<p>I think that the division by <span class="math-container">$y_t^\alpha$</span> is for the following specification (where technological progress is labour augmenting).</p>
<p><span class="math-container">$$
y_t = k_t^\alpha (Al_t)^{1 - \alpha}.
$$</span>
Taking logs gives:
<span class="math-container">$$
\ln(y_t) = \alpha \ln(k_t) + (1-\alpha) \ln(A_t) + (1-\alpha) \ln(l_t).
$$</span>
If we take the derivative with respect to time, we still have a factor <span class="math-container">$(1-\alpha)$</span> in front of the growth rate of technology <span class="math-container">$\dot A/A$</span>. To get rid of this, we can subtract <span class="math-container">$\alpha \ln(y_t)$</span> from both sides (i.e. divide the original by <span class="math-container">$y_t^\alpha$</span>):
<span class="math-container">$$
(1-\alpha) \ln(y_t) = \alpha \ln (k_t) - \alpha \ln(y_t) + (1-\alpha) \ln(A_t) + (1-\alpha) \ln(l_t)
$$</span>
Now divide both sides by <span class="math-container">$(1-\alpha)$</span>:
<span class="math-container">$$
\ln(y_t) = \frac{\alpha}{1-\alpha}\ln(k_t/y_t) + \ln(A_t) + \ln(l_t).
$$</span>
Then taking the derivative we get:
<span class="math-container">$$
\frac{\dot y_t}{y_t} = \frac{\alpha}{1-\alpha} \frac{\dot{(k_t/y_t)}}{(k_t/y_t)} + \frac{\dot A_t}{A_t} + \frac{\dot l_t}{l_t}
$$</span>
So the growth rate of technological progress is given by:
<span class="math-container">$$
\frac{\dot A_t}{A_t} = \frac{\dot y_t}{y_t} - \frac{\alpha}{1-\alpha} \frac{\dot{(k_t/y_t)}}{(k_t/y_t)} - \frac{\dot l_t}{l_t}
$$</span>
If the share of kapital over output, and labour remains constant over time, this means that technological growth is approximately equal to output growth.</p>

CC BY-SA 4.0
2021-11-09T15:49:20.033Z
I don't think that it is for this specification only...see appendix B of this paper, "Understanding Economic Growth in Ghana in Comparative Perspective". There are many like that in the literature. The argument is that TFP can also affect capital, but not sure why divide by y^alpha. In the literature, it is not model specific, like you suggest.
Emmanuel Ameyaw
CC BY-SA 4.0
2021-11-10T08:19:29.843Z
I think it depends on how you define $A$. See also [here](https://mpra.ub.uni-muenchen.de/69406/1/MPRA_paper_69406.pdf)
tdm
CC BY-SA 4.0
2021-11-10T12:49:12.207Z
Thanks. Nice article. As mentioned in there, the reason why they divide by y^alpha is that they wanna scale up the importance of TFP in the accounting process. I think that answers my question. I have to look for why they wanna scale TFP by a factor of 1/(1-alpha) and not some other factor though.
Emmanuel Ameyaw
CC BY-SA 4.0
2021-11-08T15:58:30.677Z
2021-11-08T15:58:30.677Z
tdm
<p>In <strong>standard approach</strong> to growth accounting, you decompose <span class="math-container">$y_t = A_t f(k_t, l_t)$</span> into TFP, labor and capital components.</p>
<p>In the <strong>capital-output approach</strong>, people make the argument that <span class="math-container">$A_t$</span> can also affect capital as in a solow model, where <span class="math-container">$A_t$</span> increases, which shifts the production function outward, and <span class="math-container">$k_t$</span> increases to a new steady state. So <span class="math-container">$A_t$</span> affects <span class="math-container">$k_t$</span>.</p>
<p>To account for the fact that <span class="math-container">$A_t$</span> affects <span class="math-container">$k_t$</span>, it is standard in the literature to divide both sides of <span class="math-container">$y_t = A_t k_t^{\alpha}, l_t^{1-\alpha}$</span> by <span class="math-container">$y_t^{\alpha}$</span>. I can not understand why not divide both sides by say <span class="math-container">$A_t$</span> to control for it, but instead <span class="math-container">$y_t^{\alpha}$</span>.</p>

