<p>In the case of population growth, <span class="math-container">$\dot{k}=0$</span> implies <span class="math-container">$\dot{\left(\frac{K}{N}\right)}=0$</span>. If <span class="math-container">$k=K/N$</span>, then by taking logs
<span class="math-container">$$
 \ln k = \ln K - \ln N
$$</span>
and differentiating w.r.t time
<span class="math-container">$$
\frac{\dot{k}}{k} = \frac{\dot{K}}{K} - \frac{\dot{N}}{N}
$$</span>
At the steady state, the LHS = 0, so on the RHS, the growth rate of capital <span class="math-container">$\frac{\dot{K}}{K}$</span> must equal the growth rate of labor <span class="math-container">$\frac{\dot{N}}{N}$</span>.</p>
<p>Although the capital-labor ratio <span class="math-container">$k$</span> does not change in the steady state, the level of capital <span class="math-container">$K$</span> expands at the same rate as the labor force <span class="math-container">$N$</span>. This is called &quot;capital widening&quot;.</p>
<p>Source: Jones C.I. &quot;Introduction to economic growth&quot; (1997) - page 25</p>

CC BY-SA 4.0
2024-04-03T05:56:29.607Z
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?
uninterestedacademic
CC BY-SA 4.0
2024-04-05T03:29:07.870Z
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](https://growthecon.com/StudyGuide/solow/simbgp.html#balanced-growth-and-transitional-growth) (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.
teddi
CC BY-SA 4.0
2024-04-05T03:46:14.190Z
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](https://economics.stackexchange.com/questions/58018/balanced-growth-solow-model) regarding this.
uninterestedacademic
CC BY-SA 4.0
2024-03-30T12:39:10.523Z
2024-03-30T12:39:10.523Z
teddi
<p>How is the steady state of a Solow Model defined when there's population growth and technological growth? The usual definition is to have <span class="math-container">$\dot{k} = 0$</span> (or <span class="math-container">$k_{t+1} = k_t$</span> in case of discrete time) and that would be equivalent to having <span class="math-container">$\dot{K} = 0$</span>.</p>
<p>However, when there's population and technological growth, <span class="math-container">$$\dot{k} = 0 \rlap{\quad\not}\implies \dot{K} = 0.$$</span></p>
<p>Someone help me with the definition. Cite an article, paper or a book please.</p>

CC BY-SA 4.0
2024-03-28T17:52:06.797Z
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 $N$.
teddi
CC BY-SA 4.0
2024-03-28T19:08:17.540Z
@teddi Please post answers as answers; also OP asks for a reference.
Giskard
<macroeconomics><solow>
Solow Model: Steady state when there's population and technology growth
steadystate
# Solow Model: Steady state when there's population and technology growth

(Asked by: username_0 on 2024-03-28)

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.

**Tags:** macroeconomics, solow

> **Comments:**
> **username_1**: 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 $N$.  
> **username_2**: @username_1 Please post answers as answers; also OP asks for a reference.  

---

## 1 answers

### Answer by username_1 on 2024-03-30. Score: 4

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".

Source: Jones C.I. "Introduction to economic growth" (1997) - page 25

> **Comments:**
> **username_3**: Hi, for the past few days, I was unable to comment as I lost access to my account (*username_0*). 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?  
> **username_1**: 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](https://growthecon.com/StudyGuide/solow/simbgp.html#balanced-growth-and-transitional-growth) (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.  
> **username_3**: Thank you @username_1. 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](https://economics.stackexchange.com/questions/58018/balanced-growth-solow-model) regarding this.  

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