A Ricardian Approach
2. Measuring the Effect of Environment on Production
This section develops the analytical apparatus that underlies the valuation of climate in this study. We postulate a set of consumers with well behaved utility functions and linear budget constraints. Assuming that consumers maximize their utility functions across available purchases and aggregating leads t o a system of inverse demand functions for all goods and service:
where P; and Q; are respectively the price and quantity of good i, i = 1,
..,
n, and Y is aggregate income. The Slutsky equation is assumed t o apply, so that Equation (1) is integrable.We also assume that a set of well-behaved production functions exist which link purchased inputs and environmental inputs into the production of outputs by a firm on a certain site:
Qi = Q;(K;, E), i = 1,
..,
n.
(2)In this equation, we use bold face t o denote vectors or matrices. Q; is the output of good i, K ; = (K;I,
..,
K;j,..,
K;j) where K;j is the purchased input j ( j = 1,...,
J ) in the production of good i, and E = ( E l ,..,
El,..,
E L ) where El is the exogenous environmental input 1 (1 = 1,...,
L) into the production of goods, e.g., climate, soil quality, air quality and water quality, which would be the same for different goods' production on a certain production site.Given a set of factor prices, Rj, for Kj
,
the exogenously determined level of environmental inputs, and the production function, cost minimization leads to a cost function:Here, C; is the cost of production of good i, R = (R1,
...,
RJ), and C;(*) is the cost function. Firms are assumed to maximize profits given market prices:where P; is the price of good i. This maximization leads firms to equate prices and marginal cost. Differentiating Equation (4) with respect to any purchased factor and setting the result to zero also reveals the first-order conditions pertaining to each factor used in production:
Next consider the impact of changes in the exogenous environmental variables. Assume that the environmental change is from initial point E A to new point EB. The change in value from changes in the environment are then given by:
where
J C
is the line integral evaluated between the initial vector of quan- tities and the zero vector, QA = [QI(Kl, EA),..,
Q; (K;, EA),..,
Qn(Kn, EA)], QB = [QI ( K I ,E B ) , . - 7 Qi(Ki, EB),..,
Qn(Kn, EB)], Ci(Qi,R, E A ) = Ci(Qi(Ki, E A ) , R , EA), and Ci(Qi, R , E B ) = Ci(Qi(K;, EB), R , EB). It is necessary to take this line integral as long as the environmental change af- fects more than one output. If only one output is affected, then Equation (6) simplifies to the integral of the equations for a single good. Note that as long as the Slutsky equation is satisfied, the solution to Equation (6) is path-independent and unique.The damages in Equation (6) can be decomposed into two parts. On the one hand, costs have changed for the production of good i from C;(Q;, E A )
Figure 2. The effects of an environmental change.
t o C;(Qi, EB). Second, production has changed from Q A t o Q B . The value of the lost production is the difference between the consumer surplus under the demand function and the original cost of production (see Figure 2).
The present study investigates the impact of environmental changes through their impact upon a particular factor, land. We now explicitly separate land out from the firm's profit function in Equation (4):
where L; is the amount of land used t o produce Q;, and P L E is the annual rent per unit of land given the environment E. We assume that there is perfect competition for land, which implies that entry and exit will drive pure profits t o zero:
If use i is the best use for the land given the environment E and factor prices R , the observed market rent on the land will be equal t o the annual net profits from production of good i.2
2With imperfect competition, it is possible that a farmer could pay only as much as the next highest bidder for land and that this land payment would then be less than the productivity in the best use of the land. In addition, if the land is not put to the best use, the land payment may exceed the net productivity of the land.
Let us now reexamine the measure of environmental damages with this explicit land market. If we are examining changes in the environment which will leave market prices unchanged, then Equation (6) can be expressed:
~ ( E A - E B ) = P Q B - C C ~ ( Q ~ , R , E g ) -[PQA - C C i ( Q i , R , E l ) ] (9) where P = (PI,
..,
Pi,..,
P,). Substituting Equation (8) into the above yields:~ ( E A - E B ) = ~ ( P L E B - P L E A ) L ~ 7 1
( 10) where PLEA is PLE a t EA and PLEB is PLE a t EB. Equation (10) is the definition of the Ricardian estimate of the value of environmental changes.
Under the assumptions used here, the value of the change in the environ- mental value is captured exactly by the change in land rent.
Note that all of the valuation expressions listed above implicitly assume that firms adjust their market inputs in order t o adapt t o the changing en- vironment. It is important t o recognize, however, that the measure of envi- ronmental damage incorporates this adaptive behavior. Rewriting Equation (9):
As E deteriorates from EA t o E B , one would expect that farmers would adjust their purchases of K from KiA to KiB to reduce some of the losses, although the exact form of the adaptation will generally be extremely com- plex. If one fails t o incorporate these adjustments by firms and instead assumes that K is fixed, then Equation (11) becomes:
This latter measure uses changes in gross revenues as a measure of envi- ronmental damage; it is closely related to the pduction-function approach, in which limited or no adaptation occurs. Scientific experiments where all factors are tightly controlled except for an environmental change use measure [Equation (12)l.
The Ricardian measure in Equation (lo), which includes all optimizing adaptations, is superior t o the gross revenue or production-function estimate
in Equation (12) because the former includes all adaptations. An important result, however, is that the Ricardian measure in Equation (10) will always yield an estimate of environmental damage which is less than or equal to the estimate genemted by the production-function a p p m c h in Equation (12).
This result is easily seen. The profits from adjusting all inputs and outputs optimally are clearly at least as great as the profits from not adjusting inputs or outputs at all or adjusting them incompletely. The former approach provides the estimate of the loss from the Ricardian approach while the later provides the loss from the production-function approach.
The impact of an environmental change on decisions is easily seen when there is only one input K and one environmental factor E in the production function of one good, Q = (K, E). Fully differentiating the first-order con- dition of profit maximization [Equation ( 5 ) ] with respect t o E and K and simplifying yields:
The optimal response by the firm to improvements in E will be to in- crease K if Q K E
>
0 and Q K K<
0. For example, if reduced concentrations of ozone make corn respond more positively to fertilizer Q K E>
0, then farmers would increase fertilizer use with decreased ozone. If increased car- bon dioxide decreases a plant's need for water and the marginal productivity of water Q K E<
0, then with more C 0 2 farmers will reduce irrigation. The profit function described by Equation (4) indicate adjustments of K with changes in E. If K is not permitted t o adjust, the resulting profits for each level of production must be lower so that net societal benefits must be lower.Estimates that do not allow for adjustments in purchases of market inputs, for example by measuring just changes in revenue, underestimate the value of environmental improvements (or overestimate the value of environmental damages).