Editing Production function (section)

===Homogeneous and homothetic production functions===
There are two special classes of production functions that are often analyzed. The production function <math>Q = f(X_1,X_2,\dotsc,X_n)</math> is said to be [[Homogeneous function|homogeneous]] of degree <math>m</math>, if given any positive constant <math>k</math>, <math>f(kX_1, kX_2,\dotsc,kX_n) = k^m f(X_1, X_2,\dotsc,X_n)</math>. If <math>m>1</math>, the function exhibits [[Economy of scale|increasing]] [[returns to scale]], and it exhibits [[Diseconomy of scale|decreasing]] returns to scale if <math>m < 1</math>. If it is homogeneous of degree <math>1</math>, it exhibits '''constant''' returns to scale. The presence of [[Returns to scale#Economies of scale|increasing returns]] means that a one percent increase in the usage levels of all inputs would result in a greater than one percent increase in output; the presence of decreasing returns means that it would result in a less than one percent increase in output. Constant returns to scale is the in-between case. In the Cobb–Douglas production function referred to above, returns to scale are increasing if <math>a_1+a_2+\dotsb+a_n > 1</math>, decreasing if <math>a_1+a_2+\dotsb+a_n < 1</math>, and constant if <math>a_1+a_2+\dotsb+a_n = 1</math>.

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. Moreover, in this case, if each input is paid at a rate equal to its marginal product, the firm's revenues will be exactly exhausted and there will be no excess economic profit.<ref>Chiang, Alpha C. (1984) ''Fundamental Methods of Mathematical Economics'', third edition, McGraw-Hill.</ref>{{rp|pp.412–414}}

Homothetic functions are functions whose marginal technical rate of substitution (the slope of the [[isoquant]], a curve drawn through the set of points in say labour-capital space at which the same quantity of output is produced for varying combinations of the inputs) is homogeneous of degree zero. Due to this, along rays coming from the origin, the slopes of the isoquants will be the same. Homothetic functions are of the form <math>F(h(X_1, X_2))</math> where <math>F(y)</math> is a monotonically increasing function (the derivative of <math>F(y)</math> is positive (<math> \mathrm{d}F/\mathrm{d}y >0 </math>)), and the function <math> h(X_1, X_2) </math> is a homogeneous function of any degree.