3.1 Some Numerical Results Illustrating the Model
For a given timber price and stand age, harvest is probabilistic because it depends on the values of e generated for the period. Figures 15.2 to 15.4 presents probabilities of harvest for three variations of the modified BM model (that is, the BM model with e included). All models consider the case of Loblolly Pine on a low quality site, using parameter values found in Brazee and Mendelsohn (1988). Timber growth is implied by volume parameters =12.09 and 02 =52.9 (see (3.1)). Timber prices are normally distributed with mean $167.4 per thousand board feet, and standard deviation $40.41. The discount rate is 3% (discount factor = .97). Harvest and replanting costs are $147, and 6 = 0.
The models underlying the panels of Figures 15.2 to 15.4 differ in the value of the scale parameter, %, with this value falling across the three panels. Importantly, the scale parameter of the Gumbel distribution is inversely proportional to the variance (variance= n2/6n2), and so the variance in the distribution of e is rising across the panels. In the first panel the scale parameter is especially large in the context of the model (ne= 20.0). Consequently, values of e are invariably close to zero, and so the model is, for all intents and purposes, the same as the original BM model; that is, for a given stand age, the timber is harvested if and only if the timber price is above the reservation price for the stand age. Graphically this is represented in Figure 15.2 by a distinct "probability ledge" tracing the harvest isocline of the original BM model: as the stand ages the reservation price falls (see Figure 15.1). Henceforth we refer to this model as the "virtually no variance" (VNV) model. In Figure 15.3, ne =2.0 (the "low variance" (LV) model), and so the reservation price policy no longer applies, as indicated by the transformation of the probability "ledge" of Figure 15.2 to a steep probability "hill" in Figure 15.3. An unexpected result apparent
in the figure is that harvest often occurs at very young stand ages. This is the case because growth at these ages is relatively low, and so if, at these young stand ages, the difference £1 — £0 is sufficiently high—a distinct possibility because of the relatively low value of the scale parameter—it is advantageous to harvest the stand and start over. In Figure 15.4, % = 0.2 (the "high variance" (HV) model), and the probability surface is noticeably smoother than in Figure 15.3. Essentially the harvest decision is now heavily driven by the observed values of £0 and £1 .
Figures 15.5-15.6 presents simulated data from the VNV and LV models. The data involve 500 timber stands of varying ages observed over an 80-year sequence. Initial stand ages for the sequences are pseudo-random draws from a uniform distribution in the range [1,150]. Prices for the sequences were pseudo-random draws from the price distribution (a single 80-year price sequence applies to all timber stands in the figures). The figures show that the distribution of timber harvests is very different for the two models. The VNV model has relatively few harvests, and these are concentrated in several years. The HV model has many more harvests spread fairly evenly over time, reflecting the very young rotation ages engendered by the relatively high variation
in e. Although this is not realistic (generally southern pine must be at least 30 years old to be harvested for sawtimber), the sharp contrast with results from the VNV is useful because it allows us to explore whether qualitative differences in data affect the structural estimation of discrete dynamic decision processes.
Table 15.1 presents estimation results for the simulated data. We present results for both the full simulated samples of the VNV and LV models, and for partial samples in which estimation is based on only the last 40 and 20 years of the sequences.2 For all seven estimations, the exogenous parameters - the price parameters pp and ap and the volume parameters 01 and - are fixed at their actual values.3 This leaves four potential parameters to be estimated:
2Reducing the sample in the time dimension reflects the judgment that it is easier for researchers to obtain a large cross section of timber harvest data than a large time series.
3In actual estimation, the corresponding approach is to estimate the growth function and timber price process exogenously (that is, outside the main estimation algorithm), and to use the estimated parameter values in the main estimation algorithm, thereby significantly reducing the size of the estimation problem. This approach of estimating exogenous processes outside of the main estimation algorithm is common in the literature. Alternatively, the analyst could decide, for instance, that timber owners may be using an incorrect price
3, d°,d1, But because the values of not harvesting and harvesting are linear in e° and e1, respectively, only the difference in the location parameters 01 and 9° can be identified. To see this, observe from (3.3) that the timber owner harvests if
PtW (at) - c + 3 (V (at, 1; r) - V (at, 0; r)) + ej - e° > 0. (3.8)
Defining A9 = 91 — 9°, it follows from the properties of the Gumbel distribution that the harvest decision can be cast as logistic with location parameter,
PtW (at) - c + 3 (V (at, 1; r) - V (at, 0; r)) + A9, and scale parameter significantly, only the difference A9 can be identified. This is reflected in Table 15.1, where we fix 9° at zero and estimate 91.
When the full sample is used, estimation results are generally excellent for both the VNV and the LV models. For all models except the LV model with process, in which case the parameters of the price process would be included in the set of parameters to be estimated in the main estimation algorithm. In this case an iterative estimation algorithm suggested by Rust (1994b) would be useful.
half the sample (Model 5 in Table 15.1), the estimate of the discount rate is within 0.002 of the actual value, and even for Model 5 the true value is within two standard deviations of the estimated value. This no doubt reflects the tremendous influence of the discount rate on the harvest decision. By comparison, in estimations using partial samples the estimates of % and 01 tend to be statistically different than the true values of these parameters. Are these differences significant as a practical matter? Yes and no. In an investigation of timber harvest behavior, the analyst is primarily interested in two questions: the effect of stand age and timber price on the likelihood of harvest, and the expected value of bare forestland (In particular, the nontimber value of forestland). On both counts, Models 2, 3, and 6 generate results very similar to the true model. The exception to these generally favorable estimation results is Model 5, for which the estimated discount factor is considerably lower than the true discount factor. This lower discount factor (higher discount rate) has two significant effects. First, the estimated value of bare land is much higher for Model 5 than the actual value: $9,282 per acre versus an actual value of
$1,456 per acre. Of course, the large standard error on the estimate of the discount factor in Model 5 signals the analyst that the estimate of bare land value is imprecise. Second, for timber stands that by chance mature past age 20 or so, the predicted harvest age is lower for Model 5 than for the true model.4 Figure 15.7 tells the story on this. It presents the difference in harvest probabilities between the true model and the estimated Model 5. The deep trough in Figure 15.7 beginning at about stand age 25 in the price range of roughly $200-$270, indicates that for older stands Model 5 overpredicts the probability of harvest, and so underpredicts the expected harvest age. This result is entirely consistent with the usual literature on optimal timber harvesting that indicates the
4It is worth emphasizing that in both the true model and Model 5, the odds of a timber stand reaching age
10 is extremely low. This is apparent from Figure 15.3, which shows that for ns =2.0, the probability of harvest in each of the first ten years is roughly 0.4, and Figure 15.7, which shows that in the first ten years the probability of harvest in Model 5 is virtually the same as in the true model.
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