Exhaustible Resources and Competitive Equilibrium Robert D. Cairns∗ Department of Economics McGill University 855 Sherbrooke St. W. Montreal, Canada H3A 2T7 Fax (514) 398-4938 e-mail
[email protected] December 2004
∗
Thanks to Hassan Benchekroun, Gérard Gaudet, Karine Gobert, John Hartwick, Pierre Lasserre, Ngo Van Long and Licun Xue for comments on earlier drafts, and to FCAR and SSHRCC for financial support.
Abstract Theoretical underpinnings of competitive equilibrium in an exhaustibleresource market with u-shaped costs are proposed. The analysis extends Novshek and Sonnenschein’s reconciliation, in the limit, of static general and partial equilibrium. There are three types of limiting, competitive equilibrium, all of which are equivalent in the static case and each of which has been examined by resource economists. Each leads to a qualitatively different limiting equilibrium. One of the three corresponds to the Marshallian partial equilibrium of the mine. Key Words: exhaustible resources, non-convex costs, Marshall-Walras synthesis, competitive equilbrium, noncooperative equilibrium
2
1. Introduction Much of the literature on equilibrium in a resource market is based on the presumed actions of competitive firms. What we mean by competitive equilibrium, however, is not obvious when we come to integrate the theory of the mine (Gray, 1914; Scott, 1967) into sectorial models of the type pioneered by Hotelling (1931). Gray and Scott both assume that firms face u-shaped average costs. Optimalcontrol models of the extraction path usually assume costs to be convex over the domain on which they are defined; but costs are not convex if there is an interval over which average cost declines. Eswaran, Lewis and Heaps (1983) study this problem and conclude that non-convexity precludes the existence of competitive equilibrium in such a market. Lozada (1996) confirms the nonexistence, but proposes in his conclusion that a Cournot model might be more serviceable in describing equilibrium. Novshek and Sonnenschein and others1 have made steps toward reconciling some problems in the received static, neoclassical theory of the firm. These problems involve (a) inconsistencies in the static theories of both partial equilibrium and general equilibrium, as well as (b) an inconsistency in using partial1
See the references provided by Novshek and Sonnenschein (1987).
3
equilibrium results as descriptors of individual-market equilibrium in general equilibrium. In Marshallian partial-equilibrium theory it is assumed that firms face increasing returns over a defined interval but also that there exist so many firms that none has power to influence price. If the interval is not degenerate, the integer problem arises: a zero-profit equilibrium obtains only on a set of measure zero.2 Arrow-Debreu-Mackenzie-type general-equilibrium theory, on the other hand, assumes convexity but not large numbers of firms. In summarizing their work, Novshek and Sonnenschein (1987) propose a consistent synthesis of the two. They describe a competitive equilibrium involving firms with non-convex costs as the limit of a sequence of Cournot equilibria in which the firms become small compared to their market. Their approach has great intuitive appeal, as it corresponds to economists’ predisposition that perfect competition is an ideal limiting case of imperfect markets. But their analysis has not yet been extended to dynamic models. In the present paper we explore the possibility of synthesizing the theory of the extractive firm with that of market equilibrium. The dynamics in this case are equilibrium dynamics, rather than the disequilibrium tâtonnement of Novshek and Sonnenschein’s analysis. In their single-period model, the concept of Cournot 2
See also Baumol, Panzar and Willig (1982).
4
equilibrium was suited to dealing with their proposed mutations to the market or to the firms’ average-cost curves as the number of firms increased. In a resource market, in order to be realized, any proposed equilibrium must be dynamically consistent in the sense discussed by Newbery (1981).3 Competitive (price-taking) equilibria do exist in the limit. Indeed, there are at least three different types of competitive equilibrium, each giving a different solution to the dynamic problem, but which are indistinguishable in the static case.
2. A Stylized Resource Industry Suppose that at time t = 0, the total reserve of an exhaustible resource available to society is R, each of N firms having R/N at its disposal. Let the firms face u-shaped average costs, which reach a minimum of c at quantity q = a. Firms can enter and exit freely at any time, provided only that they hold reserves. Also, let the instantaneous demand curve be (weakly) concave and stationary at all times, t, where time flows continuously from t = 0. Kimmel (1984) discusses a type of equilibrium for such a market, but an equi3
It is worth stressing at the outset that the present paper does not seek to represent an actual resource market for predictive purposes. Rather, the implications of some frequently made, highly stylized assumptions for limiting equilibria are examined.
