<p>The price $χ_L(j)$ is 1!</p>

<p>You only need to integrate a constant. That's why you have that equation.</p>

<p>(See p. 789, second paragraph, when he normalizes it to 1)</p>

CC BY-SA 3.0
2016-08-06T19:34:21.317Z
Many thanks for pointing this out, I am reading a similar paper which uses Acemoglu (2002) framework as a guide but have not seen the author mentioning the normalisation.
london
CC BY-SA 3.0
2016-08-05T19:21:58.753Z
2016-08-05T19:21:58.753Z
Belisario
<p>You may have read <a href="http://economics.mit.edu/files/4126" rel="nofollow">Acemoglu (2002)</a> on the topic of directed technical change. I am reading a similar research paper, and I could not understand how the author derived the final equation using integration. Below are details:</p>

<p><strong>Equation A:</strong></p>

<p>$Y_L=\frac{1}{1-\beta}\left [\displaystyle\int_0^{N_L}x_L(j)^{1-\beta}\,dj\right ]L^{\beta }$</p>

<p><strong>Equation B:</strong></p>

<p>$x_L=\left [\frac{p_L}{\chi_L(j)}\right ]^{1/\beta}L$</p>

<p><strong>Equation C:</strong></p>

<p>$Y_L=\frac{1}{1-\beta}p_L^{\frac{1-\beta}{\beta}}N_LL$</p>

<p>where, </p>

<ul>
<li>$N_L$ is the number of varieties of machines</li>
<li>$x_L$ is the range of machines, so $x_L(j)$ is a machine type $(j)$</li>
<li>$\chi_L(j)$ is the price of machine type $(j)$</li>
</ul>

<p>The rest are the usual macro variables.</p>

<p>The author uses the equation <strong>B</strong> in equation <strong>A</strong> and derives the equation <strong>C</strong> after integration. I was wondering if anyone could please help understand how this could be done?</p>

<macroeconomics><economic-growth><endogenous-growth><directed-technical-change>
Directed technical change, Acemoglu (2002) framework - trying to understand how an equation is derived
london
# Directed technical change, Acemoglu (2002) framework - trying to understand how an equation is derived

(Asked by: username_0 on 2016-08-05)

You may have read [Acemoglu (2002)](http://economics.mit.edu/files/4126) on the topic of directed technical change. I am reading a similar research paper, and I could not understand how the author derived the final equation using integration. Below are details:

**Equation A:**

$Y\_L=\frac{1}{1-\beta}\left [\displaystyle\int\_0^{N\_L}x\_L(j)^{1-\beta}\,dj\right ]L^{\beta }$

**Equation B:**

$x\_L=\left [\frac{p\_L}{\chi\_L(j)}\right ]^{1/\beta}L$

**Equation C:**

$Y\_L=\frac{1}{1-\beta}p\_L^{\frac{1-\beta}{\beta}}N\_LL$

where,

* $N\_L$ is the number of varieties of machines
* $x\_L$ is the range of machines, so $x\_L(j)$ is a machine type $(j)$
* $\chi\_L(j)$ is the price of machine type $(j)$

The rest are the usual macro variables.

The author uses the equation **B** in equation **A** and derives the equation **C** after integration. I was wondering if anyone could please help understand how this could be done?

**Tags:** macroeconomics, economic-growth, endogenous-growth, directed-technical-change


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## 1 answers

### Answer by username_1 on 2016-08-05. Score: 1 | Accepted answer

The price $χ\_L(j)$ is 1!

You only need to integrate a constant. That's why you have that equation.

(See p. 789, second paragraph, when he normalizes it to 1)

> **Comments:**
> **username_0**: Many thanks for pointing this out, I am reading a similar paper which uses Acemoglu (2002) framework as a guide but have not seen the author mentioning the normalisation.  

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