science Foundation of China (No. 90610031) and by the High Talent Foundation of Jiangsu University (No. 07JDG023). References [1] T. Basar, G. Olsder, Dynamic Noncooperative Game Theory, Academic Press, London, 1995. [2] J. Chermak, R. Patrick, A well based cost function and the economics of exhaustible resources: the case of natural gas, J. Environ. Econ. Manage. 28 (1995) 174–189. [3] D. Epple, J. Londregan, Strategies for modelling exhaustible resource supply, in: A. Kneese, J. Sweeney (Eds.), Handbook of Natural Resource and Energy Economics III, Elsevier, Amsterdam, 1993, pp. 1077–1107. [4] F. Groot, C. Withagen, A. de Zeeuw, Note on the open-loop von Stackelberg equilibrium in the cartel-versus-fringe model, Econ. J. 102 (1992) 1478–1484. [5] F. Groot, C. Withagen, A. de Zeeuw, Open-loop von Stackelberg equilibrium in the cartel-vs.-fringe model, Energy Econ. 22 (2000) 209–223.
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[6] T. Lewis, R. Schmalensee, On oligopolistic markets for nonrenewable natural resources, Q. J. Econ. 95 (1980) 475–491. [7] G. Loury, A theory of ‘oil’igopoly: Cournot Nash equilibrium in exhaustible resources markets with fixed supplies, Int. Econ. Rev. 27 (1986) 285–301. [8] S. Polansky, Do oil producers act as oil’ilopolists?, J. Environ. Econ. Manage. 23 (1992) 216–247. [9] J.P. Rincon-Zapatero, J. Martinez, G. Martin-Herran, New method to characterize subgame perfect Nash equilibrium in differential games, J. Optim. Theory Appl. 96 (2) (1998) 377–395. [10] S. Salo, O. Tahvonen, Oligopoly equilibria in nonrenewable resource markets, J. Econ. Dyn. Control 25 (2001) 671–702.
Equilibrium path in oligopolistic market of nonrenewable resource Ding Zhanwen∗ , Tian Lixin, Yang Honglin Faculty of Science, Jiangsu University, Zhenjiang 212013, PR China Received 8 June 2005; accepted 5 June 2007
Abstract In this paper we study the oligopoly model of nonrenewable resource in which the unit production cost is variable and depends on the resource reserve level. We consider both the open-loop strategy and the closed-loop strategy of this dynamical differential game. For the case of linear cost function we have observed that the open-loop equilibrium and the self-feedback equilibrium satisfy the same equilibrium conditions, which can be described as a dynamical system. The analysis shows that the equilibrium path of the model is the stable orbit of this system, and this result leads to further studies of the properties of the total extraction and reserve and the individual ones of each producer. For the total extraction rate and reserve, some of the properties are similar to those of most oligopoly models with fixed unit production cost. For the individual behaviors, we have found out the solution expressions of the individual extraction rate and resource reserve and got the main result that the producer with larger initial stock has a larger but declining market share and the share of each producer converges toward the average one when time approaches to infinite. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Nonrenewable resource; Differential game; Dynamical system; Stable orbit; Equilibrium path
1. Introduction On the resources market there are often several oligopoly producers who compete with each other for the price and supply. Such competition is of course similar to that of any other goods production from the view points that each producer’s strategy has influences on the market price and demands thus their profits. But there are basic differences between nonrenewable resource and other ordinary goods. Because nonrenewable resource is exhaustible, the producer must allocate his extraction along the time horizon to maximize his aggregate profits. The extraction strategy of nonrenewable resource faces not only the constraint of cost, price and demand but also the constraint of resource reserve level. The oligopoly theory of nonrenewable source is to study the extraction path of each producer who is constrained by his initial stock. Since the game of competition is played in a finite or an infinite time interval, the mathematical model can be formulated as a dynamical differential game in which every producer’s extraction choice is an optimal control problem. If each producer has his own resource stock, for N players there are just N related maximization problems in all of which all the reserve constraints are included. This brings about the difficulty of general analysis of this model. But for the case of fixed unit cost, the problem becomes simpler and easier to deal with and many related works and results are given in the recent literature (e.g. [4–7]). However, it is more realistic that the unit cost of resource ∗ Corresponding author. Tel.: +86 0511 8791467; fax: +86 0511 8791467.