CC BY-SA 4.0
2021-11-04T14:54:03.963Z
Can you give some references?
tdm
CC BY-SA 4.0
2021-11-09T15:50:24.120Z
See appendix B of "Understanding Economic Growth in Ghana in Comparative Perspective". Also, Chari uses that in his growth accounting for the USA.
Emmanuel Ameyaw
<growth-accounting>
Growth Accounting, capital-output approach
Emmanuel Ameyaw
# Growth Accounting, capital-output approach

(Asked by: username_0 on 2021-11-01)

In **standard approach** to growth accounting, you decompose $y\_t = A\_t f(k\_t, l\_t)$ into TFP, labor and capital components.

In the **capital-output approach**, people make the argument that $A\_t$ can also affect capital as in a solow model, where $A\_t$ increases, which shifts the production function outward, and $k\_t$ increases to a new steady state. So $A\_t$ affects $k\_t$.

To account for the fact that $A\_t$ affects $k\_t$, it is standard in the literature to divide both sides of $y\_t = A\_t k\_t^{\alpha}, l\_t^{1-\alpha}$ by $y\_t^{\alpha}$. I can not understand why not divide both sides by say $A\_t$ to control for it, but instead $y\_t^{\alpha}$.

**Tags:** growth-accounting
> **Comments:**
> **username_1**: Can you give some references?  
> **username_0**: See appendix B of "Understanding Economic Growth in Ghana in Comparative Perspective". Also, Chari uses that in his growth accounting for the USA.  

---

## 1 answers

### Answer by username_1 on 2021-11-08. Score: 2

I think that the division by $y\_t^\alpha$ is for the following specification (where technological progress is labour augmenting).

$$
y\_t = k\_t^\alpha (Al\_t)^{1 - \alpha}.
$$
Taking logs gives:
$$
\ln(y\_t) = \alpha \ln(k\_t) + (1-\alpha) \ln(A\_t) + (1-\alpha) \ln(l\_t).
$$
If we take the derivative with respect to time, we still have a factor $(1-\alpha)$ in front of the growth rate of technology $\dot A/A$. To get rid of this, we can subtract $\alpha \ln(y\_t)$ from both sides (i.e. divide the original by $y\_t^\alpha$):
$$
(1-\alpha) \ln(y\_t) = \alpha \ln (k\_t) - \alpha \ln(y\_t) + (1-\alpha) \ln(A\_t) + (1-\alpha) \ln(l\_t)
$$
Now divide both sides by $(1-\alpha)$:
$$
\ln(y\_t) = \frac{\alpha}{1-\alpha}\ln(k\_t/y\_t) + \ln(A\_t) + \ln(l\_t).
$$
Then taking the derivative we get:
$$
\frac{\dot y\_t}{y\_t} = \frac{\alpha}{1-\alpha} \frac{\dot{(k\_t/y\_t)}}{(k\_t/y\_t)} + \frac{\dot A\_t}{A\_t} + \frac{\dot l\_t}{l\_t}
$$
So the growth rate of technological progress is given by:
$$
\frac{\dot A\_t}{A\_t} = \frac{\dot y\_t}{y\_t} - \frac{\alpha}{1-\alpha} \frac{\dot{(k\_t/y\_t)}}{(k\_t/y\_t)} - \frac{\dot l\_t}{l\_t}
$$
If the share of kapital over output, and labour remains constant over time, this means that technological growth is approximately equal to output growth.

> **Comments:**
> **username_0**: I don't think that it is for this specification only...see appendix B of this paper, "Understanding Economic Growth in Ghana in Comparative Perspective". There are many like that in the literature. The argument is that TFP can also affect capital, but not sure why divide by y^alpha. In the literature, it is not model specific, like you suggest.  
> **username_1**: I think it depends on how you define $A$. See also [here](https://mpra.ub.uni-muenchen.de/69406/1/MPRA_paper_69406.pdf)  
> **username_0**: Thanks. Nice article. As mentioned in there, the reason why they divide by y^alpha is that they wanna scale up the importance of TFP in the accounting process. I think that answers my question. I have to look for why they wanna scale TFP by a factor of 1/(1-alpha) and not some other factor though.  

---