5
librium in which each firm’s production at all instants is constrained to be either zero or a fixed, positive quantity; average cost is minimized at the fixed level of output. In general, however, as Kimmel allows toward the end of his paper, there should be no constraint on production other than cost and availability of reserves to the firm. Firms will not necessarily produce at q = a. In Kimmel’s equilibrium, given a > 0 and N > 0, at any time some firms may be producing and others not. In considering an equilibrium without such constraints, Lozada (1996) assumes competitive (i.e., price-taking) behavior among a finite number of firms, n. Suppose, however, that at any time t, the firms (indexed by i = 1, ..., n) are producing quantities (not necessarily equal) qi ≥ 0. Total output is
P
qi and marginal rev-
P P P enue to any firm with positive output qi is p ( qi ) + qi p0 ( qi ) < p ( qi ). Price taking is not consistent with maximization under perfect information and cannot describe an equilibrium. A similar problem arises in Fisher and Karp’s (1993) approach, even in the case where a backstop technology and uncertainty about stocks apparently salvages a competitive (albeit possibly inefficient) equilibrium. Let us consider what the properties of a dynamically consistent equilibrium among the n firms would be, if it existed. As in other exhaustible-resource equilibria, the difference between marginal revenue and marginal cost would rise through 6
time at the rate of discount. In addition, however, there would be what may be called a dynamic arbitrage principle: price less average cost would rise at the rate of discount. This condition would be needed to preclude entry by firms producing in periods of low average profits but idle in periods of high average profits. Suppose that a firm is producing output q1 at time t1 and is considering shifting its output from t1 to some time t2 when it is not producing. In particular, it changes production to zero over the infinitesimal interval dt1 , saving reserves of q1 dt1 , and produces q2 over dt2 , such that q2 dt2 = q1 dt1 .4 Let its profit flow at any time be π (q). If producing at t1 is an equilibrium, the firm must not be able to gain by entering at t2 . Therefore, π (q1 ) dt1 e−rt1 ≥ π (q2 ) dt2 e−rt2 , or π (q1 ) π (q1 ) dt1 π (q2 ) dt2 e−r(t2 −t1 ) π (q2 ) −r(t2 −t1 ) = ≥ = e . q1 q1 dt1 q2 dt2 q2 If the inequality were strict, then entry could go the other way, for some firm. Thus, at times when any of the N firms is idle, discounted average profit must be constant. 4
Our argument is similar to the one used by Solow (1974) to explain Hotelling’s rule for marginal changes in output. Here, discrete changes in output, from q1 > 0 to 0 and from 0 to q2 > 0, are envisaged.
7
Both conditions cannot hold throughout the period of exploitation.5 Suppose that, in a proposed equilibrium, more than one firm is producing at the terminal time. To be explicit about the fact that this terminal time depends on (among other things) the output at which average cost is attained, a, and the number of firms, N, we write it as T (a, N). One firm could save some of its reserves from time T (a, N) to T (a, N) + ε, and capture a greater profit on those units than in the proposed equilibrium. (A slightly different point was made by Lozada.) Maximal unit monopoly profit occurs, however, when price less average cost, p (q) − A (q) = π (q) /q, is maximized. Taking both as functions of the firm’s output, and assuming that terminal output is positive, we show in the Appendix that the maximum is attained when p0 = A0 . (The condition used by Eswaran et al., that at the terminal time A (q) = a, is derived under the assumption that the firms are price takers.) If p0 = A0 at the terminal time and only one firm is producing, then there is no incentive to hold a small quantity back for production at T (a, N) + ε. If equilibrium existed, then on some interval ending at T (a, N), industry production would be less than a and carried out by a monopolist. But on that interval both conditions, p0 = A0 and price equals marginal cost, cannot 5 The conditions can hold if (i) there is a “flat-bottomed” average-cost curve (Baumol et al. 1982) and (ii) chattering controls are permitted (Eswaran et al. 1983). Neither is assumed here.
8
hold (except at the terminal time, when it is true). The non-existence of equilibrium arises from the dynamic arbitrage principle, and not from the putative jump in profit that would occur after the proposed terminal time if price were parametric to the individual firms. It holds for all the cases studied by Lozada, including situations where returns to scale are increasing everywhere and where time is discrete. Now let us follow Novshek and Sonnenschein (1987: 1290) by letting a tend to zero in the limit, and N to infinity, so as to let aN remain constant.6 The original level of industry reserves, R, also remains constant, so that R/N tends to 0. Not frequently stressed is the fact that the proposed limiting process has a side effect of convexifying the technology in the limit. As this observation is important in motivating the present analysis, we digress briefly to discuss how it arises and its implications. Novshek and Sonnenschein’s technology is described by the cost function, C(y) = C0 + v(y) for y > 0, where v0 (y) > 0 and v 00 (y) > 0. (In their definition, they allow for v00 ≥ 0, but we are ruling out a flat bottom to the average-cost curve.) Minimum average cost is at output a, and a is allowed to go to zero in the limit. For a < 1, Novshek and Sonnenschein define Ca (y) = 6
Mumy (1984) gives a hint of the type of application of Novshek and Sonnenschein’s approach to exhaustible resources used in the present paper.
