I think I have an error in my calculation, because I should get $\displaystyle s= \alpha$ at the end but I do not. Can someone please check to see where I've gone wrong?

Suppose you have: $\displaystyle c=(1-s) \left[ \frac{s}{n+x+\delta} \right]^\frac{\alpha}{1-\alpha}$

Rewrite as

$\displaystyle

c=(1-s)s^\frac{\alpha}{1-\alpha} \left[ \frac{1}{n+x+\delta} \right]^\frac{\alpha}{1-\alpha}

$

Expand the brackets:

$\displaystyle

c=(s^\frac{\alpha}{1-\alpha}-s^\frac{1}{1-\alpha}) \left[ \frac{1}{n+x+\delta} \right]^\frac{\alpha}{1-\alpha}

$

To find golden rule consumption, take the partial:

$\displaystyle

\frac{\partial c}{\partial s}=\left( \frac{\alpha}{1-\alpha}s^\frac{2 \alpha -1}{1-\alpha}-\frac{1}{1-\alpha}s^\frac{\alpha}{1-\alpha} \right) \left[ \frac{1}{n+x+\delta} \right]^\frac{\alpha}{1-\alpha}

$

Set that beastie equal to zero so we can find the local max:

$\displaystyle

\frac{\partial c}{\partial s}=\left( \frac{\alpha}{1-\alpha}s^\frac{2 \alpha -1}{1-\alpha}-\frac{1}{1-\alpha}s^\frac{\alpha}{1-\alpha} \right) \left[ \frac{1}{n+x+\delta} \right]^\frac{\alpha}{1-\alpha} = 0

$

Divide through by $\displaystyle \left[ \frac{1}{n+x+\delta} \right]^\frac{\alpha}{1-\alpha}$ to get

$\displaystyle

\left( \frac{\alpha}{1-\alpha}s^\frac{2 \alpha -1}{1-\alpha}-\frac{1}{1-\alpha}s^\frac{\alpha}{1-\alpha} \right)= 0

$

$\displaystyle

\frac{\alpha}{1-\alpha}s^\frac{2 \alpha -1}{1-\alpha} = \frac{1}{1-\alpha}s^\frac{\alpha}{1-\alpha}

$

Multiply both sides by $\displaystyle 1-\alpha$

$\displaystyle

\alpha s^\frac{2 \alpha -1}{1-\alpha} = s^\frac{\alpha}{1-\alpha}

$

Raise both sides to the power of $\displaystyle 1-\alpha$

$\displaystyle

\alpha s^{2 \alpha -1} = s^{\alpha}

$

Multiply both sides by $\displaystyle s^{-\alpha}$

$\displaystyle

\alpha s^{\alpha -1} = 1

$

$\displaystyle

s^{\alpha -1} = \frac{1}{\alpha }

$

And finish by raising both sides by $\displaystyle \frac{1}{\alpha -1}$

$\displaystyle

s = \left( \frac{1}{\alpha } \right)^\frac{1}{\alpha -1}

$

We can rewrite as

$\displaystyle

s = \alpha^\frac{1}{1- \alpha}

$

But that hardly makes it look closer to $\displaystyle s=\alpha$, which I think is the right answer. What have I done wrong?