Exercise
Determine if the following function:
f(x,y)=x^y
Is homogeneous.
Final Answer
Solution
We look at the function:
f(x,y)=x^y
By definition, a function is homogeneous of degree n if and only if the following holds:
f(tx,ty)=t^nf(x,y)
For a parameter t.
Therefore,
f(tx,ty)=
We plug in our function and get
={(tx)}^{ty}=
We open brackets and get
=t^{ty}x^{ty}=
=t^{ty}{(x^y)}^t=
And we got the following:
\neq t^nf(x,y)
For any t and any n.
In short, we got the following:
f(tx,ty)\neq t^nf(x,y)
Hence, by definition, the given function is not homogeneous.
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