9
aC(y/a) = aC0 + av(y/a). Hence, the ratio of fixed cost to variable cost (or of average fixed cost to average variable cost), C0 /v(y/a), tends to zero with a for any positive y. In the limit, then, not only does the industry equilibrium become competitive, but the technology of the individual firm becomes convex. Friedman (1977: 30-1) models a similar type of Cournot problem, but suggests that it may be more realistic to assume that variable costs are negatively related to the capital stock, and hence to the level of fixed costs. He observes that qualitative properties of the limiting competitive equilibrium may be affected by this assumption. For present purposes, what is important is that, even in the static case, one must be careful about what one means by changing conditions in order to achieve an equilibrium which corresponds to common notions of competitive. As a guide to analysis, we first consider a stylized planning problem.
3. A Stylized Planning Problem The stylization makes use of a device suggested by the work of Loury (1986) and by the properties of Cournot equilibrium summarized by Shapiro (1989). Cournot equilibrium with n symmetric firms “maximizes” a weighted average of firms’ profits, π(q) = qp (q) − Pn
i=1
Pn
i=1
C (qi ), and social surplus, W (q) =
Rq 0
p (z) dz −
C (qi ). This is true in the sense that the first-order conditions are the 10
same in the maximization as in the Cournot problem. The weighted average is [π (q) + (n − 1) W (q)] /n. Consider a planner with access to N identical deposits of an exhaustible resource at time t = 0. • Let the average cost of extraction, A (q) = C (q) /q, be u-shaped with minimum at q = a/N. Therefore, C 00 (q) < 0 for q < a/N and C 00 (q) > 0 for q > a/N. • Let the number of deposits producing, a non-negative integer, at time t be n(t) ≤ N. • Let C 0 (q) > 0 for q ≥ 0, and C(0) = 0. Because C(0) = 0, deposits not producing (qi (t) = 0) incur no cost. • Let the inverse demand p(q) be (weakly) concave. • Let R be the total initial reserve (given and fixed), divided equally among the N deposits at time t = 0: Ri (0) = R/N. • Finally, in keeping with the remarks above, let
VN =
Z
0
∞
e−rt
(
1 n(t) − 1 q(t)p [q(t)] + n(t) n(t) 11
Z
0
q(t)
p(z) dz −
N X i=1
)
C[qi (t)]
dt.
The stylized planning problem is to
max
[qi (t), n(t)]∞ t=0
VN
such that, for i = 1, ..., N,
qi (t) = −R˙ i (t); qi (t) ≥ 0; Ri (t) ≥ 0; q(t) =
N X
qi (t).
i=1
(Compare Loury’s Lemma 5.) At any time t < T (a/N, N), the terminal time, there are n(t) active deposits for which qi (t) > 0, and N − n(t) inactive deposits, where 1 ≤ n(t) ≤ N and n(t) is an integer. Since there is no equilibrium (even open loop), the function VN is not a fictitious-objective function as defined by Slade (1994), even if demand is linear (as is allowed in our model). Nevertheless, the objective VN has a maximum since it can be made bounded on a compact set in the space (q, n, T ), viz. q ∈ [0, q¯] where q¯ is the smallest value of q such that p (¯ q ) = 0; 0 ≤ n ≤ N; and T ∈ [0, R/p (˜ q )] 12
where p0 (˜ q ) = A0 (˜ q ). The full solution to this problem involves changes in integer values of n (t) and intuitive considerations relevant to Bertrand competition. We do not provide the full solution, but obtain the properties which are relevant for the present purpose. See the appendix for details. By symmetry, if there is an idle deposit, the choice between producing from it or from any other deposit that is in production is immaterial to the planner. Moreover, whenever deposit i is producing, it minimizes the planner’s cost to extract a quantity equal to that of any other producing deposit, so that qi (t) = q(t)/n(t). Therefore, the current (marginal) rent, λ, is equal across all producing deposits and hence all deposits. The current-value Hamiltonian may be written,
1 n−1 H (q, n, R, λ) = qp (q) + n n
Z
0
q
p (z) dz − nC
³q´ n
− λq.
The paths of qi (t), i = 1, ..., N, are piecewise continuous, as is the path of q(t). By the adjoint condition, Hotelling’s rule, the current rent rises at the rate of interest. Whenever qi (t) > 0, the first-order condition yields,
p [q(t)] + qi (t)p0 [q(t)] − C 0 [qi (t)] = λ.
