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In the basic New Keynesian Model most variables are described as an integral equation. Lets for Instance look at labour demand which is defined as: $$N_t=\int_{0}^{1}N_t(i)di$$

What is the intuition behind that. How does the $N_t(x)$ function look like and why do we integrate it interval $[0,1]$?

I am reading Gali's book.

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First of all, this is not something special to "the basic New Keynesian Model". It is a commonly used technique when modeling an economy with competitive environment.

In your case, the author wrote that there is a continuum of differentiated goods represented by the interval [0,1] and each firm is producing one of the goods. You can imagine that there are uncountably many different kinds of good in the market. As a result, the size of each firm is negligible (just a point on the [0,1] interval) compared to the entire economy. Each firm indexed by $i$ solves its individual profit maximization problem to obtain its labor demand $N(i)$. Then, the aggregate labor demand is simply the "sum". In this case, the sum is replaced by the integral because the variety of different firms is the whole unit interval.

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