**Introduction**

This appendix examines the analytical foundations of methods for estimating the net accumulation of timber. It begins by reviewing estimation methods for nonrenewable resources. It then reviews the standard "net depletion" method for timber resources, which generalizes methods for nonrenewable resources without taking into account the time lags that occur in the real world between timber harvests. This omission tends to cause the net depletion method to overstate both the decrease in capitalized forest value that occurs when mature forests are logged and the increase that occurs as immature forests regenerate. The appendix concludes by presenting alternative methods, based on the familiar Faustmann model of optimal forest management, that avoid these biases.

In certain respects, the analysis in this appendix parallels one by Newson and Gie (1996). We push the analysis further, however, and draw a closer comparison to estimation methods like the net price method and El Serafy’s method.

**Net accumulation of a nonrenewable resource**

The asset value of a natural resource equals the discounted sum of the net returns (resource rents) it generates over time. Net accumulation refers to the change in asset value from one period to the next. It could well be positive, especially in the case of rapidly growing renewable resources. Positive net accumulation can occur even in the case of nonrenewable resources, if for example prices of the extracted resource rise significantly and generate holding gains.

For a nonrenewable resource, asset value at time *t* is therefore

V(t) = Σ {(1+i)^{t-s}[pq(s) - C(q(s))]},(1)

where the sum is evaluated over the interval *s = t, …, T*. *i* is the discrete discount rate, *p* is the price of one unit of the extracted resource (assumed to be constant over time), *q(s)* is the quantity extracted in period *s*, *C(q(s))* is the total extraction cost, and *T* is the terminal time (the period when the resource is exhausted). We can also write (1) as

V(t) = pq(t) - C(q(t)) + V(t+1)/(1+i).(2)

We define net accumulation as

D(t) = V(t+1) - V(t).(3)

Substituting (2) for *V _{t}* in this expression, we obtain

D(t) = iV(t+1)/(1+i)-[pq(t) - C(q(t))] .(4)

Note that the absolute value of net accumulation is less than current resource rent.

Hartwick (1997) refers to the difference in discounted sums given by (3) as the *indirect method *for estimating net accumulation. Implementing this method requires projections of the future flow of rents, even if one applies (4) instead of (3). In earlier work, Hartwick (1977, 1990) derived a simpler, *direct method*. Deriving this method is easier if we switch from discrete to continuous time. Then, the asset value of the resource is

V(t) = ƒ e,^{-r(s-t)}[pq(s) - C(q(s))] ds(5)

where the integral is evaluated from *t *to *T*, and *r* is the continuous discount rate. Net accumulation is given by

D(t) = dV(t)/dt .(6)

*r* is the continuous discount rate. Applying (6) to (5), we obtain

D(t) = rV(t) - [pq(t) - C(q(t))] .(7)

This is the continuous-time analogue to (4). Under an optimal extraction program (Hotelling’s *r*-percent rule holds; see Hartwick and Hagemann 1993), dynamic programming implies that this simplifies to

D(t) = -[p - C'(q(t)] q(t).(8)

Net accumulation equals the negative of the product of quantity extracted times marginal — not average — rent. Hartwick refers to this product as Hotelling rent. Under the assumptions we have made, the product is unambiguously negative: asset value declines. The decrease in asset value due to the realization of current resource rent outweighs the increase due to the shifting of the stream of future rents toward the present. If the assumptions underlying this result hold — in particular, if price, the marginal cost schedule, and the discount rate are constant, and the resource is exploited optimally — then one can estimate net accumulation of a nonrenewable resource with only current data on price, marginal extraction cost, and quantity extracted. One does not need projections of future rents.

Optimal control techniques yield the same result. The manager of the nonrenewable resource seeks to identify the extraction program that maximizes (5), subject to the state equation

dS(t)/dt = -q(t)(9)

and the stock constraint

S(t) = ƒ q(s) ds.(10)

The integral is again evaluated from *t* to *T*. The current-value Hamiltonian for this problem is

H(t) = pq(t) - C(q(t)) + λ (t) dS(t)/dt.(11)

*λ
(t)*, the adjoint (co-state) variable or user cost, gives the capital value of a unit change in the resource. Hence, the second term in (11) equals net accumulation. The first-order condition, δ
*H(t)/δ
q(t)* = 0, yields

λ (t) = p - C' (q(t)) .(12)

Multiplying this times -*q(t)* (= *dS(t)/dt*)* *yields the same result as in (8).