E-mail address: [email protected] (D. Zhanwen). 1468-1218/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2007.06.009
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production depends on the reserve level. The hypothesis that cost increase with decreasing reserve can find its usage by researchers (e.g. [10]) and receives support in empirical studies [2,3,8]. For resource depletion, there are two concepts used by economists: Economic Depletion and Physical Depletion. For instance, Salo and Tahvonen [10] just based their work on Economic Depletion instead of Physical Depletion. The authors of this paper think that the essence of the question whether the depletion is physical or Economic arises from the question whether the unit cost is fixed or variable. If fixed, the net unit profits are, when resource is to be exhausted, the same as those of the initial stage, which means that the resource will be exhausted physically since the producer has positive profits all the time; if variable, in the stage of resource scarcity the production cost is too high for the producer to get positive net profits thus to continue his extraction, which means that he stops his extraction then and the resource is exhausted economically. The equilibrium of a dynamical game is mainly based on the open-loop or closed-loop strategy (e.g. [1]). The oligopoly theory of nonrenewable source has a lot of works in open-loop equilibrium study (including Cournot–Nash equilibrium and Von Stackelberg equilibrium). Because closed-loop strategy takes the reserve variables of all the players into the maximization problem, each producer must take into consideration not only his own stock constraint but also his rivals’ stock constraint. Because of this complexity, there are less related works in the literature. Salo and Tahvonen [10] have studied a special case where the closed-loop strategy is a linear feedback with constant coefficients. In this paper we study the oligopoly model of nonrenewable resource in which the unit production cost is variable and depends on the resource reserve level. We consider both the open-loop and closed-loop strategies and find their equilibrium equations. We pay our attention to both the aggregate level and individual level of the economic variables such as stock, extraction and reserve, i.e. the total amount of these variables on the market and the individual amount of each producer. 2. The model Suppose that there are N ( 2) producers extracting some sort of exhaustible natural resource and supplying on the same market. On the demand side, the market demand schedule is assumed to be given and characterized by the time invariant inverse demand function p = p(q), p (q) < 0, where p denotes unit price of the resource, q represents the rate at which users of the resource desire to consume when the price is p. The inverse demand function p(q) is assumed to satisfy (D1) p(0) = p¯ = c(0); (D2) (p(q)q) < 0. The assumption (D1) states that the last unit resource gains the choke price p, ¯ or the resource consumer must pay the highest price for the last unit of the resource on the market. Assumption (D2) means that the industry marginal revenue is decreasing with increasing of the resource sale q. On the supply side, all producers have the same extraction cost function c(x), where x denotes the current resource reserve. And the cost function c(x) is assumed to be increasing with decreasing of the resource reserve, i.e. c (x) < 0. For simplicity, we study the linear form of the cost function1 c(x) = p¯ − cx. Now we consider the N producers. All players are endowed with a given initial stock of the resource and choose their extraction paths so as to maximize their discounted profits. We denote the initial stock of player i by Ri , player i’s extraction rate at time t by qit (simplified by qi if not confused), and the discount rate by . Then player i’s choosing of extraction strategy qit is to solve the following maximization problem: ∞ max e−t (p(q) − c(xi ))qi dt (1) qi
s.t.
0
x˙i = −qi , qi 0, xi 0, xi (0) = Ri ,
(2) (3) (4)
1 For general linear function c(X) = c − cX (X represents reserves), Salo and Tahvonen [10] define X ¯ by c0 − cX¯ = p¯ and let x = X − X, ¯ then 0 have c(X) = p¯ − cx. They call the case physical depletion when X = 0 and the case economic depletion when x = 0.
