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August 2nd, 2016, 08:16 AM   #1
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Integration, confused

I am reading a research paper and I could not understand how the author derived the final equation using integration.

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 author uses the equation B in equation A and derives the equation C after integration. It appears, the author replaces $x_L(j)$ in equation A with the equation for $x_L$ which is the equation B. Can anyone please help understand how this could be done?
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August 2nd, 2016, 02:50 PM   #2
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Statement A is puzzling. Is j the variable of integration? The description has j a discrete variable.
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