Dynamic Indirect Production Functions

Abstract

For consideration of indirect production functions in a dynamic model, it is convenient to modify the function space of vectors of input functions (histories), introduced in [2] for the production correspondences as

$${\text{x }}\varepsilon {\text{BM}}_ + ^{\text{n}} \to \mathbb{P}\left( {\text{x}} \right){\text{ }}\varepsilon {\text{ }}{{\text{2}}^{{\text{BM}}_ + ^{\text{m}}}},{\text{u}}\varepsilon {\text{BM}}_ + ^{\text{m}} \to \mathbb{L}\left( {\text{u}} \right)\varepsilon {2^{{\text{BM}}_ + ^{\text{n}}}}$$

in which ℙ(x) represents the set of vectors of output functions (histories) obtainable from the vector x of input functions (histories), and L(u) represents the set of vectors of input functions (histories) yielding at least the vector u of output functions (histories). Each component of x and u is defined on the interval [0,+∞), for pointwise addition of two functions and pointwise multiplication by a scalar. A function f ε BM₊ is nonnegative, bounded in the sup norm and Lebesgue measurable.