Individual Firm and Market Demand Curve In Section 10.2, Monopoly Power, in Pindyck's text on Microeconomics, an example says that suppose $N$ firms produce a particular product and the market demand curve is $P = -aQ + b$. An individual firm is likely to face a demand curve which is more elastic than the market demand curve.

Problem here is, while finding the individual demand curve, we first find $Q = (b-P)/a$, which is the market demand. We divide this by $N$ to get the individual firm's demand curve, given by $Q = (b-P)/Na$

If I compute the elasticity of demand in both cases, which is just equal to $(dQ/dP)(P/Q)$, I get the same value, which is not expected. Is there some mistake in my calculation, for finding the elasticity? Could someone please explain what's going on, and possibly show the exact calculation.

A good check would be that as $N \to \infty$, there is perfect competition and hence perfectly elastic demand (right?)

Thank you!
What you're doing when you calculate $dQ/dP$ is assuming that all forms change their process by $dP$. This is the same as saying that the market price changes by $dP$ and is why you get the same number.

The main difference here is that when the book says price elasticity of demand is higher for an individual firm, it means that, holding the prices of other firms constant, a 1 percent change in a firm's price would result in a greater change in quantity demanded than when all forms change the price.