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If a production function is homogeneous of degree one, it is sometimes called "linearly homogeneous". A linearly homogeneous production function with inputs capital and labour has the properties that the marginal and average physical products of both capital and labour can be expressed as functions of the capital-labour ratio alone.
Wire-grid Cobb–Douglas production surface with isoquants A two-input Cobb–Douglas production function with isoquants. In economics and econometrics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs (particularly physical capital and labor) and ...
The production functions listed below, and their properties are shown for the case of two factors of production, capital (K), and labor (L), mostly for heuristic purposes. These functions and their properties are easily generalizable to include additional factors of production (like land, natural resources, entrepreneurship, etc.)
The CES production function is a neoclassical production function that displays constant elasticity of substitution. In other words, the production technology has a constant percentage change in factor (e.g. labour and capital) proportions due to a percentage change in marginal rate of technical substitution.
Two input Leontief Production Function with isoquants. In economics, the Leontief production function or fixed proportions production function is a production function that implies the factors of production which will be used in fixed (technologically predetermined) proportions, as there is no substitutability between factors.
If the production set Y can be represented by a production function F whose argument is the input subvector of a production vector, then increasing returns to scale are available if F(λy) > λF(y) for all λ > 1 and F(λy) < λF(y) for all λ<1. A converse condition can be stated for decreasing returns to scale.
The point of diminishing returns can be realised, by use of the second derivative in the above production function. Which can be simplified to: Q= f(L,K). This signifies that output (Q) is dependent on a function of all variable (L) and fixed (K) inputs in the production process. This is the basis to understand.
A firm's production function could exhibit different types of returns to scale in different ranges of output. Typically, there could be increasing returns at relatively low output levels, decreasing returns at relatively high output levels, and constant returns at some range of output levels between those extremes.