Answer

Solution

1. 1

   Steady-state per-capita consumption: $c^{*}(s)=(1-s)f(k^{*}(s))=f(k^{*})-\delta k^{*}$. The Golden Rule condition is $f'(k^{*})=\delta$. With $f(k)=k^{1/3}$, $f'(k)=\tfrac{1}{3}k^{-2/3}=\delta$. Combined with $sf(k^{*})=\delta k^{*}$:

2. 2
   $$s = \frac{\delta k^{*}}{f(k^{*})}=\delta\cdot k^{*2/3}=\delta\cdot\frac{1}{3\delta}=\frac{1}{3}.$$
3. 3

   This is the standard result that the Golden Rule saving rate equals the capital share $\alpha=\tfrac{1}{3}$ in a Cobb--Douglas economy.

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