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where xi = xit is i’s the resource reserve at time t, c(xi ) = p¯ − cx i is his extraction cost, and q = qt is the total quantity of resource supply on the market, q = N i=1 qi . This is a dynamical differentiable game of N players. We seek below the equilibrium conditions of the open-loop and closed-loop equilibriums. 3. Equilibrium conditions 3.1. Open-loop equilibrium For the open-loop best strategy profile (q1 , . . . , qi , . . . , qN ), player i’s strategy qi is the solution of system (1)–(4) with his opponents’ strategies qj t (j = i) being given as known. We focus on the interior solution, qi > 0 and xi > 0, so that the constraints (3) and (4) become loose. Define the Hamilton function H i = (p(q) − (p¯ − cx i ))qi − i qi , then the interior solution is characterized by the necessary conditions (5) and the Euler equation (6) jH i = p (q)qi + p(q) − p¯ + cx i − i = 0, jqi
(5)
i
jH ˙ i = − + i = −cq i + i . jxi
(6)
Noting the state equation x˙i = −qi , from (5) and (6) we have p (q)qq ˙ i + p (q)q˙i + p (q)q˙ = (p (q)qi + p(q) − p¯ + cx i )
(i = 1, 2, . . . , N ).
(7)
The equations in (7) determine the open-loop equilibrium strategies profile (q1 , q2 , . . . , qN ). Sum them to get the aggregative equation on the market (p (q)q + (N + 1)p (q))q˙ = (p (q)q + Np(q) − N p¯ + cx), where x = xt = N i=1 xit is the total quantity of the resource reserve of the N players, obviously x˙ = −q. By the marginal revenue decreasing assumption (p(q)q) < 0, holds p (q)q + (N + 1)p (q) = (p(q)q) + (N − 1)p (q) < 0. Thus (8) can be written as a dynamical system of the state variable x and q, i.e. x˙ = −q, (p (q)q + Np(q) − N p¯ + cx) q˙ = . p (q)q + (N + 1)p (q)
(8)
(9)
(10,11)
3.2. Feedback equilibrium To search for the best closed-loop strategy, we view each producer’s extraction rate as a function of the states profile x = (x1 , x2 , . . . , xN ) and the time variable t, i.e. qi = qi (x, t) (i = 1, . . . , N ). Now producer i’s problem is to solve the following optimal control program: ∞ e−t (p(q(x, t)) − c(xi ))qi (x, t) dt (1a) max qi (x,t)
s.t.
0
x˙k = −qk (x, t),
qk 0,
k = 1, 2, . . . , N,
(2a)
k = 1, 2, . . . , N,
(3a)
xk (0) = Rk , k = 1, 2, . . . , N, where q(x, t) = N i=1 qi (x, t), c(xi ) = p¯ − cx i .
(4a)
xk 0,
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Let q(x, t) = (q1 (x, t), q2 (x, t), . . . , qN (x, t)), i = (i1 , i2 , . . . , iN ), and define the Hamilton function H (t, x, q(x, t), i ) = (p(q(x, t)) − p¯ + cx i )qi (x, t) − i
N k=1
ik qk (x, t).
In a similar way by Rincon-Zapatero et al. [9], the necessary conditions of the system’s closed-loop equilibrium strategy (q1 (x, t), q2 (x, t), . . . , qN (x, t)) are given by jH i (t, x, q(x, t), i ) = 0, jqi
i = 1, . . . , N,
jH i (t, x, q(x, t), i ) i + ij , ˙ j = − jxj
(12)
j = 1, . . . , N, i = 1, . . . , N.
(13)
Let qi = qi (x, t) and q = q(x, t), then we have its partial derivatives of the Hamilton H i jH i (t, x, q(x, t), i ) = p (q)qi + p(q) − p¯ + cx i − ii , jqi
(14)
N
N
k=1
k=1
jqk jqk jH i (t, x, q(x, t), i ) jqi − ik , + (p(q) − p¯ + cx i ) = qi p (q) jxj jxj jxj jxj N
N
k=1
k=1
j = i,
jqk jqi i jqk jH i (t, x, q(x, t), i ) + cq i + (p(q) − p¯ + cx i ) − k . = qi p (q) jxi jxi jxi jxi
(15)
(15a)
If each producer’s extraction depends only on his own resource reserve xi , his feedback strategy takes the selffeedback form2 qi = qi (xi , t). For this case, holds jqk /jxi = 0 (k = i). Now from (12) to (15a), we have d(p (q)qi + p(q) − p¯ + cx i ) jqi jqi = − qi p (q) − cq i − (p(q) − p¯ + cx i ) dt jxi jxi jq i + (p (q)qi + p(q) − p¯ + cx i ) + (p (q)qi + p(q) − p¯ + cx i ) jxi or ˙ i + p (q)q˙i + p (q)q˙ = (p (q)qi + p(q) − p¯ + cx i ). p (q)qq
(16)
This is just the same equilibrium condition (7) of the open-loop strategy, and its sums give the same dynamical system (10)–(11) of the extraction q and reserve x. Our work below is based on such system. 4. Equilibrium path and properties For system (10)–(11) we have first that Proposition 1. (0, 0) is its unique equilibrium point and just a saddle point. Proof. Since p(0) = p, ¯ (0, 0) is an equilibrium point and its uniqueness is obvious. The Jaccobian at (0, 0) is given by 0 −1 0 −1 c J= , m (N + 1)p (0) 2 In reality, a producer may consider the information of his opponents when make decision. But because of the asymmetry or lack of information, it may be not easy for one to adjust his extraction promptly to be accord with his opponents’ reserve. The self-feedback strategy considered here is to focus on the bounded rationality rather than complete rationality of the decision makers. This form of feedback can be done easily by the producers and simplify our mathematical analysis.
