# Thread: Difficult maximisation problem

1. ## Difficult maximisation problem

We need to find the value of X that maximizes U,

U= [(w*X)^(1-a)-1]/(1-a)*[(s*(1-X))^(1-b)-1]/(1-b)

where 0<X<1 w>1 s>1

For special cases of a=1, we have as the first term Ln(wX); for b=1 we have Ln(s(1-X)
We are allowed to assume w and s are large such that both terms are positive

Taking the first differential using the product rule I get:

dU/dX= 0= w(w*X)^-a * [(s*(1-X))^(1-b)-1]/(1-b) +
s(s*(1-X))^-b *[(w*X)^(1-a)-1]/(1-a)

Which doesn't really seem to get me very far!

Any idea of alternative approaches or next steps ?

We can get a neater solution under special conditions w=S -> infinity; a>1; b>1, which are acceptable approximations

0 = [X^-a / (b-1)] + [(1-X)^-b / (a-1)]

X^-a / (1-b) = -(1-X)^-b / (1-a)

X^-a * (a-1) = (1-X)^-b *(b-1)

But even here I am still stuck on how to proceed.

2. ## Re: Difficult maximisation problem

Update

The following seems to give a good approximation for the last problem but I don't know why:

X= 1-1/[(a-b)(a-1)+2] as a-b -> 0 for small values of a-b the solution is almost perfect

3. ## Re: Difficult maximisation problem

It appears that there is actually no analytic solution to this problem.