August 2nd, 2016, 08:16 AM  #1 
Newbie Joined: Aug 2016 From: uk Posts: 1 Thanks: 0  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? 
August 2nd, 2016, 02:50 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,786 Thanks: 708 
Statement A is puzzling. Is j the variable of integration? The description has j a discrete variable.


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