In practice, obtaining data on marginal extraction costs is usually difficult. Direct estimation methods are still available in the absence of such data, however, if the marginal cost curve is elastic in quantity extracted. If one has data on average extraction cost (= *C(q(t))/q(t)*), then (8) can be written as

D(t) = [p - (1+ß)C(q(t))/q(t)] q(t),(13)

where *ß* is the elasticity of the marginal cost curve with respect to quantity extracted. Alternatively, if one has data on resource rent, Vincent (1997) has demonstrated that (8) equals

D(t) = [pq(t) - C(q(t))] (1+ ß)/(1+ ße.^{r(T-t)})(14)

For the special (but unlikely) case when *ß* = ∞
, this simplifies to

D(t) = [pq(t) - C(q(t))]e.^{-r(T-t) }(15)

El Serafy (1989; see also Hartwick and Hagemann 1993) proposed using the discrete-time counterpart of this formula to estimate net accumulation.

**The net depletion method for timber**

The obvious way to generalize the nonrenewable resource model in the preceding section to the case of renewable resources is to modify the state equation, (9), to include growth of the resource stock:

dS(t)/dt = g(S(t)) - q(t).(16)

Now, the Hamiltonian in (11) indicates that net accumulation is given by

D(t) = [p - C' (q(t))] [ g(S(t)) - q(t)].(17)

This says that net accumulation is the product of marginal rent times the negative of *net depletion* of the resource, where the latter is defined as the difference between current harvest and current growth. This is the method presented in Mäler (1991) and applied, with some modification, by Vincent (1997). Others, notably Repetto et al. (1989), have applied a method that differs only in terms of multiplying the negative of net depletion times average, not marginal, rent:

D(t) = [p - C(q(t))/q(t)] [q(t) - g(S(t))].(18)

In both cases, current harvest and current growth are multiplied by the same net price, either marginal rent (17) or average rent (18). If the cost function is linear in the quantity extracted, that is if *C(q(t)) = cq(t)*, then the two methods are identical.

**The El Serafy variation for timber**

The net depletion approach implicitly assumes that resource growth is immediately available for harvest. This is not the case with forest resources. Timber rotations typically span several decades. For an individual hectare of forest, the discrete-time asset value corresponding to (1) is

V,_{H}(t) = (1+i)^{t-T }[pq(T) - C(q(T))] / [1 - (1+i)^{-T}](19)

where *t* is the current age of the forest (not calendar year) and *T* is the harvest age that maximizes *V(0)* (asset value for *t* = 0). That is, *T* is the optimal rotation age: the forest is harvested every *T* years, with no intervening production. *q(s)* now represents both timber harvest and standing timber stock: all trees on the hectare are the same age, and all standing timber is harvested at maturity. This expression assumes that land remains permanently in forest use. Hartwick (1993) examined forest values and optimal rotation decisions under different land-use assumptions.

We can derive a correct method for estimating net accumulation in the model given by (19) by applying (3). To begin, consider the case when *t = T*: the forest is economically mature. An instant before harvesting, asset value equals

V._{H}(T) = [pq(T) - C(q(T))] / [1 - (1+i)^{-T}](20)

One period later (*t *= 1), it equals:

V._{H}(1) = (1+i)^{1-T}[pq(T) - C(q(T))] / [1 - (1+i)^{-T}](21)

Net accumulation for one hectare of mature forest is thus

D,_{H}(T) = V(1) - V(T)_{ }

or, after some manipulation,

D._{H}(T) = iV(0) - [pq(T) - C(q(T))](22)

This is the analogue to (4). Net accumulation reflects both the exploitation of rents from the current harvest, which decreases asset value, and the shifting of rents from future harvests toward the present, which increases asset value. Once again, current rent overstates the absolute value of net accumulation. Written out in full, (22) is

*D _{H}(T) = i (1+i)^{-T}[pq(T) - C(q(T))] / [1 - (1+i)^{-T}] - [pq(T) - C(q(T))] *,

which simplifies to

D._{H}(T) = -[pq(T) - C(q(T))] [1 - (1+i)^{1-T}] / [1 - (1+i)^{-T}](23a)

This is the forestry analogue to El Serafy’s method for nonrenewable resources (14). Hence, we refer to it as the *El Serafy variation*. It indicates that net accumulation can be calculated from current rent if one also knows the discount rate and the optimal rotation age, which are needed to calculate the last two terms in brackets.

Now, consider the case when *t* < *T*. Through steps similar to those in the preceding paragraph, one obtains

D,_{H}(t) = iV(t)

or

D._{H}(t) = [pq(T) - C(q(T))] i (1+i)^{t-T}/ [1 - (1+i)^{-T}](23b)

Like (23a), this includes per-hectare rent from harvesting the mature forest, but the discounting terms differ, and one of them includes current age (*t*) as well. Note that (23b) does *not* include current growth, even though it concerns net accumulation of an immature forest.

These results suggest a way to estimate the net accumulation of timber resources based not on changes in timber volumes, as in (17)-(18), but rather on changes in the *age class structure* of forests. For example, if *A _{T}* hectares of mature forests are harvested in a given period and

A,_{T}D_{H}(T) + Σ A_{t}D_{H}(t)(24)

where the sum in the second term is evaluated over the interval *t = 1, …, T-1*. This can be readily calculated using (23a) and (23b) if one knows, in addition to areas by age class, the per-hectare rents from harvests of mature forests, the optimal rotation, and the discount rate.

