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It makes more intuitive sense to me: as I increase the quantity, this is how much the price changes. Why is a much more complex formula used that takes into consideration the current quantity and price at a given point on the curve?

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A key feature of elasticity is that it is a unit-free measure. Suppose we are considering the price-elasticity of demand for apples which is $\frac{{\triangle}y/y}{{\triangle}x/x}$ where $y$ is quantity demanded and $x$ is price. The elasticity will be the same whether apples are measured in pounds (lb) or kilograms (kg) because both $\triangle y$ and $y$ will be in the same units. Similarly it will not matter in which currency price is measured, because $\triangle x$ and $x$ will be in the same currency units. This makes comparisons of elasticities, between goods measured in different units, or between countries with different currencies, much more meaningful than comparisons of slopes.

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The reason why elasticity is not defined as the slope of the graph is because the idea of slope is mathematically different from elasticity.

When we calculate slope we use the formula ${\triangle}y/{\triangle}x$ however in the case of elasticity we use $\frac{{\triangle}y/y}{{\triangle}x/x}$ which is the percent change in $y$ divided by percent change in $x$ which is not the same as the formula for slope.

Bottom Line:

${\triangle}y/{\triangle}x \ne \frac{{\triangle}y/y}{{\triangle}x/x}$

Hope this helps.

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Elasticities tell economists how responsive changes in price are to changes in quantity, and this is useful because it tells you whether revenue will increase or decrease. Suppose you are the owner if a firm that is interested in expanding. It will be more informative for you to know that if you expand your production by 20%, the prices of your products will decrease by only 10% than to know that if you produce two more units per week, the price will drop by $1.00. The first description is informative because it tells you whether your revenue will increase or not (in this case it will, because demand is price elastic), whereas the second is an absolute reference which requires further calculation in order to understand what it means for your business.

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