APPENDIX 6 - CLOSED ECONOMY WITH TIMBER HARVESTING AND DEFORESTATION

In this appendix we return to a model of a large closed economy, as in Appendix 3, but we introduce land clearing in addition to a pure harvesting activity. This model differs from the one in Appendix 5 in that all prices are now endogenous. Harvesting and timber growth are defined on homogeneous sub-areas. Economywide relationships are in part defined on the aggregated sub-units¹⁸. A new production activity is the clearing of forestland. This activity yields new land for agriculture plus marketable timber. The costs of clearing include the costs of felling and transporting salvaged timber. Harvesting and clearing are distinct activities and each is competitive, leaving only producer surpluses as profits. The two activities are linked because each yields marketable timber. Harvesting is assumed to be sustainable on land dedicated to forestry.

There are three elements of particular interest from an accounting perspective. First, we observe that "investment" in new agricultural land shows up in the accounts as a price gap multiplied by a stock change. The price gap is that between a unit of land in agriculture and a unit in sustainable forestry. This is a new measure of "rent."

Secondly, rent on forestland does not appear in the accounts. It is completely capitalized in timber stock rent. This in part reflects the fact that there are no explicit nontimber values in our model. It also reflects the general principle that land price is discounted rentals associated with the most profitable activity on the land.

Thirdly, we observe that land price can rise or decline during clearing, depending on the net cost of clearing a unit of land. This result is similar to the one for our small open economy. Here, however, land prices are endogenous. To see what is transpiring, one should think of a unit of forestland at date t as having two alternative uses. It can be harvested but retained as forest, which yields marginal profit [F_R-g_H]δ(S), or it can be cleared, with fraction D(t) being cleared, which yields marginal profit . The land price change, must adjust to equilibrate the current marginal profit from these two alternative uses:

= f_D - δ(S)g_H.

Whether land price rises or falls during clearing depends on whether current net marginal clearing cost f_D-δ(S)g_H is positive or negative. In a sense, the price gap adjusts to "accommodate" the current net marginal cost of clearing.

Overview of the model and associated national accounts matrix

At any date, timber production derives from harvesting H cubic meters per hectare of forestland retained under forest cover and salvaging δ(S)D per hectare cleared. δ(S) is the density of timber stocking, which varies with the stock-land ratio. Hence, timber production is

R = HL + δ(S)DL,

where L is land in forests and D is the tract cleared. Clearing yields DL of new agricultural land and diminishes L:

= - DL.

The timber stock, per hectare, changes as

= G(S) - H - δ(S)D,

where S is the current stock and G(S) is the growth increment per hectare. The sum of harvested and salvaged timber, R, enters as an input into aggregate production, along with agricultural land A. Total output F(K,N,R,A) is allocated among current aggregate consumption, C, harvesting costs, Lg(H), clearing costs Lf(D), and new investment, , in human-made capital, K. N is the constant labour force. Hence, we have

= F(K,N,R,A) - C - Lg(L) - Lf(D).

The total land area is ,with = A+L. This means that

= DL.

NDP for this economy turns out to be

where

ψ/λ = price of a hectare of land in agriculture

γ/λ = price of a hectare of land in forestry

p^R = price of a cubic meter of harvested timber

mc^H = the marginal cost of harvesting a unit of timber.

When we work through the national accounting matrix for this problem, we will see that net national income (equal to NDP) is

where r is the rental rate on capital K, and F_A is the rental rate on a unit of agricultural land. π^H(H) and π^D(D) are net profits per hectare from timber-harvesting and land clearing that remain for households. They exist because g(H) and f(D) are not constant returns to scale processes. Two points are significant: (i) land in forests has no rental flow in NNI because this income is accruing to the timber stock owners directly and not indirectly to the forest landowners, and (ii) the timber stock rents derive from gross flow G(S)L from stock SL, not from net flow [G(S) - H - δ(S)D]L. It will, however, take some steps to reach NNI via the social accounting matrix.

NDP includes an entry for "investment" in agricultural land. This corresponds to land cleared of forests at a price and turned into land in agriculture. In Ricardian parlance, it is as if we drained swampy terrain, at a cost, to create usable agricultural land. This has been referred to as investment in land. Our switching of use from forestry to agriculture is analogous to the switch of land from swamp to agriculture, at a cost. However, the price of our (marginal) piece of land may rise or fall during the switch depending on the net cost of clearing it. Equilibrium in our model involves marginal profit being zero for timber harvesting and land clearing. Formally, we have

[F_R-g_H]L - η = 0

and

[F_R-f_D/δ(S)]L + [ψ-γ]L/(δ(S)λ) - η = 0

for timber harvesting and land clearing respectively. This implies that net profit is the same in each activity, or

[F_R-g_H] = [F_R-f_D/δ(S)] + [ψ-γ]/δ(S)λ,

which reduces to the fundamental condition defining the direction of change in land prices, namely