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D. Zhanwen et al. / Nonlinear Analysis: Real World Applications 9 (2008) 1918 – 1927 q
Q+
Γ (ϕ = 0) Q− L2
E
qT
L1
0
xT
x
Fig. 1. Phase diagram of equilibrium path.
where m = c/(N + 1)p (0) < 0. So the eigenvalue
( + 2 − 4m) ( − 2 − 4m) > 0, < 0 and 2 = 1 = 2 2 which implies (0, 0) a saddle point.
Because x 0 and q 0, we need to pay attention only to the first quadrant of the x–q plane. Denote = p (q)q + Np(q) − N p¯ + cx. According to p (q)q + (N + 1)p (q) < 0 and by (11), q is increasing when < 0 and decreasing when > 0. We use the symbol to represent the curve = 0, Q+ to represent the area < 0 (in which q is increasing), Q− the area > 0 (in which q is decreasing), and E the stable orbit of the system. Apparently the stable orbit E is located in the area Q− (if not, the orbit E enters the area Q+ at some time, hence q increases thereafter and limt→∞ qt = 0, which is a contradiction to its stableness). And the norm vector of the curve defined by the function is given by (c, p (q)q + (N + 1)p (q)), where the x-coordinate is positive and the q-coordinate is negative, which means that the x-axis is located at the side of Q− (in which > 0) and the q-axis at the side of Q+ (in which < 0). In addition, there exits no more stable orbits other than E since (0, 0) is a saddle point. From these discussions, we get the phase diagram showed in Fig. 1. Theorem 1. The open-loop equilibrium path or the self-feedback equilibrium path is the stable orbit of the dynamical system (10)–(11). Proof. The system’s orbits except E must move along the path L1 or L2 showed in Fig. 1. For L1 , there exists some time T when L1 cuts x-axis at (xT , 0). With the constraint q 0, the optimal path of system (10)–(11) must end at this time T . Now the supply qT on the market is zero and the resource reserve xT is at a positive quantity. This means that there is at least a producer i whose reserve xiT > 0 and his unit profit p(qT ) − (p¯ − cx iT ) = p(0) − p¯ + cx iT = cx iT > 0, so it is profitable for him to extract his resource and supply on the market, which contradicts the zero supply when L1 cuts x-axis at (xT , 0). Therefore L1 is not the optimal path of the game. For L2 , it also cuts q-axis at (0, qT ) at some time T . By continuity, there exists a time such that when t ∈ (, T ), qt > qT /2 and cx t < p¯ − p(qT /2). In the period (, T ), each producer’s unit profit p(qt ) − (p¯ − cx it ) < p(qT /2) − p¯ + cx t < 0, which means it is not the best strategy to supply on the market in this period. But qt > 0, t ∈ (, T ), some producers have supplied. A contradiction. L2 is not the equilibrium path of the model, either. Proposition 2. On the equilibrium path, the price p and the resource extraction q satisfy (a) p˙ > 0 and q˙ < 0, i.e. p is increasing and q decreasing. (b) If x < (N − 1)p/c ¯ then p/p ˙ < .
