What are the reasons why I should advice a producer to reduce the price of his product in relation to price elasticity in demand?

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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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