Every year I teach "elasticity". And every year the students ask "Why not talk about "slope" instead?". They are familiar with "slope", but "elasticity" is a new concept. Why do we teach a new concept, if an old familiar concept would do just as well?
[For non-economists: the slope of a demand curve is the change in price divided by the change in quantity demanded; the elasticity of a demand curve is the percentage change in quantity demanded divided by the percentage change in price. Elasticity = (1/slope)x(Price/Quantity demanded).]
My normal answer is that elasticity is unit-free, while slope has the units dollars.years/tonnes squared (if price is measured in dollars per tonne and quantity in tonnes per year). So you can compare the elasticities of demand for wheat and electricity, but you can't compare the slopes, because you can't compare tonnes with kilowatts-hours.
But I think there's a better answer. "Elasticity" helps us distinguish the individual experiment from the market experiment. "Slope" doesn't. Things that look flat on one scale don't look flat on another scale (e.g. Earth).
The slope of the individual farmer's demand curve is exactly the same as the slope of the market demand curve. One extra tonne of wheat causes the price to drop by exactly the same amount whether he produces that extra tonne, or his neighbour produces that extra tonne, or a million farmers produce one extra gram each.
But the elasticity of the individual farmer's demand curve is very different from the elasticity of the market demand curve. If there are one million farmers, the individual farmer's demand curve will be one million times more elastic than the market demand curve. If one million farmers grow 1% more wheat the drop in price will be a million times bigger than if the individual farmer alone grows 1% more wheat.
When an individual farmer decides whether to grow more wheat, he treats "Price" and "Marginal Revenue" as the same thing. (Yes, I have talked to farmers.) This makes sense. There is a simple relationship between Marginal Revenue, Price, and Elasticity. MR=[1-(1/E)]P. So as E approaches infinity, MR approaches P. So MR and P are (almost) the same thing, if elasticity is (almost) infinite. He sees a flat demand curve for his wheat.
But when that same individual farmer speaks about production quotas and imports at the National Farmer's Union meeting, he doesn't see a flat demand curve for wheat. He knows that restricting production and restricting imports will increase the price.
We want to say, and we need to say, that the individual farmer's demand curve is "flatter" than the market demand curve. When we talk about "flatness" we are talking about elasticity. If we were talking about slope we would be talking nonsense.
If the farmers all colluded, and asked what would happen to their profits if they all increased or decreased output together, the elasticity would be much lower, and so MR would be much lower too, than if just one farmer increased his output. But the slope would be the same. The NFU has a lower elasticity than its members, but the same slope.
Addendum: The same is true when we move away from perfect competition.
Cournot duopolists selling identical products will each have a demand curve twice as elastic as their market demand curve, but with exactly the same slope. With differentiated products it won't be twice as elastic, but it will still be more elastic than if they colluded.
[In the Cournot-Nash equilibrium, each firm chooses output to maximise profits given other firms' outputs. I was implicitly assuming that farmers play Cournot because, well, that's mostly what they do. They decide how much wheat to plant, and then sell it at what they can get. They don't announce a price and then plant the quantity of wheat demanded at that price.]
Bertrand duopolists selling differentiated products will each have a demand curve that is more elastic than if they colluded.
[In the Bertrand-Nash equilibrium, each firm chooses price to maximise profits given other firms' prices.]
You can think of perfect competition as the limiting case of Bertrand-Nash, as the firms' products become perfect substitutes. Or as the limiting case of Cournot-Nash, as the firm's products become perfect substitues and the number of firms gets very large.