Why the price elasticity of demand is not constant along a straight-line demand curve

This week my ECONS101 class covered elasticities. The most important elasticity that we cover in that topic is the price elasticity of demand. The price elasticity of demand can be calculated as [Percentage change in quantity demanded]/[Percentage change in price], or in shorthand, [%ΔQd]/[%ΔP]. Because price and quantity demanded always move in opposite directions (because of the Law of Demand - when price goes up, people buy less, and when price goes down, people buy more), the price elasticity of demand is always a negative number.

One important aspect of the price elasticity of demand is the determination of whether demand is elastic or inelastic. Demand is elastic if the percentage change in quantity demanded is greater than the percentage change in price. In other words, the price elasticity of demand is greater than one (in absolute terms). Demand is inelastic if the percentage change in quantity demanded is less than the percentage change in price. In other words, the price elasticity of demand is less than one (in absolute terms).

Now, here's where things get a little tricky. A straight-line demand curve that is relatively more elastic will be flatter than a demand curve that is relatively less elastic. That seems pretty straightforward, but the word 'relatively' is important. That's because the price elasticity of demand is not constant for a demand curve that is a straight line. Although the slope of the curve does not change as we move along the demand curve, the price elasticity of demand does.

To see why, consider the straight-line demand curve shown in the diagram below. At the top of the demand curve, such as the point A, the quantity demanded is low and the price is high. As we move along the demand curve a little bit towards the point B, the percentage change in quantity demanded is going to be large (because it's a percentage of a small number), but the percentage change in price will be small (because it's a percentage of a large number). That means that the price elasticity of demand, [%ΔQd]/[%ΔP], will be a large number (because we're dividing a large number by a small number). So, at the top of the demand curve, demand is elastic.

Now, at the bottom of the demand curve, such as the point C, the quantity demanded is high and the price is low. As we move along the demand curve a little bit towards the point D, the percentage change in quantity demanded is going to be small (because it's a percentage of a large number), but the percentage change in price will be large (because it's a percentage of a small number). That means that the price elasticity of demand, [%ΔQd]/[%ΔP], will be a small number (because we're dividing a large number by a small number). So, at the top of the demand curve, demand is inelastic.

Now consider the same example, but with some numbers, as shown in the diagram below. As we move from point A to point B (at the top of the demand curve), the percentage change in quantity demanded is 100%, as we move from a quantity demanded of 1 to a quantity demanded of 2 (we can calculate this percentage change as [[2-1]/1]-1). [*] The percentage change in price is -10%, as we move from a price of $10 to a price of $9 (we can calculate this percentage change as [[9-10]/10]-1). So, the price elasticity of demand when we move from point A to point B is [100]/[-10] = -10. This is larger than one (in absolute terms), so demand when we move from point A to point B is elastic.

As we move from point C to point D (at the bottom of the demand curve), the percentage change in quantity demanded is 11.1%, as we move from a quantity demanded of 9 to a quantity demanded of 10 (we can calculate this percentage change as [[10-9]/9]-1). The percentage change in price is -50%, as we move from a price of $2 to a price of $1 (we can calculate this percentage change as [[1-2]/2]-1). So, the price elasticity of demand when we move from point C to point D is [11.1]/[-50] = -0.22. This is smaller than one (in absolute terms), so demand when we move from point C to point D is inelastic.

So, there you have it. Although the demand curve may be a straight line, and so the slope doesn't change, the price elasticity of demand does change. At the top of the demand curve, demand is elastic, while at the bottom of the demand curve, demand is inelastic. As we move down the demand curve, demand becomes progressively less elastic. And as we move up the demand curve, demand becomes progressively more elastic. That means that there is a point where the demand curve transitions from being elastic to inelastic. That point occurs exactly halfway along the straight-line demand curve. At that point, the price elasticity of demand will be exactly equal to -1, which we refer to as unitary elastic.

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[*] For simplicity, I'm not using the midpoint method for calculating the price elasticity of demand here. We'd get qualitatively very similar results if we did so.