[ψ-γ]/λδ(S) = f_D/δ(S) - g_H,

where the right hand side is the net marginal cost of "creating" a hectare of agricultural land. The net marginal cost comprises marginal clearing cost and marginal salvage harvesting cost. To clear a hectare, one salvage harvests δ(S) amount of timber at marginal cost g_H(H) per unit. One clears a hectare at marginal cost f_D. Clearing includes the cost of removing timber. Hence the net cost of clearing is f_D-δ(S)g_H. This represents the increase in the value of a hectare from clearing. The net clearing figure can be positive or negative. Relatively high marginal clearing costs are associated with an increase in the value of a hectare in the switch from forestry to agriculture. Relatively low marginal clearing costs are associated with mining forested land for the standing timber. In these cases, clearing is pushed into ranges in which new agricultural land is not very productive and has a value less than in forestry.

Recall that π^H and π^D are net profits accruing because harvesting costs g(H) and clearing costs f(D) are not linear in amounts harvested and cleared, respectively:

π^H(H) = Hmc^H - g(H)

π^D(D) = Df_D - f(D).

These two entries enter directly into the national accounting matrix under production activities. The zero marginal profit conditions for land clearing and harvesting become the following key entries for total profit per unit of land:

These conditions yield the respective row and column entries in the social accounting matrix for "Land clearing" and "Timber harvesting". One must keep in mind that column sums for "Land clearing" and "Timber harvesting," excluding the "Household" entry, are total costs of the respective activities.

Observe that neither harvesting nor clearing disburses rent to landowners. The rent a firm might be expected to pay to owners of forestland is paid instead to owners of the timber stock on the forestland. These two entries are [p^R-mc^H]H and [p^R-mc^H]δ(S)D respectively. Land-clearing firms buy the forestland which they intend to clear at price γ/λ per hectare and, once cleared, sell the land at price ψ/λ per hectare. They sell salvaged timber for δ(S)Dp^R dollars. Clearing costs include salvage harvesting costs. Hence clearing an area of D hectares yields δ(S)D of marketable timber, and generates a profit, at the margin, of [p^R-mc^H]δ(S)D. The standing timber is acquired by land-clearing agents when they purchase the land for clearing.

A central result of our analysis is that land rent accruing to owners of forestland is simply discounted net income (timber stock rent and surpluses, or π's) from using the forestland. Use involves both clearing and timber harvesting. Price γ/λ equals the present value of current net income,

from a hectare of land. Part of net income derives from clearing, namely [p^R-mc^H]δ(S)D + π^D(D), and part from timber harvesting, [p^R-mc^H]H + π^H(H). Recall that if clearing and harvesting were constant returns to scale (i.e., f(D) = αD and g(H) = βH), then both π^H(H) and π^D(D) would be zero. The price of a hectare of forestland would then simply be discounted timber rent associated with the removal of stock H+δ(S)D in the current period. However, the general result for land price γ/λ is

γ(S) = γ₀ + {[p^R(t) - mc^H(t)][H(t) + δ(S)D(t)] + π^HH(t) + π^D(D)(t))} e ^-(t-s)r dt.

That is, current land price is discounted net income from forested land.

The implication of this for accounting is that land rent for forestland does not appear in national income. That land rent is incorporated in timber stock rent and the surpluses or π's. If one were to record both land rents and timber stock rents into the accounts, one would be double-counting.

We are now in a position to represent the model in the form of a national accounting matrix. Compared to Table 5, the national accounting matrix in Table 7 has two new columns (and matching rows): one for land clearing, and one for agricultural land as a primary input. Forestland, we repeat, is taken account of in the rows and columns for its products, namely timber and new agricultural land. The new conceptual aspect is that agricultural land is an output of the clearing activity, and forested land is an input. The net change in land value for a cleared hectare can be interpreted as investment in land.

The column and row sums have been discussed above, following equations (1) and (2). The new agricultural land column and row are straightforward. Agricultural land is an input in the general production function, F(^.), and the rents on agricultural land are income to households. The new entry is "investment" [v^A-v^F] in agricultural land as an output of the clearing activity. Clearing has joint outputs, salvaged timber and cleared land. Agricultural land is produced, in contrast to the timber stock, which nature bestows on the households via natural growth. Recall that the value of forestland is the discounted surplus, including timber stock rent, obtainable from the forestland. v^F in the household column is the payment by the land clearing agents for the forestland that is transformed by clearing into agricultural land.

The central result for valuing primary inputs, or net national income, appears in the household row. The sum of incomes [p^R-mc^H]HL, [p^R-mc^H]δDL, and [p^R-mc^H]L reduces to [p^R-mc^H]G(S)L. That leaves the household row sum as the value of gross flows from stocks plus the value of labour services and profits, π^H(H)L + π^D(D)L. Hence, national income is the sum of the value of primary flows, gross of harvests or withdrawals from those flows, plus producer surpluses ("profits"). This sum is the standard measure of aggregate value-added in the economy.

