- Email: [email protected]
Optimal forest rotation under monopoly and competition
JOURNAL
OF ENVIRONMENTAL
ECONOMICS
AND MANAGEMENT
17, 54-65 (1989)
Optimal Forest Rotation under Monopoly and Competition’ PHILIPPE J. CRABBh Department
of Economics,
University
of Ottawa,
Ottawa,
Ontario,
Canada
KIN
6N.5
AND NGO VAN LONG Department of Economics, Australian National University, GPO Box 4, Canberra, ACT 2601, Australia
Received August 17,1987; revised April 29,1988 The two main propositions of the paper establish sufficient conditions for the optimal rotation to be invariant to market structure whether monopolistic, competitive, or maximizing the temporal sum of consumer’s surpluses. A corollary to the second proposition yields sufficient conditions for the monopolistic rotation to be actually longer than the competitive one. 0 1989 Academic Press. Inc.
1. INTRODUCTION
There has been a long-lasting concern in public policy not borne out by available scientific empirical evidence however, about privately owned forests being subjected to shorter than optimal rotations (e.g., [6, 31). This public policy concern is more relevant for countries in which forests are mainly privately owned, such as the United States or Sweden, than for countries in which they are mostly publicly owned, such as Canada, or in a state of open access as in less developed countries generally [23, 3, lo]. Monopolistic practices due to large ownership, proximity of the forest to timber markets, lack of a close substitute for a species or a quality of timber, etc. might explain why private forest owners are suspected of adopting shorter than optimal rotations. Rough intuition suggests that monopolists will tend to adopt rotations shorter than perfectly competitive ones in order to keep volumes lower and prices higher when faced with a downward sloping demand for timber. The economics of non-renewable resources has taught us, however, that intuition about monopoly-competition extraction paths comparison may be misleading. In particular, there exist circumstances under which the monopolist behaves like a perfect competitor or extracts a non-renewable resource even faster than under competition. One can, therefore, ask whether similar circumstances exist in forestry. This is the question we set ourselves to answer. The forestry economics literature has paid scant attention to comparison of rotations between monopoly and competition. We were able to find only one article which dealt, unsatisfactorily according to us, with the issue [21]. It concluded that a ‘The paper was written while the second author was visiting professor at Carleton University. It was presented at Lakehead University, The University of British Columbia, and at the Canadian Seminar on Economic Theory. Comments from F. J. Anderson, B. Forster, P. Neher, P. Pearse, and A. D. Scott were appreciated. 54 0095-0696/89 $3.00 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any fom reserved.
ROTATION
AND
MARKET
STRUCTURE
55
monopolistic rotation would be shorter than a competitive one and that the latter would not coincide with the one which maximizes the temporal sum of consumers’ surpluses. Since our methodology, assumptions, and conclusions are at variance with the corresponding ones of the aforementioned article and the several comments it generated, we will review the latter after we present our own results. Our two main propositions establish sufficient conditions for the optimal rotation to be invariant to market structure whether monopolistic, competitive, or maximizing the temporal sum of consumers’ surpluses. A corollary to the second proposition yields sufficient conditions for the monopolistic rotation to be actually longer than the competitive one. We will assume throughout that regeneration costs are nonnegative constants. The first proposition is a short-run one to the extent that the forest land area is assumed to be fiied and to have no alternative use. It does not allow for thinning. It assumes the forest to be initially uniformly regulated; i.e., the land is divided into an integer number of plots corresponding to the number of integer years in the optimal rotation, each plot being stocked with the same number of trees, the first being bare land or stocked with trees of age zero; the second, with trees of age one; the third, with trees of age two, etc., until plot T + 1, with trees of age T, the optimal rotation. Under the above assumptions both monopolists and competitors will adopt the same rotation which is the Faustmann periodic rotation. Moreover, the same timber price and the same quantity of timber will be supplied under the two market structures. The second proposition is a long-run one from both the static and dynamic points of view. In a static sense, the forest land area is assumed to be variable and subject to alternative uses. The non-fixity of the area does not per se preclude Faustmann rotations. In a dynamic sense, we assume that enough time has elapsed for a steady-state policy to set in, steady-state policy whose existence we assume as well. This steady-state policy will turn out to be the Faustmann periodic solution in the particular case where thirming is not allowed. This second proposition is independent of any initial age structure for the forest and allows the possibility of thinning. Under these assumptions, a monopolist will again adopt the same rotation as a competitor, but the timber price which will prevail and the quantity of timber supplied may vary from one market structure to the next. Finally, the third and last proposition is a corollary of the second one. Under the assumptions for the second proposition, except that land has no alternative use for the monopolist and is so plentiful as not to be able to generate a monopolistic rent, the monopolistic rotation would be longer than the competitive one.
2. SUFFICIENT CONDITIONS FOR ROTATIONS TO BE INVARIANT TO MARKET STRUCTURE WHEN FOREST LAND AREA IS FIXED, HAS NO ALTERNATIVE USE AND INITIAL FOREST IS UNIFORMLY REGULATED
A first result on rotation comparison between monopoly and competition can be obtained as a direct corollary of Theorems 3.1 and 3.2 due to Mitra and Wan ([17]). While they did not make any explicit monopoly-competition comparison, one of their results points in that direction. Thus, assuming that society somehow begins with a uniform distribution of trees with the oldest ones being at the Faustmann
56
CRABBk
AND
LONG
age, Theorem 3.1 states that the optimal policy for society is to maintain that uniform distribution (thus trees are harvested if and only if they reach the Faustmann age). In the Mitra-Wan discrete time model, the Faustmann (1849) age T* is the positive integer which maximizes
aT[f(T) - Cl/(1 - ST), where f(T) f(T) f(T) f(T) S is C >
is the growth curve for a representative tree such that
= 0 for 0 < T < b for some b 2 1 has unique maximum TM to the right of b and is continuous is increasing for b d T < TM and decreasing for T > TM the discrete-time discount factor 0 is the regeneration cost.
The Mitra-Wan theorem 3.1 holds for all concave utility functions U(q) with positive marginal utility, where q is the quantity of timber harvested. Their theorem 3.2 states that optimal stationary programs are invariant to changes in the utility (profit or benefit) function as long as it is concave and increasing and f( 7’) satisfies the above assumptions. In particular, this is true for
where P(q) is the demand curve-provided P’(q) < 0 and P(q) > 0 for all q. Clearly the competitive outcome coincides with the socially optimal outcome, since a linear function is concave and satisfies Mitra-Wan theorem 3.1 if it is increasing. Suppose that a socially managed forest, in its optimal stationary program, with a steady-state output q* (for all t), is handed over to a monopolist who wants to maximize the sum (or integral) of discounted profit, where profit is
WI) = p(dq. Then, as long as II(q) is concave and II’(q) > 0 for all q < q*, the monopolist will find it optimal to maintain the forest exactly as it is under the socially optimal program which is identical to the perfectly competitive one. In particular, the monopolistic price and output are the same as under competition. This is a direct corollary of Mitra and Wan’s Theorem 3.2. It is important to note that the Mitra-Wan result stated above is restricted to cases where the initial condition happens to be a steady-state optimal stationary forest. Moreover, Mitra and Wan assume that the surface area is fixed and that there is no thinning. * Planting and harvesting costs are zero or a positive constant [18]. This result is reminiscent of the well-known result in the economics of non-renewable resources, that the monopolistic price and output are the same as under competition if the resource stocks are the same in both cases, if extraction cost is zero, and if demand elasticity is constant [24]. *Mitra even with
and Wan conjecture that with a linear utility function [18].
forest
thinning
the Faustmann
solution
may
not
be optimal
ROTATION
AND
MARKET
57
STRUCTURE
Our first result can then be summarized as: PROPOSITION 1.
