Question

Define and explain Faustmann forestry management is about rent maximization for land

Answer 1

Faustmann’s Formulas for Forests

The canonical, Faustmannian forest is revisited to sharpen understanding of forests as forms of capital. Investment and two r-percent rules are discussed and re-interpreted. A forest’s two natural resources, the stand and the land, act in conjunction as a composite asset. Non-marketed or intangible capital is also absorbed into the single composite. If capital is comprehensively defined, there is no independent role for the concept of an internal rate of return. A forest provides real options in optimal and sub-optimal rotation patterns. Old growth has superficial similarities to a exhaustible resource, but the forest still consists of two resources that, in conjunction, behave comparably to a plantation forest.

. A remarkable analysis was provided in 1849 by Martin Faustmann. The contribution of forestry to economics, and vice versa, is a theme in Gane’s (1968) examination of papers by Faustmann (1968) and Faustmann’s contemporary, E.F. von Gehren (1968).

One of the formulas applies to both optimal and non-optimal forest management and affords a simple comparison of the two. It is possible to consider the implications of an insightful forester’s recognizing that maximum present value should be pursued rather than a sub-optimal rotation. The discussion stresses that a forest is a single asset made up of two natural resources, namely (1) the growing trees or stand and (2) the land, which comprises land area, soil, climate and general capacity to grow a stand. Both resources contribute in concert, and not individually, to the realization of value. They remain a composite even in the special situation of old growth. Faustmann’s two formulas, which are the foundation of the present paper, are well known. Re-interpretations of them are presented as two theorems about capital value. A number of further results that add to the re-interpretations are presented as corollaries to these theorems.

Faustmann (1968) only briefly mentions the choice of rotation age. His main preoccupation is with a forest for which, in modern terminology, a particular resourceallocation mechanism is in place (Dasgupta and Mäler 2000). In this case the mechanism predetermines a rotation age. That age is not necessarily optimal. Let it be represented by a > 0, the net revenues of the harvest by R(_a) and the planting cost by c. A forest may also provide flows of thinnings, amenity values and costs, ?i (t), i = 1, ..., n, at any age t ? (0, a) (e.g. Faustmann 1968, Hartman 1976, Strang 1983, Pearce 1994). Let the set of indices of internalized flows, identified as amenities or simply flows, be denoted by I ? {1, 2, ..., n}. A flow of cost, for maintenance or administration, is a negative value. Only the algebraic sum of the internalized flows is used below. Therefore, let ? (t) = S i?I ?i (t). 1 Also let U t 0 ? (s) e?rsds = A(t) denote the present value of internalized flows up to age t ? a. Let F (a) be defined to be the sum of revenues from harvesting at age a and the cumulated, internalized values of the flows:

F (a) = R (a) + e^raA (a).

Most writers, including Faustmann, have concerned themselves with determining the value of the land. If the rotation age is predetermined to be a, the present value of bare forest land is

L(a) = ?c + F (a) e^ra ? c e^ra + F (a) e^2ra ? c e^2ra + ...

= ?c + F (^a) e^ra + 1 era % ?c + F (a) era ? c era + F (a) e2ra ? c e2ra + ...&

= ?c + F (a) er^a + 1 e^raL(a).

The following were derived by Faustmann (1968: 30): L(a) = F (a) ? cera era ? 1 ; (1) L(a) = ?c + F (a) ? c era ? 1 . (2) According to equations (1) and (2) the value of the forest is discounted into the land value, L(a), no matter what the value of a. If L(a) < 0, or equivalently if F (a) ? cera < 0, for every a, the forest is not planted. In an interesting problem there is an a0, depicted in Fig. 1, such that F (a0)?cera0 = 0, and a>a0 for a forest that is planted. For the rotation age a0 the value of the land, L(a0), is zero. Equation (2) can be rearranged to reveal analytic features of the problem beyond solving for the value of the land. For any fixed rotation age a>a₀, L(a) + c = F (a) ? c e^ra ? 1 and

F (a) + L(a) = F (a) ? c + F (a) ? c e^ra ? 1 = e^raF (a) ? c e^ra ? 1 = e^ra (L(a) + c).

