Efficient multiple-use forestry may require land-use specialization: comment.
Their graphical analysis shows a classical production possibilities frontier (PPF) which is skewed when shifted either inward or outward, due to the differences in responsiveness to management effort. Because the PPFs are skewed when shifted, constant output prices imply different combinations of timber and nontimber production on two sites with different levels of management effort. VB then demonstrate that giving the sites unequal levels of effort increases total income from both sites over allocating the same level of effort to each site.
This result is very surprising for neoclassical production functions. As a general rule, two identical sites facing the same input and output prices should behave identically. For instance, let [T.sub.i] be the timber produced on site i, and [N.sub.i] be the nontimber resource produced on site i, i = 1, 2. If [E.sub.i] is the management effort used on site i, the PPF can be represented by the function [E.sub.i] = [f.sub.i] ([T.sub.i], [N.sub.i]), where [Delta][f.sub.i]/[Delta][T.sub.i], [Delta][f.sub.i]/[Delta][N.sub.i] [is greater than] 0 (increasing production of either good involves more management input); [Mathematical Expression Omitted], [Mathematical Expression Omitted] (increasing either output requires constant or increasing additional inputs); and [E.sub.1] + [E.sub.2] = E* (a total level of E* is to be allocated between the two sites). With fixed output prices [P.sub.T] and [P.sub.N], as in VB, the profit-maximization problem can be written as:
[Mathematical Expression Omitted]
which yields first-order conditions:
[P.sub.T] - [Lambda][Delta][f.sub.i]/[Delta][T.sub.i] [is less than or equal to] 0
and [T.sub.i][[P.sub.T] - [Lambda][Delta][f.sub.i]/[Delta][T.sub.i]] = 0, i =1,2;
[P.sub.N] - [Lambda][Delta][f.sub.i]/[Delta][N.sub.i] [is less than or equal to] 0
and [N.sub.i] [[P.sub.N] - [Lambda][Delta][f.sub.i]/[Delta][N.sub.i]] = 0, i = 1,2;
E* - [f.sub.1]([T.sub.1], [N.sub.1]) - [f.sub.2]([T.sub.2], [N.sub.2]) [is greater than or equal to] 0
and [Lambda][E* - [f.sub.1]([T.sub.1], [N.sub.1]) - [f.sub.2]([T.sub.2], [N.sub.2])] = 0. 
Analysis of the Hessian shows that the second-order conditions for a maximum are satisfied.
If these functions are as defined above, if an internal solution exists with positive quantities of [T.sub.i] and [N.sub.i] on both sites, and if [f.sub.1] = [f.sub.2] as postulated by VB, then the Implicit Function Theorem implies that [T.sub.1] = [T.sub.2] and [N.sub.1] = [N.sub.2], which implies that [E.sub.1] = [E.sub.2], and the two sites should be treated identically. If VB's analysis is to hold, then, at least one of the above assumptions must be violated. Alternatively, these assumptions do not underlie their graphical analysis.
This comment will clarify VB's results through several steps. First, it will show that VB's framework cannot rely on global constant returns to management effort, the assumption that allows them to compare profits between their scenarios. Next, it develops an alternative framework that replicates their diagram and results, but with somewhat different assumptions. Finally, it shows that VB's result is correct under these alternative assumptions, which are quite plausible in the real world.
I. CONSTANT RETURNS TO MANAGEMENT EFFORT AND VB's ANALYSIS
In order to compare profits under their alternative scenarios (either managing both sites identically, or managing them for dominant use), VB assume constant returns to management effort (CRME)--that is, that adding one unit of management effort to one site shifts the PPF outward to the same extent that it shifts inward when a unit of management effort is taken away. However, an assumption of CRME has properties unrecognized in their analysis. A CRME function is homogeneous of degree 1: that is, increasing both timber and nontimber by a proportion t requires that effort must also increase by proportion t. A function which is homogeneous has the property that "the slopes of the level surfaces . . . are constant along rays through the origin" (Varian 1984, 330). This property cannot be reconciled with VB's graphical argument that the slopes of the PPFs change along a ray due to the differences in the effectiveness of management effort. Indeed, the assumption of differing responses to management effort by the two outputs, depending on precisely how it is defined, may violate homogeneity properties.
