So, this is an economics question but the problem I have is a pure math problem I guess. So I have the following equation:f(x,y)
this function have the elasticity of substitution(EOS): 1/(1-beta). a,b are constants >0 and beta is also a constant, but is ≤1. So, the general equation for EOS is the following: σ =−(∂log(x/y)) / (∂log((∂f(x,y)/∂x)/(∂f(x,y)/∂y)))
From some calculus I got following result for the inner part of the denominator(without the log or derivative): (∂f(x,y)/∂x)/(∂f(x,y)/∂y)
For the numerator, I rewrite the expression to this(since it appear to be the case that we want to have x/y explicit):x/y
Taking the logarithm of this gives me following result:log(x/y)
Putting these together and insert in to the formula gives me:σ (note that the index should not be i and j, but x and y, my fault)
Okay, so this is how far as I get. And what I don't get from here is the "∂"-part, what am I suppose to take the derivative off? With respect to what variable? Even though there can be som errors in my calculations leading to that expression I still don't understand how I can take the derivative of an expression like that. The main formula for me does not make sense.
Can someone please explain how I should continue?
(Sorry for a messy post, I could not figure out if there was an effective way of writing in calculations in the post so I went with pictures instead)
Kind regards, Fridein