Homogeneity and derivatives

[Sidney]

I've been reading a book on economics and they defined a homogeneous function as: ƒ(x₁,x₂,…,x_n) such that
ƒ(tx₁,tx₂,…,tx_n)=t^kƒ(x₁,x₂,…,x_n) ..totally understandable.. they further explained that a direct result from this is that the partial derivative of such a function will be homogeneous to the degree k-1.They proved this by simply differentiating both sides of the equation. My problem arises when they differentiate the left hand side (with respect to the first argument as an arbitrary choice). They say the partial differential(of the LHS wrt x₁) is:

(∂ƒ(tx₁,tx₂,…,tx_n)/∂x₁).t

my question is where does the t come from.. ..please bear with me

 

[Mentallic]

They used the chain rule. When you take a derivative of

$$f(g(x))$$ with respect to x, you first take the derivative of f, but then you need to multiply by the derivative of g.

$$\frac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)$$

So

$$\frac{d}{dx}f(tx)=f'(tx)\cdot \frac{d}{dx}tx = f'(tx)\cdot t$$

assuming t is independent of x (constant).

 

[Sidney]

thank you :) I don't know why it seems so obvious now ..I did think of the t as being a result of the chain rule but for some reason the way they wrote it down made no sense to me and had me stuck...I think it's because they wrote (∂ƒ(tx1,tx2,…,txn)/∂x1) which to me means with respect to x( i.e. ∂x1) and not with respect to the change in the intermediate function(tx) and so it came across as meaning the complete derivative of ƒ₁ encompassing all the intermediate processes..

the way you have your functions written down is so neat. If you don't mind me asking what did you use because the way I'm doing it takes forever, is very messy and I can't write in fraction form

 

[Mentallic]

thank you :) I don't know why it seems so obvious now ..I did think of the t as being a result of the chain rule but for some reason the way they wrote it down made no sense to me and had me stuck...I think it's because they wrote (∂ƒ(tx1,tx2,…,txn)/∂x1) which to me means with respect to x( i.e. ∂x1) and not with respect to the change in the intermediate function(tx) and so it came across as meaning the complete derivative of ƒ₁ encompassing all the intermediate processes..

Things often become clear again when it's explained in simple terms :)

the way you have your functions written down is so neat. If you don't mind me asking what did you use because the way I'm doing it takes forever, is very messy and I can't write in fraction form

Check out this page:
https://www.physicsforums.com/threads/introducing-latex-math-typesetting.8997/

And what you can also do to help speed up the learning process is to quote a post and observe what the poster had written in their latex.

 

[CellCoree]

..

I would like to clarify that homogeneity and derivatives are important concepts in mathematics and economics. A function is considered homogeneous when it satisfies the property that when all its inputs are multiplied by a constant, the output is also multiplied by that constant raised to a certain power. In this case, the power is represented by the variable k.

The statement that the partial derivative of a homogeneous function will have a degree of k-1 is correct and can be proven by differentiating both sides of the equation. However, the confusion arises when the book introduces the t in the partial derivative of the left-hand side with respect to the first argument.

The t represents the constant that is multiplied to the first argument of the function. This is because when we differentiate with respect to x1, we are treating x2, x3, etc. as constants. Therefore, when the function is multiplied by t, the derivative becomes t times the partial derivative of the function with respect to x1. This is a standard mathematical technique used when differentiating with respect to a variable that is multiplied by a constant.

I hope this helps clarify the concept of homogeneity and derivatives. It is important to understand these concepts in order to fully grasp the concepts of economics and other fields that use mathematical models.