If
(a) the forest area is given but has no alternative uses, (b) the forest is uniformly regulated, (c) the cost of regeneration is a non-negative constant,
(d) there is no thinning, rotations under perfect competition and monopoly are identical to Faustmann rotations and efficient according to the criterion of intertemporal maximization of consumers plus producers’ surpluses; moreover, output and prices will be identical as well under the two market structures. 3. SUFFICIENT CONDITIONS FOR ROTATIONS TO BE INVARIANT TO MARKET STRUCTURE WHEN FOREST LAND AREA IS VARIABLE AND HAS ALTERNATIVE USES
In the present section, we wish to show that the monopolistic choice of rotation period is the same as the competitive choice, even when their outputs and prices are different. To allow for differences in output, we assume that forest land has alternative uses. The simplest way to model this is to assume that foresters can always rent land at an exogeneously determined and time-independent rental rate R (per acre per period). We introduce a non-negative cost of planting and harvesting. We also allow for thinning; in this respect our model is more general than Mitra and Wan [17, 181. The introduction of land rent (at an exogeneous rate) to the forestry problem may seem inconsistent with the Faustmann model in which land has no non-forestry use and its rent is endogeneous (see [22]). However, Comolli has shown that there is no inconsistency and has demonstrated that with exogeneous land rent the single rotation or forest mining formula coincides with the Faustmann formula, as long as the zero profit condition is taken into account and land is bare initially [4].3 This will serve as a reminder of the Faustmann formula and may be explained as follows: Given R, the exogeneous rental rate per unit per period, and C, the regeneration cost, each competitive firm does not need to solve the replanting problem; land is always available somewhere else, at the cost R. Each firm solves problem (1) for given P, the price of timber, C and R: PROBLEM (1).
Maxi+nize II = [Pf(T)
- C]exp(-rT)
= M$x[Pf(T) 3This result is quite similar to determined [22]. The forest mining Faustmann one and was mistakenly I. Fisher, Hecksher, Hotelling, Allen,
r91.
- C]exp(-rT)
- irRexp(-rt)
- frl
dt
- exp( - rT)]
that obtained by Samuelson, where land rent is endogeneously formula ignores the site value for reforestation contrarily to the adopted by such prominent economists as ThEnen, Jevons, Wicksell, and F. and V. Lutz until Gaffney proved that Faustmann was right
58
CRABBe
AND
LONG
when Max, II = 0. Therefore, Mra[Pf(T)
- C]exp(-rT)/[l
- exp(-rT)]
= 1,
(1)
where exp( .) denotes the exponential function and r > 0 is the rate of interest (here time is a continuous variable). In a competitive equilibrium, profit is zero, so that each firm is indifferent between solving (1) with rented land and buying land at the price R/r and solving the Faustmann problem (2): PROBLEM (2).
Maxiinize
[Pf(T)
= Mrm[Pf(T)
- C]exp(-rT)[l
+ exp( -rT)
- C]exp(-rT)/[l
+ exp( -r2T)
+ . .. ]
- exp(-rT)].
(2)
We will show that with a perfectly elastic supply of land, if thinning is not allowed, the Faustmann solution will obtain with any initial forest, not just with initial bare land. In this section, we allow the firm to choose different harvesting ages for trees planted at different times. Let y(t) denote the number of trees planted at time t or equivalently the number units of land on which trees are planted; each tree occupies one unit of land. A tree planted at time t and chopped down at time t + A(t) yields a flow of timber for thinning and eventually when being felled. The flow at time S, where t G s G t + A(t), is denoted by sb - t, 4th
#w)~
where s - t is the age of tree at time S, and P(t) denotes the thinning policy adopted by the firm for trees planted at time t. Without loss of generality, we assume that for given t, /3(t) is a real number to be chosen from the real interval [0, 11. Clearly, since A(t) is the harvesting age, it must be true that g(s - t, 44 g(s - t, 4th
P(t)) Nt))
= 0 a 0
ifs-t>A(t) if s - t < A(t).
(34 w
Thus, if the price of a unit of timber were unity, then the present value at time t of this flow would be
J+wT P(t)) =
r+A(f)g(s -
t, A(t), P(t))exp(
--T(s - t)) ds
(44
or
V(A(t),P(t)) = ~A”‘g(~yA(t), P(t))exp(-ra)da.
(4b)
To make our analysis comparable with models where thinning is not allowed, we take it that this is the limiting case where B = 0, and we assume that sji;o where f(a)
W(t),
B(t))
= f(dt))exp(
-4t)L
(5)
is the size of the tree at age a if no thinning takes place at earlier ages.
59
ROTATION AND MARKET STRUCTURE
The quantity of lumber harvested at time s is
Assume that our monopolist starts the optimization problem at time 0. He inherits ya(u) units of land on which y,(u) trees were planted at time u (u G 0) and are still surviving at time 0. The cost of land rent from time u to time 0 is sunk cost. He still has to decide when to chop down these old trees. If A(u) is the age at which these trees are to be chopped down, the present value of land rent incurred from time 0 to the cutting date is ‘(‘)+“Rexp(-rz)
dz = (R/r)[l
/0
- exp(-rA(u)
- ru)].
(7)
(For example, if u = - 10 and A(u) = 30, then the monopolist has to pay for 20 years of land rent, starting from time 0.) The total cost of maintaining all old trees (planted before time 0) until their cutting date is Co = (” y,(u)(R/r)[l
- exp(-rA(u)
- ru)] du.
(8)
Note that yO( u) is exogeneously given, and A(u) can be chosen at time 0. We now turn our attention to the new trees to be planted at t >, 0. Let y,(t) be the number of units of land on which new trees planted at time t. Let A(t) denote the cutting age. Then the rental cost of each block, discounted back to time I, is A(f)Rexp(-rz)
/0
dz = [l - exp(-rA(t))]R/r,
(9)
where z = s - t The present value of all the new trees is C, = imyn(t)[l
- exp(-rA(t))](R/r)exp(-rt)
dt.
00)
The quantity of timber harvested at time s, Q(S), given by (6), can be thought of as consisting of Q,-,(s) and Q,(S) (timber obtained from trees planted before and after time 0, respectively),
Q(s) = Q,(s) + Q,(s),
(11)
where Q,(s)
= (” y,(u)& J-00
Q,(s) = ~ht)ds
- 0, A(u),
s(u))
- t, A(t), p(t)) dt
du
02)
03)
60
CRABBI?
