APPENDIX 1 - BASIC MODEL WITH TIMBER, NONTIMBER VALUES AND DEFORESTATION

In this and several subsequent appendices, we construct and analyze models using optimal control techniques, which permit a rigorous derivation of national accounting relationships. The appendices contain a considerable amount of technical detail. They are intended for the specialist, not the general reader. The main text presents the principal results in a non-technical manner.

Structure of the model

The basic model is similar to one presented in Mäler (1991), although with more of a forestry focus. It contains five production activities:

F	=	forestry
W	=	wood production (logging)
N	=	household collection of nontimber products
D	=	deforestation (land conversion)
Q	=	rest of the economy

The letters will be used as subscripts on variables and superscripts on functions. Among the variables are five factors of production:

L	=	labor
K	=	capital
A	=	agricultural land
F	=	forest land
P	=	pollution (other than CO2)

Pollution is a production factor in the sense that it represents use by industry of the disposal services of the environment. These factors are used to produce four goods:

Q	=	a homogeneous good that can be consumed, invested, or used as an intermediate input
ZW	=	wood (logs)
ZN	=	nontimber products
ZD	=	newly cleared land.

These are produced according to the following production functions:

Q	=	(LQ , KQ , ZW , A , F , P , C)
ZW	=	(LW, QW , S)
ZN	=	(LN , F)
ZD	=	(LD)

( · ), ( · ), ( · ), ( · ) are functions. L_Q is labor used in the production of Q, etc. C is the concentration of atmospheric CO₂ (not the emissions of CO₂, in contrast to P, the emissions of other pollutants). S is the standing volume of timber. The signs on all variables except C are positive. The negative sign on C indicates that higher CO₂ concentrations reduce economic output, for given levels of other inputs. The inclusion of Z_W and A in ( · ) represents the use of logs and agricultural land, respectively, as production inputs. The inclusion of F represents environmental services provided by forest to the rest of the economy (e.g. watershed protection). The inclusion of S in ( · ) indicates that logging costs are lower when the timber stock is higher. Similarly, the inclusion of F in ( · ) indicates that collection costs for nontimber products are lower when forest area is greater (e.g. people do not need to walk as far to reach the forest).

The model contains six state equations:

dA/dt	=	+ ZD
dF/dt	=	- ZD
dS/dt	=	g (LF , KF , QF , S , F , P) - ZW
dC/dt	=	dCROW/dt -αdS/dt
dKQ/dt	=	IQ - δKQ
dKF/dt	=	IF - δKF

The first two equations indicate that agricultural area increases, and forest area decreases, according to the amount of newly cleared land. The third indicates that the change in the timber stock equals the difference between growth (the function g ( · )) and harvest (Z_W). Growth responds positively to forest management expenditures (L_F , K_F , Q_F), negatively to pollution (P), and in a presumably nonlinear and not necessarily monotonic way to forest area and the stock of timber (F, S). The fourth equation indicates that the change in the stock of atmospheric CO₂ equals the difference between net emissions occurring in the rest of the world and net sequestration in the country’s forests, which is proportional to the change in timber stock. For simplicity, we assume that the country does not emit any CO₂ itself, except from forests if the timber stock declines. The last two equations indicate that human-made capital employed in production of the homogeneous good and forest management changes according to the balance between investment and depreciation (δ is the depreciation rate).

Production and consumption are subject to the following constraints:

Q	=	QC + QW + QF + IQ + IF
L	=	LQ + LW + LN + LD + LF
A	=	F0 - F

The first constraint says that output of the homogeneous good is allocated among household consumption (Q_C), intermediate inputs in logging (Q_W) and forestry (Q_F), and investment (I_Q , I_F). The second says that labor is allocated among the five production sectors. Finally, the third says that the area of agricultural land is the difference between total land area (assumed to be entirely in forest initially, F₀ ) and area of forest land (F).

The objective in this model is to maximize welfare, which is a function of three variables:

u

=

u (QC , ZN , F).

Households consume two tangible goods, the homogeneous good and nontimber products, and one less tangible good, amenities associated with forests. The signs on all three variables are positive. As the purpose of the model is to examine national accounting relationships related to forest resources, we ignore disamenities associated with pollution. The objective is to be maximized with respect to the allocation of the homogenous good (Q_C , Q_W , Q_F , I_Q , I_F), labor (L_Q , L_W , L_N , L_D , L_F), the intermediate goods (Z_W , Z_N , Z_D), and pollution (P). That is, there are fourteen control variables. The current-value Hamiltonian for this problem is:

H	=	u (QC , ZN , F)
		+ pQ [f Q(LQ , KQ , ZW , F0 - F , F , P , C) - QC - QW - QF - IQ - IF]
		+ w [ L - LQ - LW - LN - LD - LF]
		+ rQ [IQ - δ KQ] + rF [IF - δ KF]
		+ pW [(LW , QW , S) - ZW] + pN [(LN , F) - ZN] + pD [(LD) - ZD]
		+ λA [ZD] + λ F [-ZD]
		+ λ S [g(LF , KF , QF , S , F , P) - ZW]
		+ λ C [dCROW/dt - α {g(LF , KF , QF , S , F , P) - ZW}]

p_Q, p_W, p_N, p_D, and w are prices, and r_Q, r_F, λ _A, λ; _F, λ _S, λ _C are adjoint variables (shadow prices).

