Maximizing Function (Something to Do with Roth-Nash)

[MATHS]

I want to know how to maximize P in the following expression:

P such that Max [ (V-B-P)^(a) * (P-W)^(1-a) ]

It is present in a paper I am reading and I've tried several times to derive the solution, which is supposedly:

P=(1-a)*(V-B) + aW

I tried calculated the derivate with respect to P and going that route...but I haven't done this sort of thing in a while so I'm stuck.

I do know that when this value of P is entered back into the expression, the expression takes the interesting form:

(aV-aB-aW)^(a) * ([1-a]V-[1-a]B-[1-a]W)^(1-a)

Can anybody show me the way?

Thanks

M.A.T.H.S.

 

[Deleted member 4993]

I want to know how to maximize P in the following expression:

P such that Max [ (V-B-P)^(a) * (P-W)^(1-a) ]

It is present in a paper I am reading and I've tried several times to derive the solution, which is supposedly:

P=(1-a)*(V-B) + aW

I tried calculated the derivate with respect to P and going that route...but I haven't done this sort of thing in a while so I'm stuck.

I do know that when this value of P is entered back into the expression, the expression takes the interesting form:

(aV-aB-aW)^(a) * ([1-a]V-[1-a]B-[1-a]W)^{(1-a)}

Can anybody show me the way?

Thanks

M.A.T.H.S.

It is fairly straight-forward but a little algebra-intensive.

Assume

\(\displaystyle f(P)\, = \, (V - B - P)^a\cdot(P-W)^{(1-a)}\)

Now take 'ln' of both sides - differentiate - equate to zero - simplify - and you are there....

show us what you got - so that we know where to begin to help you.