Elasticity and Slope with Linear Demand
The price elasticity of demand is the percentage change in the quantity of the good demanded associated with a one percent increase in the price of the good. This Demonstration lets you explore the relationship between elasticity and slope for the case of linear demand functions. Use the button bar to select either elasticity calculated at a point or the midpoint (arc) formula commonly used in introductory texts.
Snapshot 1: the steeper demand curve is less elastic at every price
Snapshot 2: linear demand functions with the same price intercept have the same elasticity at any given price
Snapshot 3: even though is flatter than , is less elastic at every price
The formula for the point elasticity of demand is
For discrete changes in price and quantity demanded, the average price and quantity demanded can be used as the base in calculating percentage changes. This "midpoint" or "arc" elasticity formula is the version used in most introductory texts.
Note that elasticity can also be expressed as .
(In the case of the midpoint formula, the average of the two prices and quantities is used.) Using this formula it is easy to show the following results.
1. Elasticity is not constant along linear demand functions. In fact, falls as you move down the demand function.
2. If two demand curves intersect at a positive price and quantity, then the steeper demand function is less elastic at every price.
3. Given two parallel linear demand functions, the one further to the right is less elastic at every price.
4. If linear demand functions have a common price intercept, then they will have the same elasticity of demand at any given price.
5. Consider two linear demand functions. Excluding the extreme case of perfectly elastic demand, the demand function with the higher price intercept is less elastic at every price.
Note that 5 implies 4, 3, and 2. To prove 5, let the price be arbitrary and calculate the slope over the interval from the origin to the quantity demanded. The quantity demanded cancels, giving the elasticity coefficient as the negative of the ratio of the price to the price intercept. Thus, an increase in the price intercept reduces the absolute value of the elasticity of demand.