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You may have read Acemoglu (2002) 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?

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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)

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  • $\begingroup$ 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. $\endgroup$
    – london
    Commented Aug 6, 2016 at 19:34

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