ABSTRACT
The discussion of two elementary cases will be sufficient to give the flavor of the first part. Consider first the three-factor production function:
F(k1, k2,m)
where k1 and k2 are capital inputs and m is labor. The problem is whether there exists an “index” of capital, that is a function G(k1, k2), and a two-factor production function H , depending only on “capital” and labor, such that:
F(k1, k2,m) = H(G(k1, k2),m) Now suppose that G and H exist and consider in the plane (k1, k2), for a given m, the family of isoquants:
F(k1, k2,m) = H(G(k1, k2),m) = y Let P ∗ = (k∗1 , k∗2) be a point in the plane (k1, k2), the slope in P ∗ of the isoquant through P ∗ is
−∂G(k ∗ 1 , k
and is therefore independent of m (this is nothing but a geometric illustration of Leontief’s theorem, see p. xvi). On the other hand, an example as simple as:
F(k1, k2,m) = k1 + k2m in which the restriction above is violated, illustrates how stringent the condition for the existence of G and H are.