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Proof. As stated above, E is located in the area Q− in which > 0, this means q˙ < 0, q is decreasing and hence p increasing. Noting that (p(q)q) < 0 (it means (p(q)q) = p (q)q + p(q) decreases with q), we have p (q)q + p(q) < p(0) = p¯ since q > 0 on the equilibrium path. Thus when x < (N − 1)p/c, ¯ holds − (N − 1)p(q) = p (q)q + p(q) + cx − N p¯ < p¯ + cx − N p¯ < 0 this means /(N − 1)p(q) < 1. And by the facts (p(q)q) < 0, p (q) < 0, q > 0 and > 0 on the equilibrium path, we have then p (q)q˙ p (q) p˙ = = p p(q) p(q)(p (q)q + (N + 1)p (q)) =
p (q) p(q)((p(q)q) + (N − 1)p (q))
<
p (q) = < . p(q)(N − 1)p (q) (N − 1)p(q)
Proof finished.
Proposition 2 means that on the oligopolistic market for nonrenewable resources the price of resource is increasing but will increase at a speed not more than the discount rate (noting that the condition of (c) will be satisfied at a moment since x approaches to zero). These results are similar to those of the models of constant production cost (cf. [6, Proposition 2, 7, Lemma 4]). For the fixed initial stock R = N i=1 Ri , we now use the variables with lower index N or in order to study the effects caused by the number of players or discount rate. The influences of the number of players N or discount rate on the equilibrium path are given with the following proposition. Proposition 3. For the equilibrium path E (a) qN (0)qM (0) and pN (0)pM (0), if N > M, (b) q (0)qϑ (0) and p (0)pϑ (0), if > ϑ. Proof. Let h=
(p (q)q + Np(q) − N p¯ + cx) = . p (q)q + (N + 1)p (q) p (q)q + (N + 1)p (q)
Obviously h is decreasing with and it can be verified easily that h is increasing with N since > 0 on the equilibrium path E. So q˙N > q ˙M when N > M. If qN (0) > qM (0), holds qN > qM . From the stable condition x(∞) = 0 we have ∞ ∞ R = 0 qN dt > 0 qM dt = R, a contradiction. Similarly p (0) > pϑ (0) is impossible. Here the result (a) of Proposition 3 differs from that of the fixed cost case (cf. [6, Proposition 3]). The difference is caused by the assumptions that the unit product cost is variable or fixed. For the fixed case, the increase of the numbers of players will increase competition and increase the current supply thus lower the current price since a change in the number of players will not alter the unit production cost. But for the variable case, when the total stock is fixed, the increase of the number of players will decrease their individual stock and hence increase their production cost, which means their extractions may decrease. However, this kind of result from the assumption of unchanged total stock is just a theoretic one. In the actual market, if all players have their own stocks, the total resource stock will be increased with his resource stock when a new player enters. 5. Individual behaviors The discussion above is about such aggregate variable as the total extraction rate q and the total resource reserve x. In this section, we pay our attention to the individual extraction rate qi and resource reserve xi . For calculating
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D. Zhanwen et al. / Nonlinear Analysis: Real World Applications 9 (2008) 1918 – 1927
convenience, we consider the linear inverse demand function p(q) = p¯ − q. Taking such function into (7) or (16) and (10)–(11), we get the linear system as c cN c q˙i = qi + x1 + · · · − xi + · · · + xN , i = 1, . . . , N, (17,18) N +1 N +1 N +1 x˙i = −qi , i = 1, . . . , N of which the linear matrix is given by ⎡ (N + 1) c ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ c ⎢ ⎢ A= N +1⎢ ⎢ − (N + 1) ⎢ c ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
(N + 1) c
−
(N + 1) c
..
.
..
.
(N + 1) c
−N
1
...
1
−N
...