AND
The monopolist’s problem is to choose r,(t), Q,(s) so as to maximize II = /omexp(-r.s)P(Q(s))Q(s)
LONG
A(t),
A(u), /3(t), p(u), (Jo(s), and
(14)
ds - C, - C,,,
where Q(S) = QO(s) + Q,(s), Cc and C, are given by (8) and (10). In general it is difficult to obtain theorems that can be given simple economic interpretation. It is unlikely that A(t) is a constant for t sufficiently close to zero. Thus, if the monopolist starts with bare land (no old trees), he may find it advantageous to harvest the first generation of trees very early rather than to supply nothing for the first few periods. If we are willing to assume that in the long run both A(t) and P(t) converge to constant values, then we can say something meaningful about these constants. Thus we assume that the monopolist has been optimizing for a long time and that at the present moment (which we can denote as t = 0 without loss of generality) all new trees to be planted after 0 will have a constant cutting age A and a constant thinning policy B. Then C, can be simplified with the help of (13). With A(t) = A and /3(t) = j3, (13) reduces to
Q,(s) = (x&)g(s Multiply
- t, A, P) dt.
(15)
(15) by exp( - rs) and integrate:
From the theory of Laplace transforms, the right-hand side of (16) can be written as lmg(n,
A, P)exp( -ra)
d$?&)exp(-rt)
dt.
(17)
Now, recall that, from (3a), g(a, A, /?) = 0 if a > A. Hence A, p)exp(-ru)
img(u,
da = /04g(u, A, P)exp(-ru)
du = Y(A, a).
(18)
Therefore, using (16), (17), and (18), lmyn(t)exp(-rt)
0
dt = [l/V(A,
B)l~mQ.(~)exp(-~~)
U!S.
(19)
From (lo), with A(t) = A, C,, = (R/r)[l
- exp(-rA)]dmyn(t)exp(-rt)
dt.
(20)
From (19) and (20), c, = [l - exp(-~A)][R/rV(A,B)l~wQ.(s)exP(-rs)
~5.
(21)
ROTATION
AND
MARKET
61
STRUCTURE
Using (21) and (14), the monopolist’s profit is II = ~~exp(-rs)P(P(s))Q(s)
- [l - exp( -rA)]
d -
[R/rY(A,
CO
P)] imQn(s)exp(
-rs)
U!S.
(22)
Clearly, to maximize (22) for any given /3, A must minimize [l - exp( -rA)]
[R/rV(A,
P)].
(23)
Thus, we have obtained the following result: The long-run optimal rotation is independent of the function P(Q) and hence is the same as the rotation under competition or the maximization of the temporal sum of consumers’ surpluses provided land rent is exogeneous. Consider now the special case where thinning is not allowed (j3 = 0). In this case, recalling (5), we can write (23) as
@/r)[l - exp(-rA)lN(A)exp(-rA). Minimizing
(24) with respect to A yields
f’(A)/f(A) = r/b - ew(-rA)l
(25)
which is identical to the Faustmann formula, if the planting and harvesting cost, C, is zero. With positive planting and harvesting cost and with no thinning (j? = 0), we still obtain the same result. For, in this case, C, in (10) is modified as c, = dw~“(t)[(R/r)(l and the counterpart
+ Cexp( -rA(t))]exp(-rt)
d
(lo’)
of (24) is
[(R/r)(l Minimizing
- exp( --d(t))
- exp(-rA))
+ Cexp(-rA)]/f(A)exp(-rA).
(24’)
(24’) with respect to A yields f’(A)/f(A)
= rexp(rA)/[exp(rA)
- 1 + Cr/R]
(25’)
which is identical to the Faustmann formula obtained under competition. Our second and main result can then be summarized as: PROPOSITION
2.
If
(a) forest land is available in unlimited quantities but has alternative (perfectly elastic supply), (b) the forest is regulated or not, (c) the regeneration cost of a tree is a non-negative constant,
uses
rotations under perfect competition and monopoly are in the long run identical and efficient according to the criterion of intertemporal maximization of consumers plw
62
CRABBlk
AND
LONG
producers ’ surpluses; however, output and prices will not generally be the same under monopoly and competition. The optimal rotation wiN be the Faustmann one if thinning is not allowed.
4. ROTATIONS UNDER MONOPOLY AND COMPETITION WHEN FOREST LAND IS VARIABLE BUT HAS NO ALTERNATtVE USES FOR THE MONOPOLIST
In the previous section we assume that the rental rate for land, R, is exogeneous. If it were endogeneous, for example, if R is the marginal product of agricultural land which depends on the quantity of land used in agriculture (which diminishes as the quantity of land used in forestry increases), then it would no longer be true that the monopolist would choose the same rotation period as the competitive one. Alternatively, one can endogenize R by assuming that forest land has no alternative use. Suppose that the monopolist’s marginal revenue, P’(Q)Q + P(Q), equals zero at some Q = Q. If the forestry land area is large enough, then the implicit land rental to the monopolist is zero (some land being left vacant).4 In this case the monopolist would be happy to maintain a forest (on a fraction of the total land area) with trees from age 0 to age TM, where TM minimizes C/rf(T) (see (24’), with R = 0). But under competition, with zero profit condition and with a larger output, all land would be used and the equilibrium land rental would be positive. The competitive rotation period would then be the Faustmann one, i.e., the one satisfying
f’(T!l[l(T)
or, substituting
= r/[l
- exp(-rT)]
for P from (l), i.e.,
P = [(R/r)(l
f’(T) fo
- z]
- exp(-rT))
= rexp(rT)/[exp(rT)
+ Cexp(-rT)]/f(T)exp(-rT) - 1 + Cr/RI.
Thus the monopolistic rotation would be longer than the Faustmann perk15 Our third and last result can then be summarized as:
rotation
PROPOSITION 3. Under monopoly and under the same assumptions as Proposition 2, except for thinning which is not allowed and for rent which is assumed to be endogeneously determined and nil, the rotation would be longer than the perfectb competitive one. 4This case is ruled out by Mitra and Wan because they assume that marginal revenue is always positive [17]. ‘To see this, note that both the monopolist and the competitive firm minim& (24’), but by assumption the monopolist’s R is zero, while the competitive firm faces a positive R. Applying the implicit function theorem to the first-order condition of the minimization problem (241, one sees that &/dR < 0.
ROTATION
AND MARKET
STRUCTURE
5. THE NAUTIYAL - FOWLER
63
PROPOSITION
The only article we were able to find in the forestry economics literature on the monopoly-competition comparison of rotations is by Nautiyal and Fowler [21]. This article generated quite a few comments [2, 19, 5, 15, 201. Nautiyal and Fowler attempted to prove that a monopolistic rotation (denoted here by 7”) is shorter than the one adopted if the monopolist were forced to behave competitively (7”). They also claimed that the former rotation is shorter than the one which the monopolist would adopt if he were forced to maximize the temporal sum of consumers’ surpluses (T,,) which, in turn, is shorter than the latter, i.e., TM G T,, G T,. The results allegedly hold for the case where time is continuous, the regeneration cost is nil, thinning is not allowed, the forest land area is given, and two alternative initial conditions hold. The first alternative is an unregulated (age-homogeneous) forest initially planted on bare land (the Faustmann model without cost but with an inverse demand function). The second alternative is a uniformly regulated forest obtained progressively by starting from bare land and planting each “year” (time is assumed to be continuous though!) a fraction of land equal to one over the best rotation corresponding to a given market structure. The results are short-run (fixed land area) in a static sense. However, they are not comparable to our first proposition since our initial condition calls for an instantaneous uniformly regulated forest at the initial time. Let us point out immediately that the assumption of progressively obtained uniformly regulated forest made by Nautiyal and Fowler is artificial and must necessarily lead to second best types of results.6 Indeed, leaving a portion of fixed land vacant during the first rotation is never optimal when regeneration is costless. This assumption alone may be responsible for the discrepancy between the rotation for the three market structures. If we come back now to the first initial condition, i.e. unregulated forest starting with bare land, this model, like the Faustmann one, is a pure point input-point output model. The “feast and famine” nature of the model is certainly not compatible with perfect competition and a stationary price. Indeed, with a stationary demand schedule, spot markets would clear at rotation dates only and a complete set of future markets could not be established except for rotation dates. The only way to remedy physical illiquidity of an asset through market liquidity is to add into the model money and an instrument which converts a title to a tree at date T into money [12]. One could object to this general equilibrium line of argument that Nautiyal and Fowler’s model is a partial equilibrium model: a monopolist is forced to adopt a competitive behavior or to maximize the temporal sum of consumers’ surpluses. However, the consumer surplus will not be defined except at rotation date in a pure point input-point output model. The only meaningful rotation comparison remaining in a partial equilibrium point input-point output model is between monopolistic behavior and price parametric behavior obtained by letting the elasticity of demand grow without bound. Thus, we agree with Nautiyal and Fowler that the rotation adopted costlessly by a monopolist planting a forest on a fixed and bare land area (Faustmann model) lengthens when one lets the price elasticity of stationary demand curve go to infinity. This result does not conflict with ours since it is based on different assumptions. However, our results demonstrate that the
6Tbis was already noticed by Femow [8].