13
(3.1)
Let the consumers’ surplus be S (q) =
Rq 0
p (z) dz − qp (q). Since A (q/n) =
C (q/n) / (q/n), the current-value Hamiltonian can be rewritten as
H (q, n, R, λ) =
Z
0
q
p (z) dz − λq − q [A (q/n) + S (q) / (nq)] .
The number of firms, n (t), producing at time t, is also a control variable and maximizes the Hamiltonian. Maximizing the Hamiltonian with respect to n is equivalent to minimizing an opportunity cost,
O(q, n) = A (q/n) + S (q) / (nq) ,
the sum of the average cost of an active firm and an opportunity cost of average surplus per firm due to oligopoly pricing. Industry average cost is scalloped near points where the cost-minimizing number of firms changes (Baumol et al. 1982). The opportunity cost O(q, n) exhibits complicated behavior. An algebraic example is provided in the appendix. ¯ A (necessary) transversality condition is that H ¯T (a/N,N) = 0, or that, for
14
t = T (a/N, N), ¸ ∙ Z q 1 p (z) dz − A (q/n) . λ= pq + (n − 1) nq 0 Equating the values of λ from this and from condition (3.1) and simplifying yield that, at t = T (a/N, N ),
p0 (q) =
S (q) dA (q/n) + (n − 1) 2 . d (q/n) q
If n [T (a/N, N)] = 1, the equilibrium on an interval at and just before time T (a/N, N) obeys the condition of the putative equilibrium discussed in Section 2: only one deposit is in production and its production path follows the monopoly solution, as given by equation (3.1) when n(t) = 1, up to the point where
p0 (q [(T (a/N, N)]) = A0 (q [(T (a/N, N)]) .
That is to say, at the terminal time the slopes of the demand curve and the average-cost curve of the remaining deposit are equated. However, the opportunity cost O (q, n) may be minimized when n [T (a/N, N)] > 1. For example, it is optimal not to have n [T (a/N, N)] = 1 if O [q, 2] < O [q, 1], 15
with q at its optimal value for t = T (a/N, N), or
C [q] > 2C
£q¤ 2
− 12 S [q] .
In such cases, p0 [q (T )] 6= A0 [q (T )] in the solution to the planning problem, whereas this condition is necessary in a non-cooperative problem. The reason is that the planner takes the consumers’ surplus into account. Finally, the number of producing firms n(t) decreases at discrete points, and, by the continuity of the shadow value λ, total production q(t) = n(t)qi (t) adjusts discontinuously at those points so as to maintain equation (3.1) before and after the jumps. Notwithstanding the fact that the marginal rent, λ, is the same across firms, average profit is not the same through time for a given deposit. The industry produces for a finite period, T (a/N, N). Let
Πi =
Z
0
a T(N , N)
{qi p [q(t)] − C [qi (t)]} e−rt dt
be the total profit attributed to deposit i. Let the planner arrange production among the deposits such that their contributions to total discounted profits diverge
16
minimally, in the sense that ³ ´ ∆(a/N, N) = max Πi − min Πi i
i
is minimized. For finite N, ∆(a/N, N) > 0. This stylized planning solution is used below in proposing a non-cooperative equilibrium to the problem with N profit-maximizing firms.
4. Modified Epsilon Equilibrium and Limiting Cases Now let each of the N firms control one deposit. As is discussed above, for finite N and u-shaped costs, we are not able even to propose an equilibrium of the sort found by Loury (1986).7 It would appear that, in the case at hand, competitive equilibrium cannot be the limit of equilibria for a sequence of games with increasing values of N. It is, however, possible to utilize a slight modification of a concept called epsilon-equilibrium (Radner 1980: 137) to generate the required sequence: “an epsilon-equilibrium is a combination of strategies, one for each player, such that 7
Loury finds that when marginal costs are constant, an open-loop equilibrium always exists. Under certain restrictive assumptions, all firms produce at all times along the equilibrium path until the resource is exhausted.