While VB are correct that the skewed PPFs they draw (labeled PPF+ and PPF-) produce different optima than the original PPF ([PPF.sup.0]), their comparison of profit levels depends on the assumption of CRME. As discussed above, they cannot assume both global CRME and unequal slopes of the PPF along rays through the origin. Thus, their original analysis cannot hold. The question remains under what circumstances their analysis can, with modified assumptions, be replicated.
II. A REINTERPRETATION OF VB's ANALYSIS
One way to reproduce their results is to redraw their diagram with linear PPFs, a polar case which assumes that both timber and nontimber have constant marginal costs of production. Assume that each additional unit of management effort can produce either 2 units of T, 1 unit of N, or a linear combination of those outputs. (Note that these figures reflect differing responsiveness to management effort.) Figure 1 presents one version of this diagram. Here, the first unit of management effort produces a PPF connecting the points (2T, 0N), and (0T, 1N). The PPFs are parallel, and the functions exhibit CRME. In contrast, Figure 2 assumes that no effort is required to produce 3N and 0T; the first unit of management effort can produce either 4N and no T, or no N and 2T; subsequent units produce an additional 2T, 1N, or a linear combination of those, as before. Now the PPFs are no longer parallel, and they replicate the skewed shape in VB's figure.
This polar case, which can produce VB's results,(1) incorporates constant returns to additional effort for each output separately after the first unit. The constant returns are not global, however, due to the initial "free" units. In our Figure 2, while T does exhibit constant returns to effort, N exhibits decreasing returns to effort: because of the free first 3 units of N, doubling effort from 1 unit to 2 units does not double output. If 2[P.sub.N] [is greater than] [P.sub.T] [is greater than] [P.sub.N] (that is, the price vector falls between the slope of the lower PPF (labeled PPF-) and the higher PPF (labeled PPF+)), then higher profits can indeed be obtained by having one area produce only N and one area produce only T, rather than both areas producing some of both N and T.
The key effect in this redrawn diagram is the fate of those initial low-cost units of nontimber when timber production starts on that site. In Figure 2 of this paper, with no negative interaction between production of timber and nontimber, one unit of management effort could produce 4 units of N and no T, or 3 units of N and 2T: production of T would not disallow the possibility of producing the original 3 costless units of N. In this case, the PPFs would appear as the dotted lines in Figure 2, whose parallel slopes reflect constant returns to management effort when the "free" first 3 units of N are not lost through T production. Instead, as the solid lines show, less of the nontimber resource is produced for any specified level of timber. Timber production thus has a negative effect on the nontimber resource, separate from the effect of substituting effort from one output to the other.(2)
Figure 2 in VB appears to show the same initial low-effort production of N with little or no production of T, and negative interaction between T and N. If their PPF- is shifted inward by the same amount as the difference between it and [PPF.sup.0], (approximately) no timber and some nontimber would be produced. Thus, at very low levels of effort, their PPF is almost vertical. Additionally, as in our linear diagram, constant returns to additional management effort would produce parallel PPFs (with a vertical portion) unless the production of timber and nontimber interact counterproductively. If, therefore, their diagram is intended to incorporate local CRME for individual outputs, then their diagram suggests negative effects on nontimber from production of timber.(3)
III. IMPLICATIONS OF THIS REINTERPRETATION
Specialization of output generally arises from nonconvexity in the social production set (Baumol, Panzar, and Willig 1982, chap. 4; Baumol 1964). The nonconvexity can arise in the production function of an individual firm (e.g., a natural monopoly), or in joint production of two goods, if joint production leads to less output than producing the goods separately (diseconomies of scope). Baumol and Oates (1988, chap. 8) argue that a negative externality imposed by one firm on another, if strong enough, will inevitably lead to a nonconvexity in the social production set.