(3) During a rotation, the value of the land is not recovered. At the end of the rotation, the land is returned intact, along with the harvested timber. Formula (3) expresses the following r-percent growth rule.

Theorem 1 Faustmann’s General Formula. Let the resource-allocation mechanism set a predetermined, repeated rotation age that is not necessarily optimal. The sum of the value of bare land and the planting cost grows at the rate of interest to the sum of the values of (a) the harvest, (b) the cumulated, internalized amenities and costs and (c) the bare land. Formula (3) is considered general because it applies to both an optimally and a non-optimally exploited forest. Although it is the formula explicitly examined by Faustmann (1968), it has not been stressed in the economics of forestry. But it does help in the interpretation of the economic properties of forest assets. An implication of formula (3) is that investment in the forest includes the value of the contribution of the bare land, in kind, in addition to the planting cost. Even though it has no alternative use, bare land has an opportunity cost: there is a choice of the planting time. The bare land is an asset committed at age t = 0 to the enterprise over the rotation period. Corollary 2 Investment. The investment in a forest is the sum of the planting cost and the value of the land. To stress the comprehensiveness of forest capital, let the total value invested be written ? (a) = c + L(a). According to formula (3), the total investment ? (a) grows at rate r to the total value obtained at the end of the rotation (including the cumulated value of flows of amenities and costs): ? (a) era = [R (a) + A (a) e^ra] + L(a) = F (a) + L(a). The r-percent growth rule (3) is not explicit in Faustmann’s analysis or in other analyses such as Samuelson’s (1976) or Hartman’s (1976). Given the chosen rotation period the total investment earns a rate of return of r.

Corollary 3 Internal rate of return. (i) The internal rate of return on the total value invested in a forest, including the value of the land, is equal to the rate of interest. (ii) A finding that the internal rate of return is greater than the rate of interest is indication that forest capital has not been defined comprehensively. Conventionally, the internal rate is given by the solution, ?, to the equation c (1 + ?) a = F (a). The internal rate is used as a criterion of investment by not considering the forest land to be invested, as if it is not scarce (has no value). Once investments and yields are comprehensively defined to include the land, the internal rate of return is redundant: total investment ? (a) grows at the external (market) rate r to F (a) + L(a). Faustmann and Samuelson define the (full) land rental to be rL(a). For any rental charge no greater than this value the rotation a can be supported, with a non-negative profit accruing to the forester. Even if there is a rental contract for the use of the land, the value c+L(a) is invested through the forester’s choice and is paid back with interest. Computation of the full rental, rL(a), presupposes computation of L(a). The essence of the analysis is the role of the land and stand as capital stocks, not how the forest is financed. Fig. 1 depicts the value of a forest with chosen rotation (resource-allocation mechanism) a1. Faustmann (1968) insists that the land value remains L(a) at all ages. The land would have value L(a) at age t ? (0, a) if it were bare. But it is not bare. During a rotation the land is not available for sale independently. Faustmann’s contemporary, Gehren (1968: 24), comes closer to realizing the point: “At the time of planting, the acre becomes forest.” On the other hand, Gehren wishes to assign a portion of the total value to the land. The interplay of land and stand is fundamental to formula (3). Once invested, the land and stand form a composite, a forest. The value of the land can be realized only by cutting the stand. The value of the stand can be realized only by restoring the bare land.

Corollary 4 A forest with standing trees is a single asset that comprises two natural resources, the stand and the land. The change in perspective on forest capital is definitional, not tautological: there is a wider, comprehensive definition of capital. The land is transformed to a forest by planting. The land and the planting cost remain invested until the harvest age a. At age a the forest is transformed to the harvest with value R (a) and bare land with value L(a). Amenities accrue at rate ? (t). The cumulated value of these flows is a part of what is returned, in addition to the value of the harvest. The flows are also attributable to the forest and not to either natural resource alone.

