Consider, as we did in much of the chapter, a downward sloping linear demand curve.
A: In what follows, we will consider what happens to the price elasticity of demand as we approach the horizontal and vertical axes along the demand curve.
(a) Begin by drawing such a demand curve with constant (negative) slope. Then pick the point A on the demand curve that lies roughly three quarters of the way down the demand curve. Illustrate the price and quantity demanded at that point.
(b) Next, suppose the price drops by half and illustrate the point B on the demand curve for that lower price level. Is the percentage change in quantity from A to B greater or smaller than the absolute value of the percentage change in price?
(c) Next, drop the price by half again and illustrate the point C on the demand curve for that new (lower) price. The percentage change in price from B to C is the same as it was from A to B. Is the same true for the percentage change in quantity?
(d)What do your answers imply about what is happening to the price elasticity of demand as we move down the demand curve?
(e) Can you see what will happen to the price elasticity of demand as we get closer and closer to the horizontal axis?
(f) Next, start at a point A′ on the demand curve that lies only a quarter of the way down the demand curve. Illustrate the price and quantity demanded at that point. Then choose a point B′ that has only half the consumption level as at A′. Is the percentage change in price from A′ to B′ greater or less than the absolute value of the percentage change in quantity?
(g) Now pick the point C′ (on the demand curve) where the quantity demanded is half what it was at B′. The percentage change in quantity from A′ to B′ is then the same as the percentage change from B′ to C′. Is the same true of the percentage change in price?
(h)What do your answers imply about the price elasticity of demand as we move up the demand curve? What happens to the price elasticity as we keep repeating what we have done and get closer and closer to the vertical intercept?
B, Consider the linear demand curve described by the equation p = A−αx.
(a) Derive the price elasticity of demand for this demand curve.
(b) Take the limit of the price elasticity of demand as price approaches zero.
(c) Take the limit of the price elasticity as price approaches A.