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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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.
For more help in Homogeneous Functions And Euler’s Theorem click the button below to submit your homework assignment