17
each player’s strategy is within epsilon in utility (e.g. average profit) of the maximum possible against the other players’ strategies.” Radner focuses exclusively on average profit over a repeated game (the full game, or, briefly, the subgame beginning at the current time) as the definition of utility. The modification to be used here is that discounted profit over the full game is the definition of utility. Suppose that the firms agree prior to time t = 0 to implement the stylized planning solution of Section 3, including the possibility of not satisfying the condition p0 [q (T )] = A0 [q (T )], and including the allocation of production resulting in minimizing of ∆(a/N, N). Because condition (3.1) holds, there is no incentive to any firm to make a marginal change in output at any time t < ∞. Now let sequence ε(a/N, N) = ∆(a/N, N). If an epsilon-equilibrium is defined such that the firms are willing to forbear from entering in periods in which they have low average profits when the maximal difference ∆(a/N, N) is no greater than ε(a/N, N), then the planning solution is an open-loop, epsilon-equilibrium. We take the limit of the sequence of these equilibria as N → ∞. The concept of epsilon-equilibrium does not involve fully rational behavior. Unlike Novshek and Sonnenschein, however, we are primarily interested in the limiting equilibrium rather than the properties of the sequential equilibria. In the sequential equilibria, the choice of which firms are to have which levels of profit 18
could be made by the firms’ agreeing to a random selection mechanism at time t = 0. Before the random mechanism is brought to bear, the expected equilibrium is symmetric. After this, the (asymmetric) epsilon equilibrium for N firms would follow from the assumed forbearance of the firms. Radner (1980: 153) gives two reasons why a firm might be satisfied using a response to other firms’ strategies which is not optimal. The implicitly co-operative flavor of epsilon equilibrium could be overcome by, for example, assuming that the cost of changing price lists and informing clients of a deviation at some later date would exceed ε(a/N, N). We prove in the Appendix that limN →∞ ε(a/N, N) = 0. Therefore, the limit of the sequence of ε(a/N, N)-equilibria has symmetrical discounted profits for all firms.
5. Limiting Competitive Equilibria It is possible to discuss at least three cases according to how the market or the technology of the firms is altered as N increases. Two of them utilize limiting epsilon-equilibria.
19
5.1. Case i. Average cost is “squeezed from the left and from the right” Imagine a sequence of ε(a/N, N)-equilibria in which an individual firm’s cost is transformed according to Novshek and Sonnenschein’s proposal, as follows:
CN (y) = C(Ny)/N, N = 1, 2, 3, ...
(This is the example of a cost function given in the appendix.) In the limit, a firm’s average-cost curve has infinite slope beyond the minimum point (zero), as well as a slope of negative infinity up to the minimum point. For finite N, the industry’s technology is non-convex. In the limit, the industry’s technology is convex and indeed has constant returns to scale: A mass of firms of infinitesimal size produce at minimum average cost. Since the price net of marginal cost and price net of average cost each rise at the rate of interest, firms are indifferent as to whether they produce at one instant or another during the time interval, [0, T ], where T = limN →∞ T (a/N, N). The entire reserve endowment is exhausted at time T . Also in the limit, the mass of firms producing at any time t < T is given by the total quantity produced in the optimal solution. Since the total mass of firms is decreasing with time, not all firms produce at all times. Rather, there are 20
inactive firms which are holding reserves but not currently producing. There is a symmetry among firms in terms of total discounted profits (the value of their deposits), as well as discounted average profit at all times. Thus, the incentive to entry arising from the dynamic arbitrage principle is eliminated in the limit. Industry output and the measure of active firms tend to zero at T . This solution has some interest in the theory of exhaustible resources. Its characteristics bear a strong resemblance to those of Schultze’s (1974) solution. The parallel indicates how the limiting solution depends on convexification of the technology: Schultze assumed that the number of firms in production was differentiable, so that, except on sets of measure zero, the “number” of firms in production was not an integer. By this assumption he inadvertently imposed constant returns to scale on the industry’s technology. Our limiting process, too, convexifies the industry’s technology, but holds to the assumption that n(t) is an integer. The resulting solution retains qualitative similarity to the ε(a/N, N)equilibrium for finite N, as well as to Kimmel’s (1984) constrained solution.
21
5.2. Case ii. Average cost beyond the minimum point is unchanged Suppose the output level at which average cost is minimized is reduced toward zero by the following modification to Novshek and Sonnenschein’s method:
CN (y) = C(Ny)/N,
y ≤ a/N
= C [y + (N − 1)a/N] , y > a/N. As N increases, the average-cost curve becomes steeper up to its minimum point, a/N. Beyond the minimum point the curve CN is obtained from C by a translation of the horizontal axis, and so remains parallel to the original for N = 1. It is easily checked that the two branches are smoothly pasted at y = a/N. The limiting equilibrium is what Eswaran, Lewis and Heaps (1983) may have had in mind but holds only for a convex technology: the difference between price and marginal cost increases at the rate of interest as firm and industry output fall toward zero at T . Because average cost differs from marginal cost, the difference between price and average cost increases at less than the rate of interest. In the limiting equilibrium, all N firms produce at all times up to time T . Each firm produces an infinitesimal flow qt , with q˙t < 0, which is such that it integrates to the infinitesimal original stock. Since there is no potential entrant at any time in
22
the limiting equilibrium, there is no market period in which discounted average profit is higher but a firm is idle. This limiting equilibrium differs qualitatively from the equilibrium of case i. Still, as N increases, the technology is convexified.