... 1
... 1
(N + 1) c By simple calculating we have two pairs of eigenvalues of A:
√ ∓ ( + 4c/(N + 1)) ∓ ( + 4c) r1,3 = and r2,4 = , 2 2
1
⎤
⎥ ⎥ 1 ⎥ ⎥ ⎥ ⎥ ... ... ⎥ ⎥ . . . −N ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
−
where r1 and r2 are negative, r3 and r4 are positive. The eigenvectors of i (i = 1, 3) are given by i = (−ri , −ri , . . . , −ri , 1, 1, . . . , 1) and the eigenvectors of i (i = 2, 4) are given by 1i = (ri , −ri , 0, 0, . . . , 0, −1, 1, 0, 0, . . . , 0) ,
2i = (ri , 0, −ri , 0, . . . , 0, −1, 0, 1, 0, . . . , 0) , .. . = (ri , 0, 0, 0, . . . , −ri , −1, 0, 0, 0, . . . , 1) N−1 i
thus there are just 2N eigenvectors with linear independence. The solution of the linear system (17) and (18) is determined by these eigenvalues and eigenvectors with 2N constants 1 , . . . , 2N . That is q N−1 1 r3 t r4 t = 1 er1 t 1 + er2 t 2 12 + · · · + N N−1 e + e + · · · + + . (19) 3 N+1 N+2 4 2N 4 2 x Now arises the question of how to decide the values of the 2N constants 1 , . . . , 2N . Of course, the initial stock conditions xi (0) = Ri (i = 1, . . . , N ) can be contributed to N of them. But we can not set the initial values of the extraction rate qi randomly since the extraction path should be interiorly determined by the model itself. However, we have got the results that the equilibrium path runs along the stable orbit on which both the variables, reserve variable x and the extraction rate variable q, approach to zero in the infinite horizon. Because r3 and r4 are the system’s positive eigenvalues, the coefficient constants 1 , . . . , 2N in (19) must be all zeroes (otherwise q or x cannot approach to zero). So the equilibrium solution of the linear takes the form q = 1 er1 t 1 + er2 t 2 12 + · · · + N N−1 . (20) 2 x
D. Zhanwen et al. / Nonlinear Analysis: Real World Applications 9 (2008) 1918 – 1927
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By the initial reserve conditions xi (0) = Ri and from the last N equations of (20) ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ R1 −1 −1 1 ⎢1⎥ ⎢ 1 ⎥ ⎢ 0 ⎥ ⎢ R2 ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢R 1⎥ 1 ⎢ ⎢ . ⎥ + 2 ⎢ . ⎥ + · · · + N ⎢ . ⎥ = ⎢ .3 ⎣ .. ⎦ ⎣ .. ⎦ ⎣ .. ⎦ ⎣ .. 1
0
1
⎤ ⎥ ⎥ ⎥. ⎥ ⎦
RN
Thus we have 1 = R/N, 2 = R2 − R/N, . . . , N = RN − R/N , where R = Ri is the total initial reserve. Taking these results into (20), we can easily get the solution of the system (17) and (18) as follows: ⎧ R r1 t R ⎪ = − e − R − r r2 er2 t , i = 1, . . . , N, q ⎪ 1 i ⎨ i N N (21,22) ⎪ ⎪ ⎩ xi = R er1 t + Ri − R er2 t , i = 1, . . . , N. N N Proposition 4. If Ri > Rj holds for the initial stocks of player i and j , then their extraction rates satisfy (a) qi > qj , limt→∞ qi /qj = 1; (b) xi > xj , limt→∞ xi /xj = 1; (c) qi /xi > qj /xj , limt→∞ qi /xi = limt→∞ qj /xj . Proof. According to r2 < r1 < 0 and from (21) and (22), all these inequalities and equalities in (a) and (b) apparently hold and the ones in (c) can be easily and straightly verified. These properties show that the producer with a larger initial resource stock will supply at a larger rate and have a larger production/reserve ratio, but when time approaches to infinity all the players will play the same role on the market. In other words, the producer with larger initial stock has a larger but decreasing market share and in the infinite time horizon his advantage declines. Summing (21) and (22) from 1 to N , we have then the solution of the total variable in this situation: q = −Rr 1 er1 t and x = Rer1 t . From their expressions of the variables qi , xi , q and x we get the expressions of qi /q and xi /x as follows: R r2 (r2 −r)t1 xi R qi 1 1 1 1 e , = + Ri − = + Ri − e(r2 −r1 )t . q N R N r1 x N R N Noting that r2 < r1 < 0, we have the following further proposition. Proposition 5. If Ri =R/N or Ri > R/N or Ri < R/N , then producer i’s market share qi /q is constant or decreasing or increasing, and remains equal to or higher than or lower than 1/N , respectively, but converges toward 1/N in each situation; and all these hold for his resource reserve share xi /x. Proposition 5 means that if the stock of player i is greater (smaller) than the average, both his market share and his reserve share are decreasing (increasing), but when t → ∞ the shares of all producers converge toward the same one, i.e. the average share 1/N. Proposition 5 together with Proposition 4 characterize the individuals’ economic roles and the relationships between each other. Using constant coefficients linear feedback strategy, Salo and Tahvonen [10] have got some similar results of this game (cf. Propositions 4 and 5 in their work). Because we have got the stable equilibrium path, we have here found out the system’s solution expressions and the results are more obvious and straight. For the case N = 2, = 0.02 and c = 0.15, the phase picture of the resource reserve x1 and x2 is referred to Fig. 2. In Fig. 2 the two dotted lines isolate two areas (q1 = 0 and q2 = 0). In the area q1 = 0, for example, because of his resource stock shortage, producer 1’s production cost is so high that his net profit is less than zero and hence gives zero
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Producer 2's resource reserve level x2
100 90
q1=0
80 70 60 50 40 30 20 10
q2=0 10
20 30 40 50 60 70 80 90 100 Producer 1's resource reserve level x1
Fig. 2. Phase diagram for two oligopolies.
