Compare the price elasticity at two parallel demand curves at a given price. This has been explained in Fig above where two demand curves AB and CD are given that are parallel to each other. The two demand curves that are parallel to each other signify that they have the same slope. Now we can prove that at price OP price elasticity of demand on the two demand curves AB and CD is different. Now draw a perpendicular from point R to the point P on Y-axis. So, at price OP the corresponding points on the two demand curves are Q and R respectively.
Elasticity of demand on the demand curve AB at point Q would be equal to QB/QA and at point R on the demand curve CD it is equal to RD/RC. Since it is right-angled triangle OAB, PQ is parallel to QB:
Hence, QB/QA= OP/PA
Hence, price elasticity at point Q on the demand curve
AB= OP/PA
At point R on the demand curve CD, price elasticity is equal to RD/RC. Because in the right angled triangle OCD, PR is parallel to OD.
Therefore, RD/RC = OP/PC
Hence, on point R on the demand curve CD, price elasticity =OP/PC
On seeing the figure it will be clear that at point Q the price elasticity OP/PA and at point R the price elasticity OP/PC aren't equal to each other. Since PC is greater than PA,
OP/PC = OP/ PA
It is hence; clear that at point R on the demand curve CD price elasticity is less than that at point Q on the demand curve AB, when two demand curves being parallel to each other have the same slope. It also follows that as the demand curve shifts to the right the price elasticity of demand at a given price goes on declining. So, as has been just seen, price elasticity at price OP on the demand curve CD is less than that on the demand curve AB.