5.3. Case iii. The market is replicated Novshek and Sonnenschein (1987: 1288) observe that in the static case, as far as the properties of the sequence of Cournot equilibria are concerned, it is equivalent (a) to replicate the market by increasing both demand and the number of finitesized firms and (b) to reduce the size of firms while keeping demand fixed. In the exhaustible-resource case, this equivalence does not hold. Let p = α − βq be the inverse-demand curve. Suppose that both N and R are fixed, but that we rotate the inverse-demand curve about its intercept on the price axis by having β fall toward zero. For example, Radner (1980: 150) allows the number of demanders to change from N1 M to N2 M when the number of firms changes from N1 to N2 , and lets demand depend on the number of firms, so that8
p (q) = α − βq/N. 8
For a non-linear demand curve one needs to imagine its being “straightened out” as it is rotated about its intercept. It is easier to think in terms of the linear demand from the start.
23
So long as β is positive, we have similar qualitative results to the ε(a/N, N)equilibrium.9 In the limit, where β is zero, we again find a qualitative difference. Price is fixed at α at all times up to T (and thereafter). All firms produce at all times (there is no potential entrant at any time), and each firm’s output falls until it reaches a at time T . Average, as well as marginal, cost declines through to T . The limiting solution corresponds to Gray’s (1914) solution and to Scott’s (1967) theory of the competitive mine.
6. Interpretation All three limiting equilibria are noncooperative equilibria to the limiting game, and not merely epsilon-equilibria. Also, under each hypothesis about changing the technology or the market, a partial social-planning optimum is obtained in the limit. Case i reproduces the competitive solution when there are constant returns to scale and an endogenous price. Case ii reproduces the competitive equilibrium when there are an endogenous price and diminishing returns to scale. 9
Increasing the market’s size by parallel shifts of the demand curve does not produce a dynamic competitive equilibrium. It would in the static case, provided that α − βN qi = c, the minimum average cost, so that there were an integral number of firms: the only relevant consideration is the quantity demanded at p = c. This is a less interesting case because demand shifts of this sort imply the addition of consumers with increasingly higher valuations of any quantity of the good, and not adding identical consumers as is done when the market is replicated.
24
Case iii reproduces the competitive solution when there are an exogenous price and u-shaped average costs. The first two retain the perspective of a given market and alter (i.e., convexify) the technology of the firm; the third fixes the firms’ technology while altering the character of the market. In two limiting equilibria, all firms produce an equal quantity at all times, as assumed by Eswaran, Lewis and Heaps, Fisher and Karp and Lozada. In equilibrium (i), the one with constant returns at the industry level, this property does not hold and the mass of firms approaches zero as the time of exhaustion approaches. Once the three cases are set out, it becomes clearer that the equilibrium that Eswaran, et al. were trying to describe combined elements of cases ii and iii, which are distinct even in the limit. In order to retain a Marshallian aspect in the limiting equilibrium, we must replicate the market with the given non-convex technology. In that limiting equilibrium, all firms produce at all times; they produce varying levels of output through time; and when exhaustion occurs each firm is producing at minimum average cost. Case iii does not give Hotelling’s (1931) prediction of a rising, rather than a fixed, price. For price to rise in case iii, there must be shifts in the demand curve such that the intercept rises. The partial-equilibrium analysis gives no reason to postulate such shifts rather 25
than downward or random shifts, or no shift at all. The r-per-cent rule of resource economics becomes not so much one of determination of the price path, as Hotelling proposed, as of the pattern of production of individual mines.10 In other words, the asset nature of resources (Solow, 1974) comes out, not in pricing, but in production, considerations. Analysis of this equilibrium is not far fetched. Since the classic study of Barnett and Morse (1963), empirical attempts to confirm Hotelling’s rule using price data have had mixed success. World markets for resource products are large, and the firms are small compared to their market but are not small in an absolute sense. More generally, the need to embed the competitive model in a dynamic framework can have unexpected implications. If one’s aim is to represent the implications of non-convex technologies of the type made famous by Marshall, the present examination recommends resort to the third conceptualization by which the size of the market is increased, rather than to what may appear to most economists to be the more natural ones in which the firms are made smaller. The recommended approach has a number of characteristics: 10 Under non-competitive conditions, the r-per-cent rule involves both pricing and production decisions.
