capital functions as inputs as a factor of production.
functions of capital as a factor of production?
inputs
for modelling the case of many outputs and many inputs, researchers often use the so-called shephard's distance functions or, alternatively, directional distance functions, which are generalizations of the simple production function in economics.





























in macroeconomics, aggregate production functions are estimated to create a framework in which to distinguish how much of economic growth to attribute to changes in factor allocation (e.g. the accumulation of physical capital) and how much to attribute to advancing technology. some non-mainstream economists, however, reject the very concept of an aggregate production function.





























section::::the theory of production functions.
production function
a simple proof assuming differentiable utility functions and production functions is the following. let w be the 'price' (the rental) of a certain factor j, let mp and mp be its marginal product in the production of goods 1 and 2, and let p and p be these goods' prices. in equilibrium these prices must equal the respective marginal costs mc and mc; remember that marginal cost equals factor 'price' divided by factor marginal productivity (because increasing the production of good by one very small unit through an increase of the employment of factor j requires increasing the factor employment by 1/mp and thus increasing the cost by w/mp, and through the condition of cost minimization that marginal products must be proportional to factor 'prices' it can be shown that the cost increase is the same if the output increase is obtained by optimally varying all factors). optimal factor employment by a price-taking firm requires equality of factor rental and factor marginal revenue product, w=pmp, so we obtain p=mc=w/mp, p=mc=w/mp.
perfect competition