A retailer of an electronic game faces constant-elasticity price-response functi

Question

A retailer of an electronic game faces constant-elasticity price-response function with an elasticity of 3.0. It costs the retailer $45 a piece to purchase the game wholesale. At what price should the retailer sell the game to maximize total contribution? After a market study, the retailer realizes that the total market for the game consists of 50,000 people, and he loses 500 people for every $1 increase in the price of the game. At what price should the retailer price the game to maximize revenue? What is the elasticity at the revenue maximizing price?

Explanation / Answer

Optimal Mark-up over cost = (1 / Absolute Value of Price elasticity of demand) x 100

Here, Absolute Value of Price elasticity of demand = 3

So, Optimal Mark-up = 1 / 3 x 100 = 33.33%

(a) Optimal price = Cost + Mark-up

= $45 x 1.3333

= $60

(b) The given information provides the inverse demand curve as follows:

Q = 50,000 - 500P Where P: Number of people & P = Price

Or,

P = (50,000 - Q) / 500

Revenue = P x Q = (50,000Q - Q2) / 500

Revenue is maximized when dTR / dQ = 0

(50,000 - 2Q) / 500 = 0

50,000 - 2Q = 0

2Q = 50,000

Q = 25,000

P = (50,000 - Q) / 500 = (50,000 - 25,000) / 500

= 25,000 / 500 = 50

Elasticity = (dQ / dP) x (P / Q)

= - 500 x (50 / 25,000)

= - 1

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