26
1. It preserves the view of competition as being a limit, albeit in a particular sense, of dynamically consistent equilibria, and hence as an idealization of the market. However, symmetry arises as a limit of asymmetric equilibria among symmetric firms. 2. It does so while preserving the technology of the individual firm and the Marshallian view, which is one object of study. 3. In so doing, it obviates the need for fanciful notions such as instantaneous fractional-firm equilibria or the differentiability of the positive integers. 4. However, its resort to the notion of an unboundedly large market in the limit is compatible with a Walrasian general-equilibrium model only if the entire economy is allowed to grow unboundedly large.
7. The Marshall-Walras Synthesis for Exhaustible Resources Further exploration of the use of limiting equilibria as a means of conceptualizing competitive equilibrium indicate the usefulness of the general approach for a particular type of dynamic market, but with qualifications. In the static case, three types of manipulation of the conditions of industry equilibrium lead to equivalent limiting solutions and, indeed, an equivalent sequence of Cournot equilibria. 27
Consideration of an exhaustible-resource market allows for a simple first attempt to introduce dynamic considerations into the model in an intuitive way. A main qualification is that the equilibrium for elements of the sequences in all three cases up to the limit exists only as an epsilon-equilibrium. Moreover, in each case, the nature of the limiting equilibrium depends on the limit process. Each displays characteristics of three different types of competitive equilibrium which have been discussed in the literature on exhaustible resources and of the limiting equilibria used by Novshek and Sonnenschein. If the technology is manipulated, what happens to the convex part of the technology — whether the average-cost curve is “squeezed from the left and the right” or “pushed up against the axis from the minimum point” — matters qualitatively to the limiting competitive equilibrium. Also, both of these methods convexify the firms’ technology, albeit in ways that are consistent with Novshek and Sonnenschein’s analysis. As in their model, such convexification is problematic when the aim of the exercise is to synthesize work involving two strands, one of which holds that technology is not convex because average costs are u-shaped. On the other hand, (a) replicating the market by a rotation of the demand curve and (b) altering the technology are not equivalent in the exhaustible-resource market studied in this paper. Replication results in a model which reproduces the 28
partial-equilibrium results of the received theory of the individual, competitive mine. Replication also produces a synthesis that preserves both the non-convex nature of the technology and consistency with the theory of competitive general equilibrium in an infinitely large economy. The case of exhaustible resource also stresses the notion that, in a finite economy, competitive general equilibrium can obtain only with a convex technology. The proposed limiting process produces desired general-equilibrium properties only through the atrophy of the partial-equilibrium ones.
References [1] Barnett, Harold and Chandler Morse (1963), Scarcity and Growth, JohnsHopkins Press, Baltimore. [2] Eswaran, Mukesh, Tracy R. Lewis and Terry Heaps (1983), “On the Nonexistence of Market Equilibria in Exhaustible Resource Markets with Decreasing Costs,” Journal of Political Economy 91, 1, February, 154-167. [3] Fisher, Anthony C. and Larry S. Karp (1993), “Nonconvexity, Efficiency and Equilibrium in Exhaustible-Resource Depletion”, Environmental and Resource Economics 3, 97-106. 29
[4] Friedman, James W. (1977), Oligopoly and the Theory of Games, NorthHolland, Amsterdam. [5] Gray, Lewis C. (1914), “Rent under the Assumption of Exhaustibility,” Quarterly Journal of Economics 28, 466-489. [6] Hotelling, Harold (1931), “The Economics of Exhaustible Resources,” Journal of Political Economy 39, 137-175. [7] Kimmel, Sheldon (1984), “A Note on Extraction with Non-Convex Costs,” Journal of Political Economy 92, 6, December, 1158-1167. [8] Lozada, Gabriel A. (1996), “Existence of Equilibria in Exhaustible Resource Industries: Nonconvexities and Discrete vs. Continuous Time”, Journal of Economic Dynamics and Control 20, 433-444. [9] Loury, Glenn C. (1986), “A Theory of ‘Oil’igopoly: Cournot Equilibrium in Exhaustible Resource Markets with Fixed Supplies,” International Economic Review 27, 2, 285-301. [10] Mumy, Gene E. (1984), “Competitive Equilibria in Exhaustible Resource Markets with Decreasing Costs: A Comment on Eswaran, Lewis and Heaps’s
30
Demonstration of Nonexistence,” Journal of Political Economy 92, 6, December, 1168-1174. [11] Newbery, D.M.G. (1981), “Oil Prices, Cartels and the Problem of Dynamic Consistency,” Economic Journal 91, September, 617-646. [12] Novshek, William and Hugo Sonnenschein (1987), “General Equilibrium with Free Entry: A Synthetic Approach to the Theory of Perfect Competition,” Journal of Economic Literature XXV, 3, September, 1281-1306. [13] Radner, Roy (1980), “Collusive Equilibria in Noncooperative EpsilonEquilibria of Oligopolies with Long but Finite Lives,” Journal of Economic Theory 22, 136-154. [14] Schultze, William D. (1974), “The Optimal Use of Non-Renewable Resources: The Theory of Extraction,” Journal of Environmental Economics and Management 1, 53-74. [15] Scott, Anthony D. (1967), “The Theory of the Mine under Conditions of Certainty,” in Extractive Resources and Taxation, Mason M. Gaffney, ed., University of Wisconsin Press, Madison, 25-62.
