description of that technology (2). The intriguing question is, therefore, what the precise rela-
tionship is between (1) and (2).8 Can a function of type (1) with property (Y) be shown to be
implied by (2) or is (1) incompatible with (2)? In the latter case, one would need to discard the
technology (1) since the materials-balance principle cannot be dispensed with.
To see what the link between (1) and (2) is like we now scrutinize the comprehensive tech-
nology (2) to elicit its major properties in several steps.
Proposition 1:
(i)
(ii)
Define Y : = { (y, e, £, m)∈ B4| z = z (v) for z = y,e, £,m and v ∈ Ω } , where z (v)
is the component z of vector v = ( a, e, £, £ a, £ y, m, ma, my,ra ,ra ,ry,y ) ∈ B1+2 and where
Ω : = {v∣ v satisfies (2a) - (2i) } . The set Y is convex.9
If (2a) and (2b) hold as equalities there is a mapping G : (e, £,m ) → y implied by the
production system (2) whose image is set valued.
To establish proposition 1(i), it suffices to prove convexity of the set Ω, since if Ω is con-
vex, its projection Y into the subspace of all vectors (y,e, £,m) is convex, too. Consider
α∈[0, 1], vα : = αv1 +(1 - α ) v2 and vi : =( a1,e1, £ i, £ 1a, Py,m, ,m,a,m,y,r,ayr'a 2 ,r,yr,y' ) for i =
1, 2. By definition, Ω is convex if for any pair v1 ,v2 ∈Ω it is true that vα ∈Ω for all
α∈[0,1] . Hence we need to show that (2a) - (2i) is satisfied for vα. Consider ryα, mαy, yα and
confirm that these variables satisfy (2b) by calculating
mαy -yα = αm1y+(1-α)my2-αy1-(1-α)y2=α(m1y-y1)+(1-α)(m2y-y2)=
= αry1+(1-α)ry2 =ryα
Quite obviously, the same procedure can be applied to all linear equations in (2). It therefore
only remains to show that (2a) and (2c) are satisfied for vα. This is easily established by ob-
serving that yα = α y1 +(1 - α) y2 ≤ αF (£1y, my ) + (1 - α) F (£2y, m2y ) ≤ F (£ay, mαy ), where the
last inequality is due to the assumption that F is a concave function. Since the function A is
also assumed concave, convexity of Ω is proved.
8 An early discussion on the relationship between technological concepts similar to (1) and (2) can be found in
Siebert et al. (1980), where the technology of type (1) is referred to as net-emissions approach and that of type
(2) as gross-emissions approach. Yet the technological concepts employed in Siebert et al. (1980) are not in line
with the materials-balance principle. See also Pethig (2003).
9 The sets Ω and Y are also closed since (2) doesn't contain inequalities excluding the equality sign.