<p>To answer the question it is necessary to remember the <strong>behaviour of the variables in the steady state</strong>, so of course it is necessary to have solved previously the model for the steady state.<span class="math-container">$^1$</span></p>
<p>In your question we have the <strong>Solow model with labour augmenting technical</strong> progress, that is the production function is</p>
<p><span class="math-container">$Y_t=f(K_t,A_t L_t)$</span></p>
<p>where <span class="math-container">$A_t$</span> represents technical progress and <span class="math-container">$A_t L_t$</span> is the so-called <em>effective labour</em>.</p>
<p><strong>Solving the model for the steady state</strong>, we obtain that <strong>the variables per effective labour are constant</strong>, that is <span class="math-container">$Z_t\equiv \frac {K_t}{A_tL_t}= \overline {Z}$</span> and  <span class="math-container">$y_t \equiv \frac {Y_t}{A_tL_t}$</span> are constant, where <span class="math-container">$\overline {Z}$</span> is the constant steady state value of <span class="math-container">$Z_t$</span>.</p>
<p>Therefore we have, in steady state:</p>
<p><span class="math-container">$y_t\equiv \frac {Y_t}{A_tL_t}= f(\overline {Z})\;\;\;\;\quad (1)$</span></p>
<p>from which</p>
<p><span class="math-container">$ Q_t\equiv \frac {Y_t}{L_t}= A_tf(\overline {Z})\qquad (2)$</span></p>
<p>But, by assumption in the model, we have <span class="math-container">$^2$</span>:</p>
<p><span class="math-container">$A_t= A_0 e^{gt}\qquad \;\;\;\;\;\;\;\;\;\;\;(3)$</span></p>
<p>where <span class="math-container">$g$</span> is the rate of growth of technology <span class="math-container">$A_t$</span> and <span class="math-container">$A_0$</span> is the initial value of <span class="math-container">$A_t$</span>.</p>
<p>Therefore, substituting in <span class="math-container">$(2)$</span>, we obtain:</p>
<p><span class="math-container">$ Q_t\equiv \frac {Y_t}{L_t}= A_0 f(\overline {Z})e^{gt}.\qquad (4)$</span>
<span class="math-container">$$\;$$</span></p>
<hr />
<p><span class="math-container">$^1$</span> Of course, the solution  of the model is too long to be reported here, you can look at a good text of macroeconomics, as Romer, <em>Advanced macroeconomics</em>  or at some standard references on economic growth, as the books of Acemoglu or Barro &amp; Sala i Martin.</p>
<p><span class="math-container">$^2$</span> In the model there is the assumption that <span class="math-container">$A_t$</span> grows at constant rate <span class="math-container">$g$</span>. Remember, very important, that the exponential form with <span class="math-container">$e^{at}$</span>, as in <span class="math-container">$(3)$</span>, is the mathematical expression to denote a growth at constant rate <span class="math-container">$a$</span> of a variable. If you calculate the growth rate in that case you can realize it.</p>

CC BY-SA 4.0
2023-03-27T23:58:12.983Z
Thanks a lot for the effort. (The website doesn't allow me to upvote you, so sorry about that, but your answer is extremely helpful.)
imbalancedgrowth
CC BY-SA 4.0
2023-03-27T13:33:21.623Z
2023-03-28T01:00:55.137Z
2023-03-28T01:00:55.137Z
BakerStreet
<blockquote>
<p>Suppose a Solow-Swan model with <span class="math-container">$Y_t = F(K,AL)$</span> as the production function. Suppose the growth rate of <span class="math-container">$A$</span> and <span class="math-container">$L$</span> are <span class="math-container">$g &gt; 0$</span> and <span class="math-container">$n &gt; 0$</span> respectively. It is also given that <span class="math-container">$\dot{K} = sY - \delta K$</span> where <span class="math-container">$\delta$</span> is the rate of depreciation and <span class="math-container">$s$</span> is the savings rate. If <span class="math-container">$Z = \frac{K}{AL}$</span> and <span class="math-container">$Q_t = \frac{Y_t}{L_t}$</span>, show that <span class="math-container">$\overline{Q_t} = A_t f(\overline{Z}) e^{gt}$</span> where <span class="math-container">$\overline{Q}$</span> is the balanced growth path value of <span class="math-container">$Q$</span>, <span class="math-container">$\overline{Z}$</span> is the steady state value of <span class="math-container">$Z$</span> and <span class="math-container">$f(Z) := F(Z,1)$</span>.</p>
</blockquote>
<p><strong>Attempt:</strong></p>
<p><span class="math-container">$Q_t = A_t \frac{F(K_t,A_tL_t)}{A_tL_t} = A_t f(Z_t) = A_t f(Z_t)$</span>. At the steady state, <span class="math-container">$\frac{\dot{Z}}{Z} = \frac{s f(Z)}{Z} - (n + \delta + g) = 0 \implies f(\overline{Z}) = \frac{n+\delta+g}{s}\overline{Z}$</span>.</p>
<p>Along the balanced growth path, let's assume <span class="math-container">$\frac{\dot{Y}}{Y} = g_Y$</span>.</p>
<p>How do I proceed from here?</p>

