The focus of this appendix is valuation of the timber stock. Timber is harvested and used as an input by other sectors of the economy. Valuation is simplified by: (a) the absence of foreign trade, (b) the absence of changes in land use, and (c) the absence of nontimber values associated with forests. There is no explicit government sector, and the timber stock S(t) is homogeneous (no variation in species, age classes, etc.). In subsequent appendices we discuss how to make the analysis more realistic (and we have already considered, in Appendix 2, how to deal with age-class issues). Valuation is done by connecting a model of a dynamic competitive economy to an accounts system. It turns out that a version of the classic social accounting matrix (a case we prefer to label as a national accounting matrix) captures all the accounting matters we analyze.
The analysis focuses on the economic appreciation or depreciation of the timber stock. From a broader perspective, changes in the timber stock are inextricably linked to changes in land use (whether areas are in forest or have been deforested during the accounting period). In that case, forestry accounting turns on both the economic depreciation of the timber stock and on the valuation of the land currently deforested. It involves simultaneous timber stock change valuation and land-use change valuation. In this appendix, there is no land-use change. We are starting with a simplified case.
Overview of the model and associated national accounts matrix
Before presenting the details of the model, we will present the principal results, relating them to the national accounts matrix in Table 1. Table 5 presents the matrix in more analytical detail than Table 1.
Aggregate production derives from the use of machine capital K, labour N, and timber H. Aggregate output F(K,N,H) is used to produce new machine capital 
, consumption goods, C, and harvesting services g(H). That is,
The production function is constant returns to scale. Thus one can think of F(.) as comprising three types of firms, aggregated to three industries,
F(KM, NM, HM) =
F(KC, NC, HC) = C
F(KH, NH, HH) = g(H),
where
KM + KC + KH = K
NM + NC + NH = N
HM + HC + HH = H.
Each unit of , C, and g(H) has a price of unity. Given constant returns to scale, we have
rKM + wNM + pHM =
rKC + wNC + pHC = C
rKH + wNH + pHH = g(H)
where r=FK, w=FN, and p=FH. The accounting is quite straightforward so far. In Table 5, row sums (receipts or revenues) for machines and consumer goods equal column sums (expenditures or costs). Total incomes from K and N (column sums for K-capital and labour) also equal total expenditures for using K and N (row sums for K-capital and labour) by firms.
We move on to the harvesting sector. The timber has net natural growth over the accounting period of G(S) and net total growth of
= G(S)-H,
where H is the current harvest. Note that, for simplicity, we are using the net-depletion model, which ignores age classes. Each unit of stock commands a marginal rent (marginal stumpage value), p-mc. That is, if the owner of the timber harvests one more unit, she will get p for it and her cost of harvesting (and selling) is mc. The remaining revenue is marginal profit or rent or stumpage value. If a harvesting firm does the cutting and selling, it must pay p-mc to the forest owner for the right to remove one more unit of timber. In a competitive economy, harvesting firms will be in a zero-profit position,
pH = [p-mc]H + g(H) + π(H)
where π(H) = mcH - g(H), is net profit for harvesters. This net profit remains because we are assuming that harvest costs g(H) are not linear (constant returns to scale) in "output", H. That is, harvest costs have rising marginal costs, a standard formulation.
The harvesting row in Table 5 is total revenue to the harvesting firms marketing their harvest, H. The column is total expenditure by the harvesting firms. Recall that total costs of harvesting are rKH + wNH + pHH, equal to g(H). The size of the rent term is determined by expectations about future timber values. The discount rate matters.
There remain two steps for completing the accounts. First, total rent paid
to forest owners by harvesting firms, [p-mc]H (in the column sum for
harvesting), must be received as income by forest owners. But forest owners'
wealth rises or falls by implicit net income [p-mc].
This entry appears as net income to forest owners in the "Forest" column of
Table 5. The corresponding entry in the row "Forest" is economic depreciation
of the timber stock. Here one should think of the households doing implicit
saving or dissaving in amount [p-mc].
For positive, households
(forest owners) are implicitly saving. For
negative, there is implicit dissaving in the forest sector. Dissaving is associated
with economic depreciation as distinct from appreciation. The column
sum for households is "green" NDP.15
The corresponding row sum for households in Table 5 is also adjusted NDP. It
comprises the value of total flows of services from primary factors into the
economy. This sum is also referred to as total value-added for the economy.
Of importance for green national accounting is that the two entries [p-mc]
and [p-mc]H combine to leave
[p-mc]G(S)
as the contribution of the forest stock to current national economic activity. This is the value of the services the timber stock (nature) provides, before human harvesting activity "intervenes". This is, in fact, the intuitively correct entry for the service flow from the timber stock in the economy's value-added.
From a national accounting point of view, timber and timber harvesting raise some somewhat novel issues. A correct value of rent on a unit of the timber stock must be obtained. Economic depreciation of the timber stock must be entered as part of national dissaving or saving. And value-added for the timber stock must be correctly arrived at (the combination of rental income accruing to timber stock owners and the net change in the value of the stock to the owners).
Details of the model
The objective is
max
subject to
K(0) = K0
S(0) = S0.
The variables are defined as follows:
C = composite consumption
K = stock of human-made capital
N = labour force (constant; i.e., no population growth)
H = timber harvest (net of defect)
S = stock of standing timber (net of defect)
G(S) = growth increment of timber (net of defect)
g(H) = cost of harvesting H, in terms of the composite good.
The current-value Hamiltonian is
H(t) = U(C) + λ(t)[F(K,L,H) - g(H)-C] + ψ(t)[G(S)-H].
The first-order conditions are:
In the last expression, FH-gH is p-mc in Table 5. The adjoint equations are:
- pλ⇒
- pψ⇒
The translation of the current-value Hamiltonian into NDP involves dividing by "price" UC and substituting for ψ and λ .16 One gets
This is written in the "Household" column of Table 5 as
C +
