# Is the steady state of $k$ enough to show $Y$ grows at the rate $n+g$?

To provide some context for the question: I was approached by a student needing help for a an empirical thesis based on Solow-Swan model. She had trouble solving the theoretical model she wanted to showcase in her work. She already put a lot of efforts and had a problem just with last steps of derivations so I helped her to solve simplified version of Solow-Swan model where:

$$k^*= \left( \frac{s}{d+ n + g} \right)^{\frac{1}{1-\gamma}}$$

where $$k$$ is the capital per effective labor, $$n$$ population growth and $$g$$ technology growth and $$d$$ depreciation, and told her that this results implies that $$K$$ is growing at the rate $$n+g$$ which implies due to the assumption of constant returns that $$Y$$ grows at the same rate. Since $$g$$ and $$n$$ are exogenous the growth must be exogenous as well.

But her supervisor told her that the above is not enough to show that economic growth is driven by exogenous factors and she has to expand her model.

Hence my question is: was my statement incorrect or is her supervisor making mistake? If my advice was incorrect whats the missing link that unambiguously shows that economic growth depends on $$n+g$$?

• Strange I woulden't expect this comment from an advisor given the model is so canonical.
– EconJohn
Nov 18, 2020 at 1:18
• @EconJohn yes I think that her advisor was just being purposely difficult but then my imposter syndrome kicked in so I needed to confirm this
– 1muflon1
Nov 18, 2020 at 1:19
• Are you sure its not qualms about data?
– EconJohn
Nov 18, 2020 at 1:27
• @EconJohn well I never spoken to that advisor directly but afterwards I helped the girl to put there more details (if I recall correctly we added about 3 more equations) and later it was ok and she passed, although she might have also some issues with empirical part the models did had some problems here and there. Like there was likely some overfitting on some subsamples...
– 1muflon1
Nov 18, 2020 at 1:35

Based on the steady state your production function is Cobb-Douglas.Taking logs and derivatives wrt time of $$Y$$, $$\frac{Y}{L}$$ and $$\frac{Y}{AL}$$ in the steady state yields the desired result: $$K$$ grows with $$n+g$$ on the BGP.
• Yes you are correct the production function was standard Cobb Douglas everything was expressed per unit of effective labor so it was $f(k)=k^{\alpha}$ she showed me his email with comments and he literary said that she puts too much effort in deriving steady state but no effort in showing the growth depends on those variables. Should I just tell her to explicitly add the time derivatives?