64
CRABBe
AND LONG
Nautiyal-Fowler conclusion that monopolistic rotations are shorter than competitive ones is a specialized one and cannot be generalized. It does not hold either when forest land is limited and when the forest is uniformly regulated initially or, in the long run, when the supply of land is perfectly elastic. 6. CONCLUDING
REMARKS
Intuition suggests that the monopolist will tend to adopt rotations shorter than the perfectly competitive ones to keep volumes lower and prices higher when faced with a downward-sloping demand for timber. As in the area of non-renewable resources, intuition about monopoly-competition output paths comparison may be misleading. One reason for the relative neglect of this problem in the forestry literature is that the optimization problem facing foresters is a lot more complicated than the one faced by miners. This is especially true when the benefit (or profit) function is not linear in the quantity of timber harvested as is the case under monopoly. The major stumbling block is that the initial conditions, i.e., the initial age distribution of the trees, may prevent the optimal rotation from converging to a stationary solution such as the Faustmann periodic rotation even if the latter exists and is unique [17, 181. We derived two sets of sufficient conditions which lead to identical and efficient rotations under the two market structures. The first set applies to uniformly regulated forests whose area is fixed and has no alternative use. It follows directly as a corollary to two theorems proven by Mitra and Wan [17]. The second applies in the long run to any forestry land whose area is in perfectly elastic supply at a stationary opportunity cost. Our conclusion, that in the long run, the monopolistic rotation period is the same as the competitive one, or the one corresponding to the maximization of the temporal sum of consumers’ plus producers’ surpluses if rent is exogeneous, has its parallel in the theory of product durability (see, for example, [l or 141). In both cases, market structure is irrelevant (in the long run) to the choice of the optimal economic life of the asset: this choice is essentially a problem of cost minimization, provided that production can be independently adjusted. The proof of Proposition 2 relies on a separability property of Laplace transforms. Global asymptotic stability which is required for Proposition 2 has been conjectured by Kemp and Moore who have shown it to hold in several examples [13]. Mitra and Wan are rather skeptical about the conjecture holding for discrete time forestry models with positively discounted concave utilities [18]. For optimal programs with discounted utilities, they point out that the conjecture (the “turnpike property”) has been proven to be true if the discount rate is not too large [16, Theorems 8.3 and 8.41. However, they emphasize that these growth models have less structure than the forestry ones. An extension of our results to the case where the forest yields nontimber benefits could be envisaged [ll]. However, this extension may not be trivial for Proposition 1, since utility becomes a function of two arguments instead of one. REFERENCES 1. A. Abel, Market structure and the durability of goods, Rev. Econom. Stud. SO, 625-637 2. F. J. Anderson, Optimal rotation: Comment, Lund Econom. 57, 293-294 (1981).
(1983).
ROTATION
AND MARKET
STRUCTURE
65
3. P. Berck, The economics of timber: A renewable resource in the long run, Bell J. Econom. 10, 441-462 4.
5. 6. 7.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
(1979).
P. M. Comolli, Principles and policy in forestry economics, Bell J. Econom. 12, 300-305 (1981). P. M. Comolli, A proposition on imperfect-stumpage markets, Land Econom. 60, 297-300 (1984). R. T. Ely, “An Introduction to Political Economy,” 1st ed., Eastons & Mains, New York (1889). M. Faustmann, Berechmmg des Werthes, welchen Walboden sowie noch nicht haubare Holzbestande ftir die Waldwirtschaft besitzen, Allgemeine Forst Jagd Zeitung 15,441-455 (1849); Engl. Tmsl. in “Martin Faustmann and the Evolution of Discounted Cash Flow” (Gane, Ed.), Oxford Commonwealth Forestry Institute Paper 42 (1968). B. E. Femow, “Economics of Forestry,” T. .I. Crowell, New York (1902). M. M. Gaffney, “Concepts of Financial Maturity of Timber and Other Assets,” A.E. Information Series No. 62, Department of Agricultural Economics, North Carolina State College, Raleigh, NC (1957). Government of Canada, “Selected Forestry Statistics,” Canadian Forestry Service (1986). Hartman, The harvesting decision when a standing forest has value, Econ. Inquiry 14, 52-58 (1976). J. Hirshleifer, Liquidity, uncertainty, and the accumulation of information, in “Uncertainty and Expectations in Economics” (Carter and Ford, Eds.), Blackwell, Oxford (1972). M. C. Kemp and E. J. Moore, Biological capital theory: A question and a conjecture, Econom. Letr. 4, 141-144 (1979). J. M. Malcomson, Replacement and the rental value of capital equipment subject to obsolescence, J. Econom. Theory 10, 24-41 (1975). K. McConnell and T. Phipps, Optimum forest rotation in an imperfect stumpage market: Comment, Land Econom. 60, 215-217 (1984). L. W. McKenzie, Optimal economic growth, turnpike theorems and comparative dynamics, in “Handbook of Mathematical Economics” (Arrow and Intriligator, Eds.), North-Holland, New York (1986). T. Mitra and H. Y. Wan, Jr., Some theoretical results on the economics of forestry, Rev. Econom. Stud. 52,263-282(1985). T. Mitra and H. Y. Wan, Jr., On the Faustmann solution to the forest management problem, J. Econom. Theory 40, 229-249 (1986). J. C. Nautiyal, Optimum rotation: Reply, Land Econom. 57, 295 (1981). J. C. Nautiyal, Optimum forest rotation in an imperfect stumpage market: Reply, Land Econom. 60, 218 (1984).
21. J. C. Nautiyal and K. S. Fowler, Optimum forest rotation in an imperfect stumpage market, Land Econom. 56,213-226 (1980). 22. P. A. Samuelson, Economics of forestry in an evolving society, Econom. Inquiry 14,466-492 (1976). 23. R. A. Sedjo and M. Clawson, Global forests, in “The Resourceful Earth” (Simon and Kahn, Eds.), Blackwell, New York (1984). 24. J. Stiglitz, Monopoly and the rate of extraction of exhaustible resources, Amer. Econom. Rev. 66, 655-661 (1976).