Continuous-time versions of (23a) and (23b) can be obtained by substituting *e ^{rx}* for

**The net price variation**

The method presented in the preceding section would seem to have little in common with the net depletion method. Substitution of the Faustmann condition for optimal forest management into (23a) and (23b) converts the expressions into versions that can be more readily compared to (17) and (18).

The discrete-time Faustmann condition states that the forest should be harvested when

[p - C'(q(T))] q' (T) / [pq(T) - C(q(T))] = i / [1 - (1+i).^{-T}](25)

That is, the forest should be harvested when the rate of growth in forest value equals the opportunity cost of funds, where the latter is adjusted for the effect of the current harvest decision on the timing of future harvests. Rearranging, we obtain

[pq(T) - C(q(T))] / [1 - (1+i),^{-T}] = [p - C' (q(T))] q' (T) / i

which can be substituted into (23a) and (23b) to yield

D_{H}(T) = -[p - C' (q(T))] q' (T) [1 - (1+i)^{1-T}] / i(26a)

D._{H}(t) = [p - C' (q(T))] q' (T) (1+i)^{t-T}(26b)

These are the analogues to the direct method for nonrenewable resources given by (8), i.e. to the net price method. We therefore refer to them as the *net price variation*.

Note that both (26a) and (26b) include marginal, not average, rent. Hence, the net depletion method as formulated in (17) would seem to be more likely than the version given by (18) to yield equivalent expressions when the cost function in nonlinear. Upon disaggregation into harvest (mature forest) and growth (immature forest) components, the formulation in (17) yields

D_{H}(T) = -[p - C' (q(T))] q(T)(27a)

D._{H}(t) = [p - C' (q(T))] q' (t)(27b)

Two differences are immediately obvious: first, (27a) includes the harvest volume at the optimal rotation age, while (26a) includes growth at the optimal rotation age; and second, (27b) includes current growth, while (27b) includes growth at the optimal rotation age. The comparison is complicated, however, by the fact that both (26a) and (26b) also include discounting terms. Let us consider the two pairs, (26a) and (27a) for mature forests and (26b) and (27b) for immature forests, more carefully in turn.

From (22), we already know that total rent overstates loss in value of mature forests due to harvesting. A comparison of (26a) and (27a) suggests that Hotelling rent (i.e., (27a)) does as well. The ratio of (27a) to (26a) is

{q(T) / q' (T)} {i / [1 - (1+i).^{1-T}]}

Solving (25) for* q(T)* and substituting the result into this expression, we obtain

{[p - C' (q(T))] / [p - C(q(T))/q(T)]} {[1 - (1+i).^{-T}] / [1 - (1+i)^{1-T}]}(28)

For the linear cost function *C(q(t)) = cq(t)*, this simplifies to just the second term in curly brackets,

[1 - (1+i),^{-T}] / [1 - (1+i)^{1-T}]

which is greater than one. Hence, the Hotelling rent associated with timber harvests overstates the absolute value of net accumulation in mature forests when the cost function is linear. Given that the discrepancy is due to the exclusion of just one *(1+i)* term in (27a), however, the magnitude of the discrepancy will not be very large unless the discount rate is large.

When the cost function is nonlinear, the bias can in principle be in either direction, as the first term in curly brackets in (28) is less than one (marginal rent is less than average rent). All we can say is that the bias will tend to be upward — (27a) will tend to be greater than (26a) — when the discrepancy between marginal and average costs (the first curly bracket) is small relative to the discount rate (the second curly bracket), and downward when the opposite relationship holds. Forest economists usually assume that marginal and average logging costs do not vary within individual logging units, especially small ones. That is, they assume that the cost function is linear at the micro-level. To the extent that this assumption is valid, (28) will be greater than one in practice, and (27a) will unambiguously overstate the absolute value of net accumulation in mature forests.

The value of (27b) is also likely to be larger than the value of (26b). Economists usually assume that the function relating timber volume to age (*q(s)*) has either a concave or logistic shape (Hartwick 1993). In the latter case, they usually assume that the inflection point occurs at a relatively young age. Data from actual forests generally supports these assumptions. Hence, for all ages in the case of concave volume functions, and most ages in the case of logistic functions, *q'(t) > q'
(T)*: the timber growth rate (the current annual increment to foresters) declines as the forest grows to maturity. For this reason alone, (27b) would tend to be larger than (26b), but the latter expression is reduced further by the inclusion of the *(1+i) ^{t-T}* term. Hence, we expect the net depletion method to overstate the increase in value of immature forests due to timber growth, e
xcept in very young forests with a logistic volume-age relationship. This result does not depend on the linearity or nonlinearity of the cost function. However, calculating (27b) by using average rent instead of marginal rent, in accordance with (18), would worsen the bias in the net depletion method when the cost function is in fact nonlinear.