CC BY-SA 4.0
2023-03-27T11:49:37.930Z
Are you sure that in the text is $\overline{Q_t} = A_t f(\overline{Z}) e^{gt}$  and not $\overline{Q_t} = A_0 f(\overline{Z}) e^{gt}$ , where $A_0$ is the initial value of $A_t$?
BakerStreet
CC BY-SA 4.0
2023-03-27T11:51:00.113Z
@BakerStreet Exactly what came to my mind. I still don't know how to get there but I realized that $A_t = A_0 e^{gt}$ would probably mean that $\bar{Q_t} = A_0 e^{gt} f(\bar{Z})$. I don't understand the balanced growth path - as in, which all variables are supposed to be constants and which all should be equal to one another. Can you please suggest me a reference and maybe answer this too if time permits? Thank you.
imbalancedgrowth
CC BY-SA 4.0
2023-03-27T11:52:43.140Z
It could be a typo. $A_t$ is strange in this formula.
BakerStreet
CC BY-SA 4.0
2023-03-27T11:54:47.787Z
@BakerStreet Very possibly so. Please read my previous comment; I have edited it. You don't have to solve the problem; you can just add a sketch of how the variables should be like along the balanced growth path. Thanks.
imbalancedgrowth
CC BY-SA 4.0
2023-03-27T12:00:55.407Z
In the steady state the ratio $Y_t/A_tL_t$, that is the output per effective worker is constant. It means that the output per worker $Q_t$ grows at the rate of growth of $A_t$, that is $g$. The exponential form of $\overline {Q} = A_0f(\overline ({Z}) e^{gt}$ means exactly that.
BakerStreet
CC BY-SA 4.0
2023-03-27T12:14:04.167Z
In the steady state it is not true that all the variables are constant, only the per capita variables are constant. Without tecnical progress the output per capita is constant, with tecnical progress (labour augmenting as in your question) the output per *effective* labour $Y/(AL)$ is constant. So that the overall output  $Y_t$ is growing.
BakerStreet
CC BY-SA 4.0
2023-03-27T12:15:13.370Z
Standard references are the books on growth of Acemoglu or Barrow, and Sala i Martin,  or a good text of macroeconomics, as Romer. Your model is quite standard.
BakerStreet
CC BY-SA 4.0
2023-03-27T12:23:22.953Z
@BakerStreet Thanks for the suggestions. Can we say that $\frac{\dot{Y}}{Y} = \frac{\dot{K}}{K}$ in the steady state? Or is it that $\frac{\dot{Y}}{Y}$ and $\frac{\dot{K}}{K}$ are both constants, but not equal to one another?
imbalancedgrowth
CC BY-SA 4.0
2023-03-27T12:30:49.703Z
In the steady state all absolute varlables grow at rate of growth of population, if the model is without technical progress, or at the rate of growth of population plus the rate of growth  of technology if there is labour augmenting technical progress.
BakerStreet
CC BY-SA 4.0
2023-03-27T12:32:25.577Z
If I have time later I will give an answer, but unfortunatly I'm very busy, so I fear I can't.
BakerStreet
<macroeconomics><solow>
Solow-Swan Model: Show that $\overline{Q_t} = A_t f(\overline{Z}) e^{gt}$ along the balanced growth path
imbalancedgrowth
# Solow-Swan Model: Show that $\overline{Q_t} = A_t f(\overline{Z}) e^{gt}$ along the balanced growth path

(Asked by: username_0 on 2023-03-27)

> Suppose a Solow-Swan model with $Y\_t = F(K,AL)$ as the production function. Suppose the growth rate of $A$ and $L$ are $g > 0$ and $n > 0$ respectively. It is also given that $\dot{K} = sY - \delta K$ where $\delta$ is the rate of depreciation and $s$ is the savings rate. If $Z = \frac{K}{AL}$ and $Q\_t = \frac{Y\_t}{L\_t}$, show that $\overline{Q\_t} = A\_t f(\overline{Z}) e^{gt}$ where $\overline{Q}$ is the balanced growth path value of $Q$, $\overline{Z}$ is the steady state value of $Z$ and $f(Z) := F(Z,1)$.

**Attempt:**

$Q\_t = A\_t \frac{F(K\_t,A\_tL\_t)}{A\_tL\_t} = A\_t f(Z\_t) = A\_t f(Z\_t)$. At the steady state, $\frac{\dot{Z}}{Z} = \frac{s f(Z)}{Z} - (n + \delta + g) = 0 \implies f(\overline{Z}) = \frac{n+\delta+g}{s}\overline{Z}$.