OF ENVIRONMENTAL
ECONOMICS
AND MANAGEMENT
17, 54-65 (1989)
Optimal Forest Rotation under Monopoly and Competition’ PHILIPPE J. CRABBh Department
of Economics,
University
of Ottawa,
Ottawa,
Ontario,
Canada
KIN
6N.5
AND NGO VAN LONG Department of Economics, Australian National University, GPO Box 4, Canberra, ACT 2601, Australia
Received August 17,1987; revised April 29,1988 The two main propositions of the paper establish sufficient conditions for the optimal rotation to be invariant to market structure whether monopolistic, competitive, or maximizing the temporal sum of consumer’s surpluses. A corollary to the second proposition yields sufficient conditions for the monopolistic rotation to be actually longer than the competitive one. 0 1989 Academic Press. Inc.
1. INTRODUCTION
There has been a long-lasting concern in public policy not borne out by available scientific empirical evidence however, about privately owned forests being subjected to shorter than optimal rotations (e.g., [6, 31). This public policy concern is more relevant for countries in which forests are mainly privately owned, such as the United States or Sweden, than for countries in which they are mostly publicly owned, such as Canada, or in a state of open access as in less developed countries generally [23, 3, lo]. Monopolistic practices due to large ownership, proximity of the forest to timber markets, lack of a close substitute for a species or a quality of timber, etc. might explain why private forest owners are suspected of adopting shorter than optimal rotations. Rough intuition suggests that monopolists will tend to adopt rotations shorter than perfectly competitive ones in order to keep volumes lower and prices higher when faced with a downward sloping demand for timber. The economics of non-renewable resources has taught us, however, that intuition about monopoly-competition extraction paths comparison may be misleading. In particular, there exist circumstances under which the monopolist behaves like a perfect competitor or extracts a non-renewable resource even faster than under competition. One can, therefore, ask whether similar circumstances exist in forestry. This is the question we set ourselves to answer. The forestry economics literature has paid scant attention to comparison of rotations between monopoly and competition. We were able to find only one article which dealt, unsatisfactorily according to us, with the issue [21]. It concluded that a ‘The paper was written while the second author was visiting professor at Carleton University. It was presented at Lakehead University, The University of British Columbia, and at the Canadian Seminar on Economic Theory. Comments from F. J. Anderson, B. Forster, P. Neher, P. Pearse, and A. D. Scott were appreciated. 54 0095-0696/89 $3.00 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any fom reserved.
ROTATION
AND
MARKET
STRUCTURE
55
monopolistic rotation would be shorter than a competitive one and that the latter would not coincide with the one which maximizes the temporal sum of consumers’ surpluses. Since our methodology, assumptions, and conclusions are at variance with the corresponding ones of the aforementioned article and the several comments it generated, we will review the latter after we present our own results. Our two main propositions establish sufficient conditions for the optimal rotation to be invariant to market structure whether monopolistic, competitive, or maximizing the temporal sum of consumers’ surpluses. A corollary to the second proposition yields sufficient conditions for the monopolistic rotation to be actually longer than the competitive one. We will assume throughout that regeneration costs are nonnegative constants. The first proposition is a short-run one to the extent that the forest land area is assumed to be fiied and to have no alternative use. It does not allow for thinning. It assumes the forest to be initially uniformly regulated; i.e., the land is divided into an integer number of plots corresponding to the number of integer years in the optimal rotation, each plot being stocked with the same number of trees, the first being bare land or stocked with trees of age zero; the second, with trees of age one; the third, with trees of age two, etc., until plot T + 1, with trees of age T, the optimal rotation. Under the above assumptions both monopolists and competitors will adopt the same rotation which is the Faustmann periodic rotation. Moreover, the same timber price and the same quantity of timber will be supplied under the two market structures. The second proposition is a long-run one from both the static and dynamic points of view. In a static sense, the forest land area is assumed to be variable and subject to alternative uses. The non-fixity of the area does not per se preclude Faustmann rotations. In a dynamic sense, we assume that enough time has elapsed for a steady-state policy to set in, steady-state policy whose existence we assume as well. This steady-state policy will turn out to be the Faustmann periodic solution in the particular case where thirming is not allowed. This second proposition is independent of any initial age structure for the forest and allows the possibility of thinning. Under these assumptions, a monopolist will again adopt the same rotation as a competitor, but the timber price which will prevail and the quantity of timber supplied may vary from one market structure to the next. Finally, the third and last proposition is a corollary of the second one. Under the assumptions for the second proposition, except that land has no alternative use for the monopolist and is so plentiful as not to be able to generate a monopolistic rent, the monopolistic rotation would be longer than the competitive one.
2. SUFFICIENT CONDITIONS FOR ROTATIONS TO BE INVARIANT TO MARKET STRUCTURE WHEN FOREST LAND AREA IS FIXED, HAS NO ALTERNATIVE USE AND INITIAL FOREST IS UNIFORMLY REGULATED
A first result on rotation comparison between monopoly and competition can be obtained as a direct corollary of Theorems 3.1 and 3.2 due to Mitra and Wan ([17]). While they did not make any explicit monopoly-competition comparison, one of their results points in that direction. Thus, assuming that society somehow begins with a uniform distribution of trees with the oldest ones being at the Faustmann
56
CRABBk
AND
LONG
age, Theorem 3.1 states that the optimal policy for society is to maintain that uniform distribution (thus trees are harvested if and only if they reach the Faustmann age). In the Mitra-Wan discrete time model, the Faustmann (1849) age T* is the positive integer which maximizes
aT[f(T) - Cl/(1 - ST), where f(T) f(T) f(T) f(T) S is C >
is the growth curve for a representative tree such that
= 0 for 0 < T < b for some b 2 1 has unique maximum TM to the right of b and is continuous is increasing for b d T < TM and decreasing for T > TM the discrete-time discount factor 0 is the regeneration cost.
The Mitra-Wan theorem 3.1 holds for all concave utility functions U(q) with positive marginal utility, where q is the quantity of timber harvested. Their theorem 3.2 states that optimal stationary programs are invariant to changes in the utility (profit or benefit) function as long as it is concave and increasing and f( 7’) satisfies the above assumptions. In particular, this is true for
where P(q) is the demand curve-provided P’(q) < 0 and P(q) > 0 for all q. Clearly the competitive outcome coincides with the socially optimal outcome, since a linear function is concave and satisfies Mitra-Wan theorem 3.1 if it is increasing. Suppose that a socially managed forest, in its optimal stationary program, with a steady-state output q* (for all t), is handed over to a monopolist who wants to maximize the sum (or integral) of discounted profit, where profit is
WI) = p(dq. Then, as long as II(q) is concave and II’(q) > 0 for all q < q*, the monopolist will find it optimal to maintain the forest exactly as it is under the socially optimal program which is identical to the perfectly competitive one. In particular, the monopolistic price and output are the same as under competition. This is a direct corollary of Mitra and Wan’s Theorem 3.2. It is important to note that the Mitra-Wan result stated above is restricted to cases where the initial condition happens to be a steady-state optimal stationary forest. Moreover, Mitra and Wan assume that the surface area is fixed and that there is no thinning. * Planting and harvesting costs are zero or a positive constant [18]. This result is reminiscent of the well-known result in the economics of non-renewable resources, that the monopolistic price and output are the same as under competition if the resource stocks are the same in both cases, if extraction cost is zero, and if demand elasticity is constant [24]. *Mitra even with
and Wan conjecture that with a linear utility function [18].
forest
thinning
the Faustmann
solution
may
not
be optimal
ROTATION
AND
MARKET
57
STRUCTURE
Our first result can then be summarized as: PROPOSITION 1.
