just economic interpretation: Is every steady state a golden rule steady state?

Question

Is every steady state a golden rule steady state in solow model?

I know this question’s exact answer. I only want to ask for the following,

When know that at steady state $\dot{k}=0$

And in solow swan model $$\dot{k}= sk_t^a-(\delta + n+g)k_t$$

Since this equation is zero, I obtain the $$k_{ss}=(\frac{s}{\delta + n+ g})^{1/1-a}$$

When I maximize $$c_{ss}=f(k_{ss})-(\delta + n+g)k_{ss}$$ with respect to $k_{ss}$

I obtain the golden rule level $k_{gr}=(\frac{a}{\delta + n+ g})^{1/1-a}$

So,

$$k_{gr}=k_{ss} \iff a=s$$

That’s, saving rate is equal to capital share of income.

Otherwise, under different conditions, they are not the same.

What i want to ask that to what extent such a expression in the view of an economist is logical? This is true in terms of mathematical expression, but is it logical in terms of economic intuition as well? If correct, What does this result say to me intuitively?

Answer 1

In the Solow-Swan model, no optimization takes place with respect to the savings rate. It is treated as exogenous. The result found by the OP says that, if the saving rate $s$ is set equal to the capital share of income $a$, then it will so happen that the steady state capital will be equal to the golden rule capital.

We know that under intertemporal optimization with discounting of the future (Ramsey model), the optimal (in terms of intertemporal utility maximization) savings rate will be variable and at the steady state will be less than the level required to have the steady state capital equal to the golden rule capital.

Returning to the Solow model, the result $s=a <=> k_{ss} = k_{gr}$ has a certain intuitive appeal: we get maximum steady-state consumption if we are investing (and not consuming) all rewards to capital.

Answer 2

No. There are many steady state levels of per-capita consumption. The maximum of these is then defined to be the golden rule (steady state level of per-capita consumption).