The slope of a straight-line demand curve, one with a constant slope, has constantly changing elasticity. It includes all five elasticity alternatives--perfectly elastic, relatively elastic, unit elastic, relatively inelastic, and perfectly inelastic. No two points on a straight-line demand curve have the same elasticity.
The price elasticity of demand is different at each point on a demand curve with constant slope. The reason is that slope and elasticity are different concepts. Slope measures the steepness or flatness of a line in terms of the measurement units for price and quantity. Elasticity measures the relative response of quantity to changes in price. The diagram below shows the different elasticities.
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