If
(a) the forest area is given but has no alternative uses, (b) the forest is uniformly regulated, (c) the cost of regeneration is a non-negative constant,
(d) there is no thinning, rotations under perfect competition and monopoly are identical to Faustmann rotations and efficient according to the criterion of intertemporal maximization of consumers plus producers’ surpluses; moreover, output and prices will be identical as well under the two market structures. 3. SUFFICIENT CONDITIONS FOR ROTATIONS TO BE INVARIANT TO MARKET STRUCTURE WHEN FOREST LAND AREA IS VARIABLE AND HAS ALTERNATIVE USES
In the present section, we wish to show that the monopolistic choice of rotation period is the same as the competitive choice, even when their outputs and prices are different. To allow for differences in output, we assume that forest land has alternative uses. The simplest way to model this is to assume that foresters can always rent land at an exogeneously determined and time-independent rental rate R (per acre per period). We introduce a non-negative cost of planting and harvesting. We also allow for thinning; in this respect our model is more general than Mitra and Wan [17, 181. The introduction of land rent (at an exogeneous rate) to the forestry problem may seem inconsistent with the Faustmann model in which land has no non-forestry use and its rent is endogeneous (see [22]). However, Comolli has shown that there is no inconsistency and has demonstrated that with exogeneous land rent the single rotation or forest mining formula coincides with the Faustmann formula, as long as the zero profit condition is taken into account and land is bare initially [4].3 This will serve as a reminder of the Faustmann formula and may be explained as follows: Given R, the exogeneous rental rate per unit per period, and C, the regeneration cost, each competitive firm does not need to solve the replanting problem; land is always available somewhere else, at the cost R. Each firm solves problem (1) for given P, the price of timber, C and R: PROBLEM (1).
Maxi+nize II = [Pf(T)
- C]exp(-rT)
= M$x[Pf(T) 3This result is quite similar to determined [22]. The forest mining Faustmann one and was mistakenly I. Fisher, Hecksher, Hotelling, Allen,
r91.
- C]exp(-rT)
- irRexp(-rt)
- frl
dt
- exp( - rT)]
that obtained by Samuelson, where land rent is endogeneously formula ignores the site value for reforestation contrarily to the adopted by such prominent economists as ThEnen, Jevons, Wicksell, and F. and V. Lutz until Gaffney proved that Faustmann was right
58
CRABBe
AND
LONG
when Max, II = 0. Therefore, Mra[Pf(T)
- C]exp(-rT)/[l
- exp(-rT)]
= 1,
(1)
where exp( .) denotes the exponential function and r > 0 is the rate of interest (here time is a continuous variable). In a competitive equilibrium, profit is zero, so that each firm is indifferent between solving (1) with rented land and buying land at the price R/r and solving the Faustmann problem (2): PROBLEM (2).
Maxiinize
[Pf(T)
= Mrm[Pf(T)
- C]exp(-rT)[l
+ exp( -rT)
- C]exp(-rT)/[l
+ exp( -r2T)
+ . .. ]
- exp(-rT)].
(2)
We will show that with a perfectly elastic supply of land, if thinning is not allowed, the Faustmann solution will obtain with any initial forest, not just with initial bare land. In this section, we allow the firm to choose different harvesting ages for trees planted at different times. Let y(t) denote the number of trees planted at time t or equivalently the number units of land on which trees are planted; each tree occupies one unit of land. A tree planted at time t and chopped down at time t + A(t) yields a flow of timber for thinning and eventually when being felled. The flow at time S, where t G s G t + A(t), is denoted by sb - t, 4th
#w)~
where s - t is the age of tree at time S, and P(t) denotes the thinning policy adopted by the firm for trees planted at time t. Without loss of generality, we assume that for given t, /3(t) is a real number to be chosen from the real interval [0, 11. Clearly, since A(t) is the harvesting age, it must be true that g(s - t, 44 g(s - t, 4th
P(t)) Nt))
= 0 a 0
ifs-t>A(t) if s - t < A(t).
(34 w
Thus, if the price of a unit of timber were unity, then the present value at time t of this flow would be
J+wT P(t)) =
r+A(f)g(s -
t, A(t), P(t))exp(
--T(s - t)) ds
(44
or
V(A(t),P(t)) = ~A”‘g(~yA(t), P(t))exp(-ra)da.
(4b)
To make our analysis comparable with models where thinning is not allowed, we take it that this is the limiting case where B = 0, and we assume that sji;o where f(a)
W(t),
B(t))
= f(dt))exp(
-4t)L
(5)
is the size of the tree at age a if no thinning takes place at earlier ages.
59
ROTATION AND MARKET STRUCTURE
The quantity of lumber harvested at time s is
Assume that our monopolist starts the optimization problem at time 0. He inherits ya(u) units of land on which y,(u) trees were planted at time u (u G 0) and are still surviving at time 0. The cost of land rent from time u to time 0 is sunk cost. He still has to decide when to chop down these old trees. If A(u) is the age at which these trees are to be chopped down, the present value of land rent incurred from time 0 to the cutting date is ‘(‘)+“Rexp(-rz)
dz = (R/r)[l
/0
- exp(-rA(u)
- ru)].
(7)
(For example, if u = - 10 and A(u) = 30, then the monopolist has to pay for 20 years of land rent, starting from time 0.) The total cost of maintaining all old trees (planted before time 0) until their cutting date is Co = (” y,(u)(R/r)[l
- exp(-rA(u)
- ru)] du.
(8)
Note that yO( u) is exogeneously given, and A(u) can be chosen at time 0. We now turn our attention to the new trees to be planted at t >, 0. Let y,(t) be the number of units of land on which new trees planted at time t. Let A(t) denote the cutting age. Then the rental cost of each block, discounted back to time I, is A(f)Rexp(-rz)
/0
dz = [l - exp(-rA(t))]R/r,
(9)
where z = s - t The present value of all the new trees is C, = imyn(t)[l
- exp(-rA(t))](R/r)exp(-rt)
dt.
00)
The quantity of timber harvested at time s, Q(S), given by (6), can be thought of as consisting of Q,-,(s) and Q,(S) (timber obtained from trees planted before and after time 0, respectively),
Q(s) = Q,(s) + Q,(s),
(11)
where Q,(s)
= (” y,(u)& J-00
Q,(s) = ~ht)ds
- 0, A(u),
s(u))
- t, A(t), p(t)) dt
du
02)
03)
60
CRABBI?
AND
The monopolist’s problem is to choose r,(t), Q,(s) so as to maximize II = /omexp(-r.s)P(Q(s))Q(s)
LONG
A(t),
A(u), /3(t), p(u), (Jo(s), and
(14)
ds - C, - C,,,
where Q(S) = QO(s) + Q,(s), Cc and C, are given by (8) and (10). In general it is difficult to obtain theorems that can be given simple economic interpretation. It is unlikely that A(t) is a constant for t sufficiently close to zero. Thus, if the monopolist starts with bare land (no old trees), he may find it advantageous to harvest the first generation of trees very early rather than to supply nothing for the first few periods. If we are willing to assume that in the long run both A(t) and P(t) converge to constant values, then we can say something meaningful about these constants. Thus we assume that the monopolist has been optimizing for a long time and that at the present moment (which we can denote as t = 0 without loss of generality) all new trees to be planted after 0 will have a constant cutting age A and a constant thinning policy B. Then C, can be simplified with the help of (13). With A(t) = A and /3(t) = j3, (13) reduces to
Q,(s) = (x&)g(s Multiply
- t, A, P) dt.