The model contains no separate variable for biodiversity, which is assumed to be implicitly captured by forest area (in accordance with island biogeography theory). Incorporating biodiversity more explicitly would be straightforward, as long as one is content with a fairly crude representation. One could model the stock of biodiversity, B, as a nonrenewable resource. Decreases in the area of forest (F) and the stock of timber relative to forest area (S/F, a proxy for forest age) reduce B. Since biodiversity is nonrenewable, however, increases in the two variables do not increase B. That is, there is an asymmetry in the model. Biodiversity could then be added as a fourth variable in the welfare function (to represent, say, existence values) and as an eighth variable in the production function for Q (to represent the contribution of genetic resources).

Derivation of adjusted NDP

Following Mäler (1991), we set NDP equal to the linearized Hamiltonian. In combination with results of the fourteen first-order conditions, this yields:

Adjusted NDP	=	pQ (QC + IQ + IF)	} Conventional GDP
		+ uNZn + uFF	} Nonmarket values to be added to GDP
		- pQ δKQ - pQ δKF	} Depreciation of human-made capital

	+ λAZD - λ FZD
	+ λ S [g( · ) - ZW]
	+ λ C [dCROW/dt - α { g( · ) - ZW}]

Conventional GDP is the sum of expenditures on consumption and investment in human-made capital. GDP, and therefore NDP, ought to include as well the nonmarket value of household consumption of nontimber products and forest amenities (u_N , u_F are marginal utilities). Conventional NDP is the difference between GDP and depreciation of human-made capital. NDP ought to include as well the net accumulation of natural resources, given by the last three lines.

Adding nonmarket values to GDP and net accumulation of natural resources to NDP requires information beyond that provided by market prices. From the first-order conditions and the adjoint equations, we can show:

uNZN	=	wLN + uNF = wLN ·
λ A	=
λ F	=	λ A - pD
λ S	=	pQ - pW + α λ C
λ C	=

The value of household consumption of nontimber products, u_NZ_N, is related to the value of labor used in collection: the wage rate, w, times the amount of labor, L_N. The two are exactly equal only if forests are used for collection up to the point where the marginal product of the forest for collection ( i.e., ) equals zero. This occurs when households have free access to the forest. When access is restricted, due to formal or informal property rights or fees levied on the products collected, the value is larger than the opportunity cost of labor.

The marginal value of land converted to agriculture, λ _A, equals the discounted sum of future returns to agriculture, where returns equal the marginal product of agricultural land, , times the price of output. The marginal value of forestland, λ _F, equals the difference between price of agricultural land (λ _A) and cost of land clearing (p_D). The marginal value of the timber stock, λ _S, equals marginal net price (stumpage value) adjusted for marginal carbon sequestration value: the marginal value of logs as a production input (p_Q ) minus the sum of logging cost (p_W) and the damage caused by CO₂ released due to logging (α λ _C ; λ _C is negative). Damage caused by CO₂, λ _c, equals the discounted sum of the value of reduced economic output; is the marginal impact of CO₂ on output and p_Q is the price of output.

Environmental services and related matters

These expressions indicate that we can use information on marketed products and inputs to value nonmarket elements of NDP. The same is not true for forest amenities, however, at least in the model as we have constructed it. To value u_FF, one needs to use a direct valuation technique like contingent valuation.

The apparent exclusion of environmental services of forests from NDP is perhaps surprising. But in fact, NDP, and GDP as well, already reflect these services. First, NDP reflects these services in the expression for the net accumulation of forestland. The adjoint equation for λ _F implies:

λ _F =

The marginal value of forestland equals the discounted sum of benefits provided by forestland. These benefits include: forest amenities (u_F); environmental services

( p_Q ), net of the opportunity cost of retaining land as forest instead of converting it to agriculture ( p_Q ); nontimber products ( p_N ); and growth of timber (λ _S g_F ; g_F is wood increment) plus the value of sequestered carbon (α λ _C S_F). Note that g_F implicitly reflects the effects of pollution on timber growth; there is no need to make a separate accounting.

If markets work perfectly, then λ _F defined by this expression and λ _F defined as the difference between price of agricultural land and cost of land conversion will be exactly equal. In the absence of perfect markets, however, we expect the latter expression (i.e. λ _A - p_D) to understate the value of forestland. For the national accounts, the former expression gives the more accurate indication of the value of changes in forestland, but it is of course more difficult to apply.

Second, conventional GDP already reflects environmental services of forests, and therefore NDP does too. From above, the expression for conventional GDP is:

Conventional GDP = p_Q (Q_C + I_Q + I_F)

If we substitute ( · ) - Q_W - Q_F - I_Q - I_F for Q_C, and linearize ( · ), we obtain

VA denotes value added. This is the income approach to GDP. For heuristic reasons, we have separated production of the homogeneous good into manufacturing and agricultural components.

We assume here that it is the manufacturing component that generates pollution, and the agricultural component that benefits from the environmental services provided by forests.

Value added in manufacturing equals the payments to labor (wL_Q) plus profits (the term in parentheses). Profits include not just the marginal value product of capital (p_QK_Q), but also the free pollution disposal services that the environment provides (p_QP). Value added in agriculture equals just profits, as we did not distinguish in the model between manufacturing and agricultural labor. As in the case of manufacturing, profits include not just the marginal value product of the fixed factor, in this case agricultural land (p_QA), but also a free environmental service, in this case environmental services provided by forests (p_QF). A portion of agricultural profits is therefore attributable to the forest. Hence, conventional GDP already reflects forest-related environmental services that affect production, although it does not identify them as such.

The last two lines show that value added in logging and forestry is given by the difference between value of output (of logging services in the case of logging, of stumpage in the case of forestry) and use of intermediate inputs. The expression for forestry value added indicates that one should deduct only the value of intermediate inputs, not inputs of labor (L_F) or investment in forestry capital (I_F).