Details of the model

The timber stock, S(t), has a net natural increment G(S) per unit of forestland. Timber is obtained from harvesting on forestland and from clearing DL hectares. The density of timber per unit of land is δ(S). L hectares are in forest currently. Thus, per hectare, we have

= G(S) - H-Dδ(S),

(VI.1)

and

= -DL.

(VI.2)

Land in agriculture is A = -L. Thus

= DL.

(VI.3)

There are per-hectare harvesting costs g(H) in sustained forests, distinct from per-hectare clearing costs of f(D). Human-made capital K is invested according to

= F(K,N,A,R) - C - Lf(D) - Lg(H)

(VI.4)

where F(^.) is a constant returns to scale, neoclassical production function. N is labour services, a constant; R is the sum of harvested timber, LH, and salvaged timber, δ(S)DL; and C is aggregate final consumption.

Market equilibrium derives from (corresponds to) the solution of the optimal planning problem, to maximize

(VI.5)

subject to VI.1, VI.2, VI.3, VI.4 and initial conditions

K(0) = K0	(VI.6)
L(0) = L0
S(0) = S0.

The current-value Hamiltonian for this problem is

H = U(C) + λ[F(K,N,A,R) - C - Lf(D) - Lg(H)] + [ψ-γ]DL
+ π [G(S) - H - Dδ(S)],	(VI.7)

where

λ = until price of a unit of human-made capital

ψ = until price of a unit of land in agriculture

γ = until price of a unit of land in forestry

η = until price of a unit of the timber stock.

Until prices become monetary prices when divided by the until price of a unit of consumption, namely U_C(C).

The necessary conditions for an optimum are

(VI.8)

(VI.9)

(VI.10)

(VI.11)

(VI.12)

(VI.13)

(VI.14)

There are two zero-profit arbitrage conditions to note. First, the net marginal cost of obtaining timber from harvesting and land clearing must be the same. This is captured in (VI.9) and (VI.10). Each activity pays timber owners their rent on timber harvested. The marginal profit net of rent must be equal for the two activities. That is,

λδ(S)[F_R-g_H] = λδ(S)[F_R-f_D/δ(S)] + [ψ-γ].

This reduces to the fundamental condition

f_D-δ(S)g_H = [ψ-γ]/λ.

At any date, the wedge in the price of a unit of land in agriculture versus forestry is the net marginal cost f_D-δ(S)g_H. Note that f_D is the marginal clearing cost per hectare, including tree felling costs, whereas g_H is the marginal cost of harvesting H volume of timber. D of a hectare corresponds to δD volume of timber felled or cleared. This is different from volume H being harvested from a hectare of forestland. Thus f_D -δ(S)g_H derives from two different cost functions and from marginal costs for two different margins, one at D and one at H.

In brief, the equilibrium land price spread must equal the marginal cost of transforming one type of land to the other type. Here the process is only allowed to go "one way," from forest to agricultural use. But even with the process only moving one way, two possibilities remain: a higher price in agriculture or a lower price in agriculture, depending on the relative magnitudes of current marginal clearing costs and marginal harvesting costs, per hectare. We argued earlier that in the early stages of development, as with Canada in the eighteenth century, land in forestry should command a relatively low price whereas much later, even as clearing continues, land in agriculture would command a lower price. Our model can display this switch from clearing "leading to" higher priced land in agriculture to clearing "leading to" lower priced land in agriculture.

Second, the price of land in forestry must be linked to the value of its current and alternative uses. The current use is timber harvesting, and the alternate use is agriculture. The price of land γ/λ is reflected in VI.13. is the price change. If one substitutes VI.9 and VI.10 into VI.13, one gets

or

Land price γ(t) is discounted surplus (rent and net profit) per hectare, where surplus derives from current land clearing and timber harvesting. The definitions of net profits (producer surpluses) are

π^H(H) = Hg_H - g(H)

and

π^D(D) = Df_D - f(D)

for harvesting and clearing, respectively. Note that π^D(D) derives from clearing per se, not from timber cut and sold from the cleared land. Clearing will in general involve more than simply felling or harvesting the Dδ(S) timber removed.

Our land price "formula" for forested land is a variant of the classic result that price is discounted net timber rents (Samuelson 1976). In a sense the value of the land is the profit remaining when the land is employed in its most profitable activity. This has significant implications for accounting and valuation in general. Careful and comprehensive accounting of rent for timber stocks will make the accounting of land rent in forestry redundant. If we do the accounts correctly, including timber stock rents, we will have dealt with land rent for forested land, en passant. To be specific, when forest harvesting firms pay Hη/λ to timber stock owners for the timber currently harvested, and land clearing agents pay δ(S)Dη/λ to timber stock owners for the timber currently harvested, they are implicitly covering off much of the land rent that the land in forests commands. The bottom line is then: the land value of forestland derives from the timber stock rental corresponding to the timber currently removed from the land in question.