Homogeneous and Homothetic Production Functions
There are two special classes of production functions that are often analyzed. The production function is said to be homogeneous of degree n, if given any positive constant, . If, the function exhibits increasing returns to scale, and it exhibits decreasing returns to scale if . If it is homogeneous of degree 1, it exhibits constant returns to scale. The presence of 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, decreasing if, and constant if .
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.
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 where is a monotonically increasing function (the derivative of is positive ), and the function is a homogeneous function of any degree.
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“If we Americans are to survive it will have to be because we choose and elect and defend to be first of all Americans; to present to the world one homogeneous and unbroken front, whether of white Americans or black ones or purple or blue or green.... If we in America have reached that point in our desperate culture when we must murder children, no matter for what reason or what color, we don’t deserve to survive, and probably won’t.”
—William Faulkner (1897–1962)
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