Along the balanced growth path, let's assume $\frac{\dot{Y}}{Y} = g\_Y$.

How do I proceed from here?

**Tags:** macroeconomics, solow

> **Comments:**
> **username_1**: Are you sure that in the text is $\overline{Q_t} = A_t f(\overline{Z}) e^{gt}$  and not $\overline{Q_t} = A_0 f(\overline{Z}) e^{gt}$ , where $A_0$ is the initial value of $A_t$?  
> **username_0**: @username_1 Exactly what came to my mind. I still don't know how to get there but I realized that $A_t = A_0 e^{gt}$ would probably mean that $\bar{Q_t} = A_0 e^{gt} f(\bar{Z})$. I don't understand the balanced growth path - as in, which all variables are supposed to be constants and which all should be equal to one another. Can you please suggest me a reference and maybe answer this too if time permits? Thank you.  
> **username_1**: It could be a typo. $A_t$ is strange in this formula.  
> **username_0**: @username_1 Very possibly so. Please read my previous comment; I have edited it. You don't have to solve the problem; you can just add a sketch of how the variables should be like along the balanced growth path. Thanks.  
> **username_1**: In the steady state the ratio $Y_t/A_tL_t$, that is the output per effective worker is constant. It means that the output per worker $Q_t$ grows at the rate of growth of $A_t$, that is $g$. The exponential form of $\overline {Q} = A_0f(\overline ({Z}) e^{gt}$ means exactly that.  
> **username_1**: In the steady state it is not true that all the variables are constant, only the per capita variables are constant. Without tecnical progress the output per capita is constant, with tecnical progress (labour augmenting as in your question) the output per *effective* labour $Y/(AL)$ is constant. So that the overall output  $Y_t$ is growing.  
> **username_1**: Standard references are the books on growth of Acemoglu or Barrow, and Sala i Martin,  or a good text of macroeconomics, as Romer. Your model is quite standard.  
> **username_0**: @username_1 Thanks for the suggestions. Can we say that $\frac{\dot{Y}}{Y} = \frac{\dot{K}}{K}$ in the steady state? Or is it that $\frac{\dot{Y}}{Y}$ and $\frac{\dot{K}}{K}$ are both constants, but not equal to one another?  
> **username_1**: In the steady state all absolute varlables grow at rate of growth of population, if the model is without technical progress, or at the rate of growth of population plus the rate of growth  of technology if there is labour augmenting technical progress.  
> **username_1**: If I have time later I will give an answer, but unfortunatly I'm very busy, so I fear I can't.  

---

## 1 answers

### Answer by username_1 on 2023-03-27. Score: 2 | Accepted answer

To answer the question it is necessary to remember the **behaviour of the variables in the steady state**, so of course it is necessary to have solved previously the model for the steady state.$^1$

In your question we have the **Solow model with labour augmenting technical** progress, that is the production function is

$Y\_t=f(K\_t,A\_t L\_t)$

where $A\_t$ represents technical progress and $A\_t L\_t$ is the so-called *effective labour*.

**Solving the model for the steady state**, we obtain that **the variables per effective labour are constant**, that is $Z\_t\equiv \frac {K\_t}{A\_tL\_t}= \overline {Z}$ and $y\_t \equiv \frac {Y\_t}{A\_tL\_t}$ are constant, where $\overline {Z}$ is the constant steady state value of $Z\_t$.

Therefore we have, in steady state:

$y\_t\equiv \frac {Y\_t}{A\_tL\_t}= f(\overline {Z})\;\;\;\;\quad (1)$

from which

$ Q\_t\equiv \frac {Y\_t}{L\_t}= A\_tf(\overline {Z})\qquad (2)$

But, by assumption in the model, we have $^2$:

$A\_t= A\_0 e^{gt}\qquad \;\;\;\;\;\;\;\;\;\;\;(3)$

where $g$ is the rate of growth of technology $A\_t$ and $A\_0$ is the initial value of $A\_t$.

Therefore, substituting in $(2)$, we obtain:

$ Q\_t\equiv \frac {Y\_t}{L\_t}= A\_0 f(\overline {Z})e^{gt}.\qquad (4)$
$$\;$$

---

$^1$ Of course, the solution of the model is too long to be reported here, you can look at a good text of macroeconomics, as Romer, *Advanced macroeconomics* or at some standard references on economic growth, as the books of Acemoglu or Barro & Sala i Martin.

$^2$ In the model there is the assumption that $A\_t$ grows at constant rate $g$. Remember, very important, that the exponential form with $e^{at}$, as in $(3)$, is the mathematical expression to denote a growth at constant rate $a$ of a variable. If you calculate the growth rate in that case you can realize it.

> **Comments:**
> **username_0**: Thanks a lot for the effort. (The website doesn't allow me to upvote you, so sorry about that, but your answer is extremely helpful.)  

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