Reduce a system of nonlinear equations (CES utility maximization in economics)

Question

I need to reduce a simple system of 3 equations in 3 variables. Reduce can't handle it directly, but works if I combine the first two equations.

Here is the code:

Reduce[{z == x^a, z == y^a, x + y == b, a > 0, b > 0, x > 0, y > 0, z > 0}, 
    {x, y, z} , Reals, Backsubstitution ->True]

Reduce[{x^a == y^a, x + y == b, a > 0, b > 0, x > 0, y > 0}, 
{x,y} , Reals, Backsubstitution -> True]

It's even easy to solve by hand! I spent days adding assumptions without success.

I need to reduce the full 3 equation system for the toolkit I'm creating. Is there a way to reduce the full system by adding options or assumptions?

I'm writing a public Mathematica toolkit that provides analytical solutions to a wide set of problems in economics and finance. Many problems in economics and finance (iso elastic utility aka CES utility) require solving systems such as the one above. I need Reduce[] and not Solve[] because I need to add (linear) inequalities.