als of type 1, and where all material input used in the process of abatement is turned into an-
other kind of abatement residuals, called abatement residuals of type 2. Denoting the amounts
of abatement residuals by ra1 and ra2 , respectively, our simple technological assignment is
a= r and m= r.
a1 aa2
The technology of producing consumer goods is introduced in form of the production function
y = F ( £ y,my ) ,
where £ y denotes labor input, my denotes material input and y is the amount of consumer
good produced.
Properties (F): F : R + → R + is concave and satisfies F (0,my ) = F (£y, θ) = 0,
Fl > 0, Fm ∈] 0,1 [. FU < 0 Fmm < 0 and Fcm > 0.6
The constraint Fm ∈ ] 0,1[ is absent from conventional production functions and therefore
demands an explanation. Our simple production model assumes that there is one and only one
material input whose quantity my is transformed into at least two different outputs, the con-
sumer good (quantity y) and some production residuals (specified below), since the entropy
law prevents the full transformation of material into the desired output. Since we conceive of
y as a material output whose units are of constant weight, it follows immediately that
<1
(ly, my )
for my = 0 and £ y ≥ 0.
When this property of F is combined with the (conventional) assumption Fmm < 0, we con-
clude that Fm < 1 holds on the entire domain of the function F.7
From the preceding discussion it is obvious that y = F (£y, my ) is not a complete description
of the production technology. After all, F allows us to maintain some level (and weight) of
output y while varying the amount of material input. A minimum requirement for satisfying
6 Note that the materials-balance principle also implies ∫0y F£ (λy, my ) dλy ≤ my for all my ≥ 0, since the output
is made up of material my only and the weight per unit of output y is constant. This constraint is compatible
with the assumptions Y£ > 0 and Yu < 0 if and only if lim F££ = 0.
£y →∞
7 The upper bound Fm < 1 is usually absent in textbook treatments of production functions, and it is violated, in
particular, by the popular Cobb-Douglas function.