Our reinterpretation of VB's diagram exhibits clear diseconomies of scope. In this case, once an additional unit of effort is put into a site, it is more profitable to put that effort toward timber as long as PT [is greater than] 1/2[P.sub.N]. However, because no or little effort is needed for one site to produce some nontimber resources, it is likely to be worth-while to devote one site to that low-cost initial production of the nontimber resource.
The diseconomy of scope identified here is scale-dependent: in particular, it diminishes as the PPFs shift out with more effort. As noted in footnote 2, the equation for the PPFs presented here is:
N = 3 + E* - [3 + E*/2E*]T.
As E* becomes large, [MRT.sub.T,N] approaches 1/2. In this version, the production function for nontimber is analogous to that for a natural monopoly with high fixed costs and constant marginal costs, except that in this case the "fixed costs" are benefits. Just as average fixed costs shrink as output increases, the effect of this "fixed benefit" on the returns to scale (and diseconomies of scope) is reduced. In the limit, the constant returns to additional management effort dominate, and the PPFs are essentially parallel. Clearly, though, at low levels of management effort, this effect cannot be ignored.
VB's policy conclusion, then, is correct, but under slightly different circumstances than they originally describe. Some specialization might be more profitable than identical behavior on two identical areas, not if one output responds more vigorously to management effort than does the other, but instead, if joint production is less efficient than separate production.
This assumption matches reality to some degree in the case of timber and nontimber outputs. Many nontimber outputs, such as some wildlife habitat protection and wilderness recreation, can be obtained with little or no management inputs, and can be lost with even small degrees of timber production; in contrast, private prices (not including social values) usually favor timber. In this case, giving little or no management effort to some areas and concentrating effort on timber in other areas may provide higher social value than having multiple use on all acres.
1 In this linear case, the spacing between the PPFs is not constant along rays from the origin, even though the spacing is constant along the axes. In this way, our case differs from that of VB. Perhaps curving the PPFs makes equal spacing possible.
2 The equations that produce these results are: N = 3 + [E.sub.N]; T = 2[E.sub.T]; and [E.sub.N] + [E.sub.T] = E*. With no negative interaction between the inputs, the formula for the PPF is N = 3 + E* - T/2; the marginal rate of transformation of timber for nontimber is constant at [MRT.sub.T,N] = 1/2. With negative interaction, though, the expression for the PPF becomes N = 3 + E* -[(3 + E*)/(2E*)]T.
3 Alternatively, their PPFs with skewed slopes can be reproduced by assuming different returns to scale in production of each output: for instance, assuming [Mathematical Expression Omitted] (decreasing returns to scale), and T = [E.sub.T] (constant returns to scale) produces PPFs very similar to VB's diagram. In this case, though, identical firms/sites should act identically to maximize joint profits.
Baumol, William J. 1964. "External economies and second order optimality conditions." American Economic Review 54 (4):358-72.
Baumol, William J., and Wallace E. Oates. 1988. The Theory of Environmental Policy, 2d ed. New York: Cambridge University Press.
Baumol, William J., John C. Panzar, and Robert D. Willig. 1982. Contestable Markets and the Theory of Industry Structure. New York: Harcourt Brace Jovanovich, Inc.
Varian, Hal. 1984. Microeconomic Analysis, 2d ed. New York: W. W. Norton and Co.
Vincent, Jeffrey R., and Clark S. Binkley. 1993. "Efficient Multiple-Use Forestry May Require Land-Use Specialization." Land Economics 69 (Nov.):370-76.
Gloria E. Helfand and Marilyn D. Whitney are assistant professors, Department of Agricultural Economics, University of California, Davis.
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|Title Annotation:||Comment and Reply; response to Jeffrey R. Vincent and Clark S. Binkley, Land Economics, vol. 69, p. 370, November 1993|
|Author:||Helfand, Gloria E.; Whitney, Marilyn D.|
|Date:||Aug 1, 1994|
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