31
[16] Shapiro, Carl (1989), “Theories of Oligopoly Behavior,” in Handbook of Industrial Organization, Vol. 1, Richard Schmalensee and Robert D. Willig, eds., North-Holland, Amsterdam, 329-414. [17] Slade, Margaret E. (1994), “What does an Oligopoly Maximize?”, Journal of Industrial Economics 42, 45-61. [18] Solow, Robert M. (1974), “The Economics of Resources or the Resources of Economists,” American Economic Review 84, May, 1-14.
8. Appendix 8.1. Condition at terminal time, putative contestable equilibrium In the proposed solution, either qi (t) ≥ qT or qi (t) = 0. Clearly, there will not be intervals of zero total production (q (t) = 0) for t < T . Therefore, q(t) ≥ qT for all t ≤ T . Therefore, T ≤ R/qT < ∞. By a (necessary) transversality condition, ˜ |T = (qp(q) − C(q) − λi q) |T = 0. Using this and (3.1), we have, at time T , H 1 [qp(q) − C(q)] = λi (T ) = p + qp0 (q) − C 0 (q), or q C 0q − C d p (q) = = q2 dq 0
32
µ
¶ C(q) . q
8.2. Concavity of Hamiltonian with respect to output By the convexity of average cost,
d2 [C (q) /q] 2 0
1, the Hamiltonian is concave at all points of continuity of q. Since the Hamiltonian is maximized, costs are minimized globally, and hence each firm produces the same output. In this case, using the same observations as above,
H = pq + (n − 1) ∂ 2H ∂q2
Z
q
0
p (z) dz − nC (q/n) − λq;
= (n + 1) p0 + p00 q − C 00 (q/n) /n ≤ (n + 1 − 2/n) p0 + p00 q 33
< 0.
8.3. Algebraic Example A cost function with the desired properties is ³ ´ 1 2 2 C (q) = h q + q , q > 0, h > 0. This implies that there are u-shaped average costs. In fact, ³ ´ ´ ³ 1 3 A (q) = h q + q− 2 and A0 (q) = h 1 − 12 q− 2 ; A0 (q) = 0 at q ≈ .63. Let k > 0 and let the demand function be given by
p = k (1 − q) , 0 ≤ q ≤ 1.
Then S (q) =
Z
0
q
k (1 − x) dx − kq (1 − q) =
34
k 2 q . 2
Therefore,
O (q, n) = A
³q´ n
+
³ q ´− 12 S (q) q ; = (2h + k) + h nq 2n n
∙ ³ q ´− 32 ¸ 1 . Oq (q, n) = 2h + k − h 2n n The switch from n (t) = 2 to n (t) = 1 occurs when O (q, 1) = O (q, 2), or when ¸2 ∙ 4(X2−1)h 3 q= . Also, 2h+k ³ q ´− 32 ¶ ´ 1µ 1³ − 32 Oq (q, 1) − Oq (q, 2) = 2h + k − hq 2h + k − h − 2 4 2 ³ ´ 3 3 3 2h + k h − 2q − 2 − 2 2 q− 2 = 4 4 h i 1 − 32 2h + k + 2 (X2 − 1) hq = 4 > 0.
As q decreases, the relative opportunity cost of having one as opposed to two firms decreases. It is possible to have n (T ) > 1 for high values of k/h (so that the switch point is not attained). As an example of the cost function in Section 5 (i), let ³ ´ 1 2 2 CN (q) = C (Nq) /N = h Nq + (q/N) . 35
Then ´ ³ 1 AN (q) = CN (q) /q = h Nq + (Nq)− 2 and i h 3 A0N = Nh 1 − 12 (Nq)− 2 = NA0 (Nq) . 8.4. Limiting value of ε(a/N, N) Let the minimum value of average cost be c, and π m = maxq q [p(q) − c] = q m [p (qm ) − c], a finite constant, be the instantaneous monopoly profit for this demand curve if cost were cq. For any N, 0 ≤ ∆(a/N, N) < π m [Ri (0) /qm ]. (The last expression is the total profit that firm i would obtain if it had a monopoly for Ri (0) /q m periods and the discount rate were zero.) But π m Ri (0) /q m = (π m /qm ) (R/N), all N, and R/N → 0 as N → ∞.
36