The optimal rotation age aˆ, which is assumed to be unique, maximizes the net present value (discounted cash flow) of the forest: L(ˆa) = max_a F (a) ? ce^ra e^ra ? 1 > 0. (4)

Setting L (ˆa)=0 yields that F (ˆa) = F (ˆa) ? c eraˆ ? 1 reraˆ ; (5)

F (ˆa) = r (F (ˆa) + L(ˆa)). (6)

Sometimes equation (6) is considered to be Faustmann’s formula. Faustmann (1968) makes allusions to the possibility of doing better than following the resourceallocation mechanism that is in place. Although it has been emphasized in the economics of forestry, Faustmann does not solve for the optimal rotation. Another way to express equation (6) is to

let V (a) = F (a)+L(a). Since L (ˆa)=0, V (ˆa) = F (ˆa) + L (ˆa) = F (ˆa) = r (F (ˆa) + L(ˆa)) = rV (ˆa). (7)

Optimal management consists of maximizing the value of the forest, which is equivalent to maximizing the value of the forest land (finding aˆ to maximize L(a)) at the beginning of a rotation. If a different harvest age, a˜ say, is chosen, value is lost. Equations (1), (2) and (3) hold for any value of a, not just for aˆ (not just for an optimum). That is to say, if the resource-allocation mechanism in place prescribes that a = ˜a 9= ˆa (such as Gehren’s eighty years), then the investment at age 0, ? (˜a), grows at rate r to L(˜a) + F (˜a) (total value, given the choice of the rotation period) at age a˜. The value (and the market price) of bare forest land in the context of the resource-allocation mechanism is L(˜a). Suppose that a >˜ aˆ and that the resource-allocation mechanism, followed by all foresters in the industry, is to harvest at age a˜. In this case, Faustmann’s admonishment to Gehren, that the forest should be valued at its discounted remaining value from a˜, is not correct. Faustmann is right if a <˜ aˆ. The resource-allocation mechanism can be interpreted as being one that seems to most foresters as being incontrovertible. The ability to recognize an option that others do not recognize is an asset. (Think of the legendary figures of securities lore.) If that ability is scarce it commands a positive rent. If it is not scarce, as is implicitly assumed in the analysis of optimal management, its rent is zero. The option is to choose the most advantageous age to harvest from among a set of alternatives, and the decision may be to stop immediately rather than to continue to follow a convention that does not maximize value. In particular, the optimal policy is to wait until t = ˆa if t < aˆ or to cut immediately if t > aˆ (subject to Strang’s qualification). If t > aˆ, the option value if one forester among many recognizes the option for a potential increase in value is

? (t, aˆ)=[L(ˆa) + F (t)] ? [L(˜a) + F (˜a)] e?r(˜a?t) > 0.

CONCLUSION

Although Faustmann’s analysis is well over a century and a half old, it is deep enough to enable a re-interpretation of the exploitation of a forest in terms of modern economic concepts. For either optimal or non-optimal management of a stylized forest, there are two natural resources, stand and land. The combination of those resources is a composite asset that returns the going rate of interest over the chosen exploitation period. Other assets may also be involved; the paper has considered a scarce, intangible, non-marketed asset, management or entrepreneurship. Since each asset’s internal rate of return is exactly the interest rate, there is, in the forest at least, no need for this concept when all assets are enumerated comprehensively. Forestry is at its base a repeated point-input, point-output problem. The fundamental decision is the timing of the harvest and the implications of timing for capital value. In an optimal or non-optimal program, there is a real option throughout the rotation period. Even though the analysis is under certainty, there is a choice to wait or to strike. In particular, there may be an option to cut early and to obtain the sum of the realizable values of the two assets. The option value applies to the forest and is not attributable to any one of the resources that make up the forest. It provides the incentive to the decision maker to make an optimal intertemporal decision.