Homogeneous Functions And Eulers Theorem

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Homogeneous Functions And Euler’s Theorem

Many of the functions that are useful in economic analysis share the property of being homogeneous.

Definition. A function z = f (x,y) is said to be homogeneous of degree n ( n being a constant ) if , for any real number λ,

f (λ x, λ y) = λn f (x,y).

That is, if both x and y are multiplied by the same real number, then the resulting function value is a power of the number times the function value f (x,y).

For example, if f (x.y) = 2x2y + xy2 – y3, then
f (λ x,λ y)   = 2 (λx)2 (λy) + (λx) (λy)2 - (λy)3
                 = 2λ3 x2 y + λ3 xy2 - λ3 y3
                 = λ3 (2x2y + xy2 - y2)
                 = λ3 f (x,y) .

Thus f is homogeneous of degree 3.

Notice that the function f (x,y) is a polynomial in x and y such that the degree of each term is 3, which is the degree of homogeneity of the function. In general, we have the following remark for such functions.

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