Elasticity of Demand

Question

For the following demand equation compute the elasticity of demand and determine whether the demand is elastic, unitary, or inelastic at the indicated price. (Round your answer to three decimal places.)

 

x=(-9/4)p+19; p=7

 

 

Answer 1

At price =7, quantity purchased = (-9/4)(7)+19 = --64/4 + 19 = 76/4 - 64/4 = 12/4 = 3

x'(p) = (-9/4), for an increase in p, x decreases. It's an inverse relation

p goes up 1 and x goes down 9/4 = 2.5 elasticity = .5/3 all over 1/7 = 5/30 x 7 = -7/6 = -1.67 which is elastic, less than 100%

price elasticity of demand = % change in x divided by % change in p

as p changes from 7 to 14, a 100% increase in price, x changes from 3 to 0, a 100% decrease. -50/100 =-1/2 >-1, which is inelastic,

take the x(p) equation, x=-(9/4)p +19, and solve for p=(4)(19)/9 -4x/9 = 76/9 -(4/9)x

multiply p by x = revenue = xp= (76/9)x-(4/9)x^2

take the derivative and set equal to zero, solve for x

R' = (px)' = 8 4/9 - (8/9)x = 0

x = (9/8)(76/9)= 9.5

when x=9.5, p = (4/9)(19) - 4(9.5)/9 = 4 2/9 = 4.222...

that point is the point elasticity of 1 (or -1, the negative sign is assumed) = unit elasticity

p= 4.222... with x = 9.5.

p<4.222... is elastic

p>4.222 ... is inelastic

p=4.222.... is unitary elasticity

7 > 4.222... so at that point, the demand curve is inelastic

4.222... is the midpoint between maximum price & zero

where quantity demanded is the midpoint between 19 and zero.