Suppose we have a Solow model:
$$ Y(t)=C(t)+I(t) $$
$$ I(t)=sY(t) $$
$$ \dot K=I(t)-δK(t) $$
With a given Cobb-Douglas:
$$ Y(t)=Z(t)K^aL^{1-a} $$
$$ y(t)=Y(t)/L(t) $$
$$ k(t)=K(t)/L(t) $$
$$ y=Zk^a $$
We also know these:
$$ L=L(t), \dot L/L=n $$
$$ Z=Z(t), \dot Z/Z=g $$
Therefore we have:
$$ L(t)=L(0)e^{nt} $$
$$ Z(t)=Z(0)e^{gt} $$
We get to this point:
$$ \dot k=sZk^a-(n+δ)k $$
In the steady state:
$$ k=k^* $$
$$ k^*=\left(\frac{sZ}{n+δ}\right)^{1/(1-a)} $$
The question has to do with the time dimension in Z. Time enters the final expression if we substitute Z with what we found above, but I'm not sure what to make of this.
$$ k^*=\left[\frac{sZ(0)e^{gt}}{n+δ}\right]^{1/(1-a)} $$
Surely there's something this neophyte is mixing up. I thought that in a steady state capital per worker remains the same.