(15)
(15) by exp( - rs) and integrate:
From the theory of Laplace transforms, the right-hand side of (16) can be written as lmg(n,
A, P)exp( -ra)
d$?&)exp(-rt)
dt.
(17)
Now, recall that, from (3a), g(a, A, /?) = 0 if a > A. Hence A, p)exp(-ru)
img(u,
da = /04g(u, A, P)exp(-ru)
du = Y(A, a).
(18)
Therefore, using (16), (17), and (18), lmyn(t)exp(-rt)
0
dt = [l/V(A,
B)l~mQ.(~)exp(-~~)
U!S.
(19)
From (lo), with A(t) = A, C,, = (R/r)[l
- exp(-rA)]dmyn(t)exp(-rt)
dt.
(20)
From (19) and (20), c, = [l - exp(-~A)][R/rV(A,B)l~wQ.(s)exP(-rs)
~5.
(21)
ROTATION
AND
MARKET
61
STRUCTURE
Using (21) and (14), the monopolist’s profit is II = ~~exp(-rs)P(P(s))Q(s)
- [l - exp( -rA)]
d -
[R/rY(A,
CO
P)] imQn(s)exp(
-rs)
U!S.
(22)
Clearly, to maximize (22) for any given /3, A must minimize [l - exp( -rA)]
[R/rV(A,
P)].
(23)
Thus, we have obtained the following result: The long-run optimal rotation is independent of the function P(Q) and hence is the same as the rotation under competition or the maximization of the temporal sum of consumers’ surpluses provided land rent is exogeneous. Consider now the special case where thinning is not allowed (j3 = 0). In this case, recalling (5), we can write (23) as
@/r)[l - exp(-rA)lN(A)exp(-rA). Minimizing
(24) with respect to A yields
f’(A)/f(A) = r/b - ew(-rA)l
(25)
which is identical to the Faustmann formula, if the planting and harvesting cost, C, is zero. With positive planting and harvesting cost and with no thinning (j? = 0), we still obtain the same result. For, in this case, C, in (10) is modified as c, = dw~“(t)[(R/r)(l and the counterpart
+ Cexp( -rA(t))]exp(-rt)
d
(lo’)
of (24) is
[(R/r)(l Minimizing
- exp( --d(t))
- exp(-rA))
+ Cexp(-rA)]/f(A)exp(-rA).
(24’)
(24’) with respect to A yields f’(A)/f(A)
= rexp(rA)/[exp(rA)
- 1 + Cr/R]
(25’)
which is identical to the Faustmann formula obtained under competition. Our second and main result can then be summarized as: PROPOSITION
2.
If
(a) forest land is available in unlimited quantities but has alternative (perfectly elastic supply), (b) the forest is regulated or not, (c) the regeneration cost of a tree is a non-negative constant,
uses
rotations under perfect competition and monopoly are in the long run identical and efficient according to the criterion of intertemporal maximization of consumers plw
62
CRABBlk
AND
LONG
producers ’ surpluses; however, output and prices will not generally be the same under monopoly and competition. The optimal rotation wiN be the Faustmann one if thinning is not allowed.
4. ROTATIONS UNDER MONOPOLY AND COMPETITION WHEN FOREST LAND IS VARIABLE BUT HAS NO ALTERNATtVE USES FOR THE MONOPOLIST
In the previous section we assume that the rental rate for land, R, is exogeneous. If it were endogeneous, for example, if R is the marginal product of agricultural land which depends on the quantity of land used in agriculture (which diminishes as the quantity of land used in forestry increases), then it would no longer be true that the monopolist would choose the same rotation period as the competitive one. Alternatively, one can endogenize R by assuming that forest land has no alternative use. Suppose that the monopolist’s marginal revenue, P’(Q)Q + P(Q), equals zero at some Q = Q. If the forestry land area is large enough, then the implicit land rental to the monopolist is zero (some land being left vacant).4 In this case the monopolist would be happy to maintain a forest (on a fraction of the total land area) with trees from age 0 to age TM, where TM minimizes C/rf(T) (see (24’), with R = 0). But under competition, with zero profit condition and with a larger output, all land would be used and the equilibrium land rental would be positive. The competitive rotation period would then be the Faustmann one, i.e., the one satisfying
f’(T!l[l(T)
or, substituting
= r/[l
- exp(-rT)]
for P from (l), i.e.,
P = [(R/r)(l
f’(T) fo
- z]
- exp(-rT))
= rexp(rT)/[exp(rT)
+ Cexp(-rT)]/f(T)exp(-rT) - 1 + Cr/RI.
Thus the monopolistic rotation would be longer than the Faustmann perk15 Our third and last result can then be summarized as:
rotation
PROPOSITION 3. Under monopoly and under the same assumptions as Proposition 2, except for thinning which is not allowed and for rent which is assumed to be endogeneously determined and nil, the rotation would be longer than the perfectb competitive one. 4This case is ruled out by Mitra and Wan because they assume that marginal revenue is always positive [17]. ‘To see this, note that both the monopolist and the competitive firm minim& (24’), but by assumption the monopolist’s R is zero, while the competitive firm faces a positive R. Applying the implicit function theorem to the first-order condition of the minimization problem (241, one sees that &/dR < 0.
ROTATION
AND MARKET
STRUCTURE
5. THE NAUTIYAL - FOWLER
63
PROPOSITION
The only article we were able to find in the forestry economics literature on the monopoly-competition comparison of rotations is by Nautiyal and Fowler [21]. This article generated quite a few comments [2, 19, 5, 15, 201. Nautiyal and Fowler attempted to prove that a monopolistic rotation (denoted here by 7”) is shorter than the one adopted if the monopolist were forced to behave competitively (7”). They also claimed that the former rotation is shorter than the one which the monopolist would adopt if he were forced to maximize the temporal sum of consumers’ surpluses (T,,) which, in turn, is shorter than the latter, i.e., TM G T,, G T,. The results allegedly hold for the case where time is continuous, the regeneration cost is nil, thinning is not allowed, the forest land area is given, and two alternative initial conditions hold. The first alternative is an unregulated (age-homogeneous) forest initially planted on bare land (the Faustmann model without cost but with an inverse demand function). The second alternative is a uniformly regulated forest obtained progressively by starting from bare land and planting each “year” (time is assumed to be continuous though!) a fraction of land equal to one over the best rotation corresponding to a given market structure. The results are short-run (fixed land area) in a static sense. However, they are not comparable to our first proposition since our initial condition calls for an instantaneous uniformly regulated forest at the initial time. Let us point out immediately that the assumption of progressively obtained uniformly regulated forest made by Nautiyal and Fowler is artificial and must necessarily lead to second best types of results.6 Indeed, leaving a portion of fixed land vacant during the first rotation is never optimal when regeneration is costless. This assumption alone may be responsible for the discrepancy between the rotation for the three market structures. If we come back now to the first initial condition, i.e. unregulated forest starting with bare land, this model, like the Faustmann one, is a pure point input-point output model. The “feast and famine” nature of the model is certainly not compatible with perfect competition and a stationary price. Indeed, with a stationary demand schedule, spot markets would clear at rotation dates only and a complete set of future markets could not be established except for rotation dates. The only way to remedy physical illiquidity of an asset through market liquidity is to add into the model money and an instrument which converts a title to a tree at date T into money [12]. One could object to this general equilibrium line of argument that Nautiyal and Fowler’s model is a partial equilibrium model: a monopolist is forced to adopt a competitive behavior or to maximize the temporal sum of consumers’ surpluses. However, the consumer surplus will not be defined except at rotation date in a pure point input-point output model. The only meaningful rotation comparison remaining in a partial equilibrium point input-point output model is between monopolistic behavior and price parametric behavior obtained by letting the elasticity of demand grow without bound. Thus, we agree with Nautiyal and Fowler that the rotation adopted costlessly by a monopolist planting a forest on a fixed and bare land area (Faustmann model) lengthens when one lets the price elasticity of stationary demand curve go to infinity. This result does not conflict with ours since it is based on different assumptions. However, our results demonstrate that the
6Tbis was already noticed by Femow [8].
