# Solow Model: Steady state when there's population and technology growth

How is the steady state of a Solow Model defined when there's population growth and technological growth? The usual definition is to have $$\dot{k} = 0$$ (or $$k_{t+1} = k_t$$ in case of discrete time) and that would be equivalent to having $$\dot{K} = 0$$.

However, when there's population and technological growth, $$\dot{k} = 0 \rlap{\quad\not}\implies \dot{K} = 0.$$

Someone help me with the definition. Cite an article, paper or a book please.

• In the case with population growth, $\dot{k}=0$ implies $\dot{\left(\frac{K}{N}\right)}=0$. Although the capital-labor ratio does not change in the steady state, the level of capital $K$ can grow with the same rate of growth as $N$. Commented Mar 28 at 17:52
In the case of population growth, $$\dot{k}=0$$ implies $$\dot{\left(\frac{K}{N}\right)}=0$$. If $$k=K/N$$, then by taking logs $$\ln k = \ln K - \ln N$$ and differentiating w.r.t time $$\frac{\dot{k}}{k} = \frac{\dot{K}}{K} - \frac{\dot{N}}{N}$$ At the steady state, the LHS = 0, so on the RHS, the growth rate of capital $$\frac{\dot{K}}{K}$$ must equal the growth rate of labor $$\frac{\dot{N}}{N}$$.
Although the capital-labor ratio $$k$$ does not change in the steady state, the level of capital $$K$$ expands at the same rate as the labor force $$N$$. This is called "capital widening".
• Hi, for the past few days, I was unable to comment as I lost access to my account (steadystate). I was waiting for my "reputation" (as they call it) on my new account (basicsubset) to be 50 in order to post a reply to your answer. First of all, thank you. From what I understand, the $\dot{k} = 0$ defines the steady state. Could you please confirm the same? Could you please additionally explain me how balanced growth is related to this? Commented Apr 3 at 5:56
• Hi. Indeed, $\dot{k}=0$ defines the steady state of the capital-labor ratio. In this simple example, there is no technological progress, so the balanced growth is zero (or you can say the growth rate of output per capita is zero in the steady state/ on the BGP). You can read more about BGP here (written by Vollrath, who co-authored with Jones C.I. in the third edition of the book I mentioned). It is a very elucidated read. Commented Apr 5 at 3:29
• Thank you @teddi. I just checked it. I couldn't conclude anything to be honest. They say that BGP is defined by four characteristics, which are the stability of $g_y, I/Y, K/Y, \alpha_L$. What does stable mean here? Variables being constants? In the steady state, these characteristics are equivalent to one another. Is it the same here? I made a post here regarding this. Commented Apr 5 at 3:46