(1) $ C = (1-s)Y $
(2) $ \frac{C}{L} = (1-s)\left(\frac{Y}{L}\right) $
Along an economy's balanced-growth path:
(3) $ \left(\frac{Y}{L}\right)^{*} = {\left(\frac{K}{Y}\right)^{*}}^\left(\frac{\alpha} {1-\alpha}\right)E $
(4) $ \left(\frac{Y}{L}\right)^{*} = \left(\frac{s}{n+g+\delta}\right)^\left(\frac{\alpha} {1-\alpha}\right)E $
(5) $ \left(\frac{C}{L}\right)^{*} = \left(1-s\right) \left(\frac{s}{n+g+\delta}\right)^\left(\frac{\alpha} {1-\alpha}\right)E $
(6) $ ln\left(\frac{C}{L}\right)^{*} = ln\left(1-s\right) + \left(\frac{\alpha}{1-\alpha}\right)ln(s) - \left(\frac{\alpha}{1-\alpha}\right)ln(n+g+\delta) + ln(E) $
(7) $ \frac{d\left(ln\left(\frac{C}{L}\right)^{*}\right)} {ds} = \frac{-1}{1-s} + \left(\frac{\alpha}{1-\alpha}\right)\left(\frac{1}{s}\right) $
(8) $ \frac{d\left(ln\left(\frac{C}{L}\right)^{*}\right)} {ds} = 0 ⇒ \frac{1}{1-s} = \left(\frac{\alpha}{1-\alpha}\right) $
(9) $ s = \alpha ⇒ \left(\frac{C}{L}\right)^{*} $ is maximized
What then happens to (C/L)* and (Y/L)*?