There are two ways to make more money selling things: Sell more things, or
sell the things at a higher price. You'd like to be able to do both, but by the
Law of Demand, raising the price means reducing the quantity demanded; so you
must choose which strategy to adopt.

How do you choose? That's where the price elasticity of demand comes in. If
demand is highly elastic, you should lower the price and try to sell more
things, because a small reduction in price will produce a large increase in the
quantity you can sell. But if demand is highly inelastic, you should raise the
price, because a large increase in price will only produce a small decrease in
the quantity you can sell.

That's the basic intuition; I can also formalize this a bit.

Your revenue is price *P* times quantity *Q*, *P*Q*.
Ignoring costs for the time being, you want to maximize revenue. Therefore set
the derivative equal to zero.

d[P*Q]/dP = 0 = P dQ/dP + Q

Rearranging this slightly, we have:

P dQ/dP = -Q

dQ/dP P/Q = -1

And this is simply the price elasticity of demand, sometimes written this way
(a slight abuse of calculus):

(dQ/Q)/(dP/P) = -1

Thus, an elasticity of -1 (sometimes just said "1", but it's really -1) is the
point of maximum revenue.

If elasticity is *bigger* than -1, you're in a zone where you could
increase revenue by reducing price.

If elasticity is *smaller* than -1, you're in a zone where you could
increase revenue by increasing price.

Only when those two paths converge at exactly -1 have you set the
revenue-maximizing price.

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