## 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.

f (λ x, λ y) = λ

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) = 2x

f (λ x,λ y) = 2 (λx)

= 2λ

= λ

= λ

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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**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) = λ

^{n}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) = 2x

^{2}y + xy^{2}– y^{3}, thenf (λ x,λ y) = 2 (λx)

^{2}(λy) + (λx) (λy)^{2}- (λy)^{3}= 2λ

^{3}x^{2}y + λ^{3}xy^{2}- λ^{3}y^{3}= λ

^{3}(2x^{2}y + xy^{2}- y^{2})= λ

^{3}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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