supply until the price grows up to the point where he begins to have positive profits. After entering the area between the two dotted lines, the two producers play on the equilibrium path. When their initial stock equals, their reserve levels are on the symmetry turnpike (the 45◦ bold line). If not, one with the higher stock keeps the higher market share. But all the equilibrium paths are tangent to the symmetry turnpike at the origin point (0,0), which means the differences between the two producers disappear gradually in an infinite time. 6. Concluding remarks In this paper we have reconsidered the game model of nonrenewable resource in which the unit production cost is variable and depends on the resource reserve level. For the case of linear cost, we find that the equilibrium conditions of open-loop and self-feedback strategies can be described as the same dynamical system for the case of linear cost. Our main mathematical result of this system is that its unique stable orbit is just the equilibrium path of the game. Based on this result, we have got the equilibrium properties of the main variables such as price, the total reserve and the total supply of resource. Most properties we give here are very similar to those of other oligopoly models with fixed unit production cost. It is also from this mathematical result that we have got the solution expressions of each producer’s maximization problem and some concrete and detailed properties are given. Although the method of this paper is different to that of Salo and Tahvonen [10], the main equilibrium properties of the individual behaviors are very similar to their works. Acknowledgements This work is financially supported by the National Natural Science Foundation of China (No. 90610031) and by the High Talent Foundation of Jiangsu University (No. 07JDG023). References [1] T. Basar, G. Olsder, Dynamic Noncooperative Game Theory, Academic Press, London, 1995. [2] J. Chermak, R. Patrick, A well based cost function and the economics of exhaustible resources: the case of natural gas, J. Environ. Econ. Manage. 28 (1995) 174–189. [3] D. Epple, J. Londregan, Strategies for modelling exhaustible resource supply, in: A. Kneese, J. Sweeney (Eds.), Handbook of Natural Resource and Energy Economics III, Elsevier, Amsterdam, 1993, pp. 1077–1107. [4] F. Groot, C. Withagen, A. de Zeeuw, Note on the open-loop von Stackelberg equilibrium in the cartel-versus-fringe model, Econ. J. 102 (1992) 1478–1484. [5] F. Groot, C. Withagen, A. de Zeeuw, Open-loop von Stackelberg equilibrium in the cartel-vs.-fringe model, Energy Econ. 22 (2000) 209–223.
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[6] T. Lewis, R. Schmalensee, On oligopolistic markets for nonrenewable natural resources, Q. J. Econ. 95 (1980) 475–491. [7] G. Loury, A theory of ‘oil’igopoly: Cournot Nash equilibrium in exhaustible resources markets with fixed supplies, Int. Econ. Rev. 27 (1986) 285–301. [8] S. Polansky, Do oil producers act as oil’ilopolists?, J. Environ. Econ. Manage. 23 (1992) 216–247. [9] J.P. Rincon-Zapatero, J. Martinez, G. Martin-Herran, New method to characterize subgame perfect Nash equilibrium in differential games, J. Optim. Theory Appl. 96 (2) (1998) 377–395. [10] S. Salo, O. Tahvonen, Oligopoly equilibria in nonrenewable resource markets, J. Econ. Dyn. Control 25 (2001) 671–702.