Price Elasticity of Demand

Question

An electronic manufacturer determines that in order to sell 𝑥units of a Bluetooth speaker, its price per unit must be 𝑝=−0.2𝑥+350. If the current price of the speaker is $250, analyze the demand and the revenue if this price is decreased or increase. Also, determine the percentage change in demand if the price were to be increased by 4%and the values of 𝑝for which the demand is inelastic.

Answer 1

p = -0.2x + 350 = -x/5 + 350

250 = 350-100

x/5 = 100

x =500 when price = $250

quantity demanded when price is 250 is 500

revenue = price times quantity = px = -x^2/5 +350

Revenue = R= px = 250(500) = 125,000 when price = $250

p =-x/5 + 350

-x/5 = p-350

x = 1750 -5p

x' = dx/dp = -5p

price elasticity of demand = percentage change in quantity demanded divided by percentage change in price

point price elasticity of demand = (dx/dp)/(x/p) = (dx/dp)/(x/p) = -5(p/x) = -5p/x = -5(250)/500= -5/2=-2.5

it's unit elasticity = 1 at the midpoint of the demand curve. graph p=-x/5 + 350, x intercept is 1750, y intercept = 350. midpoint is half of each or the point (875, 175)

elasticity =-5(175)/875 =-1. Demand is inelastic for price below that midpoint p=$175. It's elastic above p=$175.

Revenue = R = px = -x^2/5 + 350x

R' = -2x/5 +350 = 0

2x/5 = 350

x =(5/2)(350 = 5(175) = 875 = revenue maximizing output level, p=175 is the revenue maximizing price

maximum revenue = (175)(875) = $153,125

at p=250, R= 250(500) = $125,000, below the max revenue

if price were increased by 4%, price would increase by 10, from 250 to 260, quantity demanded would decrease by 50, from 500 to 450. Revenue would decrease to 260(450) = 117,000, a decrease of 8,000

price elasticity= percentage change in quantity divided by percentage change in price = -10%/4% = -2.5

if price decreased by 4%, price would decrease from 250 to 240 and quantity demanded would increase from 500 to 550, a 10% increase. elasticity = 10%/-4% = -2.5. Revenue would increase to (240(550) = 132,000