64
CRABBe
AND LONG
Nautiyal-Fowler conclusion that monopolistic rotations are shorter than competitive ones is a specialized one and cannot be generalized. It does not hold either when forest land is limited and when the forest is uniformly regulated initially or, in the long run, when the supply of land is perfectly elastic. 6. CONCLUDING
REMARKS
Intuition suggests that the monopolist will tend to adopt rotations shorter than the perfectly competitive ones to keep volumes lower and prices higher when faced with a downward-sloping demand for timber. As in the area of non-renewable resources, intuition about monopoly-competition output paths comparison may be misleading. One reason for the relative neglect of this problem in the forestry literature is that the optimization problem facing foresters is a lot more complicated than the one faced by miners. This is especially true when the benefit (or profit) function is not linear in the quantity of timber harvested as is the case under monopoly. The major stumbling block is that the initial conditions, i.e., the initial age distribution of the trees, may prevent the optimal rotation from converging to a stationary solution such as the Faustmann periodic rotation even if the latter exists and is unique [17, 181. We derived two sets of sufficient conditions which lead to identical and efficient rotations under the two market structures. The first set applies to uniformly regulated forests whose area is fixed and has no alternative use. It follows directly as a corollary to two theorems proven by Mitra and Wan [17]. The second applies in the long run to any forestry land whose area is in perfectly elastic supply at a stationary opportunity cost. Our conclusion, that in the long run, the monopolistic rotation period is the same as the competitive one, or the one corresponding to the maximization of the temporal sum of consumers’ plus producers’ surpluses if rent is exogeneous, has its parallel in the theory of product durability (see, for example, [l or 141). In both cases, market structure is irrelevant (in the long run) to the choice of the optimal economic life of the asset: this choice is essentially a problem of cost minimization, provided that production can be independently adjusted. The proof of Proposition 2 relies on a separability property of Laplace transforms. Global asymptotic stability which is required for Proposition 2 has been conjectured by Kemp and Moore who have shown it to hold in several examples [13]. Mitra and Wan are rather skeptical about the conjecture holding for discrete time forestry models with positively discounted concave utilities [18]. For optimal programs with discounted utilities, they point out that the conjecture (the “turnpike property”) has been proven to be true if the discount rate is not too large [16, Theorems 8.3 and 8.41. However, they emphasize that these growth models have less structure than the forestry ones. An extension of our results to the case where the forest yields nontimber benefits could be envisaged [ll]. However, this extension may not be trivial for Proposition 1, since utility becomes a function of two arguments instead of one. REFERENCES 1. A. Abel, Market structure and the durability of goods, Rev. Econom. Stud. SO, 625-637 2. F. J. Anderson, Optimal rotation: Comment, Lund Econom. 57, 293-294 (1981).
(1983).
ROTATION
AND MARKET
STRUCTURE
65
3. P. Berck, The economics of timber: A renewable resource in the long run, Bell J. Econom. 10, 441-462 4.
5. 6. 7.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
(1979).
P. M. Comolli, Principles and policy in forestry economics, Bell J. Econom. 12, 300-305 (1981). P. M. Comolli, A proposition on imperfect-stumpage markets, Land Econom. 60, 297-300 (1984). R. T. Ely, “An Introduction to Political Economy,” 1st ed., Eastons & Mains, New York (1889). M. Faustmann, Berechmmg des Werthes, welchen Walboden sowie noch nicht haubare Holzbestande ftir die Waldwirtschaft besitzen, Allgemeine Forst Jagd Zeitung 15,441-455 (1849); Engl. Tmsl. in “Martin Faustmann and the Evolution of Discounted Cash Flow” (Gane, Ed.), Oxford Commonwealth Forestry Institute Paper 42 (1968). B. E. Femow, “Economics of Forestry,” T. .I. Crowell, New York (1902). M. M. Gaffney, “Concepts of Financial Maturity of Timber and Other Assets,” A.E. Information Series No. 62, Department of Agricultural Economics, North Carolina State College, Raleigh, NC (1957). Government of Canada, “Selected Forestry Statistics,” Canadian Forestry Service (1986). Hartman, The harvesting decision when a standing forest has value, Econ. Inquiry 14, 52-58 (1976). J. Hirshleifer, Liquidity, uncertainty, and the accumulation of information, in “Uncertainty and Expectations in Economics” (Carter and Ford, Eds.), Blackwell, Oxford (1972). M. C. Kemp and E. J. Moore, Biological capital theory: A question and a conjecture, Econom. Letr. 4, 141-144 (1979). J. M. Malcomson, Replacement and the rental value of capital equipment subject to obsolescence, J. Econom. Theory 10, 24-41 (1975). K. McConnell and T. Phipps, Optimum forest rotation in an imperfect stumpage market: Comment, Land Econom. 60, 215-217 (1984). L. W. McKenzie, Optimal economic growth, turnpike theorems and comparative dynamics, in “Handbook of Mathematical Economics” (Arrow and Intriligator, Eds.), North-Holland, New York (1986). T. Mitra and H. Y. Wan, Jr., Some theoretical results on the economics of forestry, Rev. Econom. Stud. 52,263-282(1985). T. Mitra and H. Y. Wan, Jr., On the Faustmann solution to the forest management problem, J. Econom. Theory 40, 229-249 (1986). J. C. Nautiyal, Optimum rotation: Reply, Land Econom. 57, 295 (1981). J. C. Nautiyal, Optimum forest rotation in an imperfect stumpage market: Reply, Land Econom. 60, 218 (1984).
21. J. C. Nautiyal and K. S. Fowler, Optimum forest rotation in an imperfect stumpage market, Land Econom. 56,213-226 (1980). 22. P. A. Samuelson, Economics of forestry in an evolving society, Econom. Inquiry 14,466-492 (1976). 23. R. A. Sedjo and M. Clawson, Global forests, in “The Resourceful Earth” (Simon and Kahn, Eds.), Blackwell, New York (1984). 24. J. Stiglitz, Monopoly and the rate of extraction of exhaustible resources, Amer. Econom. Rev. 66, 655-661 (1976).