Question

You are the manager of a shoe producer. Your company specializes in basketball shoes and a cleaning product for basketball shoes. While the shoes bring in more revenue ($500,000 per year), the cleaning product is a strong seller as well ($150,000 of revenue per year). You are considering a 2% decrease in the price of your company's basketball shoes. Assume that the shoes have an own price elasticity of demand of -1.8 and the shoes and cleaning product have a cross-price elasticity of demand of -1.4. How much do you estimate your company's total revenue to change if you go forward with the proposed 2% shoe price decrease?

Answer 1

Data provided in the question:

Revenue from the sale of shoes= $ 500,000

Revenue from the sale of cleaning product= $150,000

Price elasticity of demand for shoes= -1.8

Cross elasticity between shoes and cleaning product= -1.4

Decrease in price= 2%

Solution:

Price elasticity of demand = (% change in quantity demanded) / (% change in price)

Let x be the original price, therefore, new price is x-2% of x

Total revenue= price X No. of units

No. of units= Total revenue / price

Before price change (old quantity),

No. of units (shoes)= $ 500,000 / x

After price change (new quantity),

No. of units (shoes)= Total revenue / (x-2% of x)= Total revenue / 0.98x

% change in quantity demanded = [(new quantity-old quantity) / old quantity] X 100

% change in quantity demanded = {[(Total revenue / 0.98x)-($ 500,000 / x)] / ($ 500,000 / x)} X 100

Taking x common and eliminating it from numerator and denominator, we get,

% change in quantity demanded = {[(Total revenue /0.98)-$ 500,000] / $ 500,000} X 100

% change in quantity demanded = [(Total revenue-$ 490,000) / $ 490,000] X 100

Substituting values in Price elasticity of demand = (% change in quantity demanded) / (% change in price), we get,

-1.8 = ([(Total revenue-$ 490,000) / $ 490,000] X 100) / -2%

by eliminating % sign,

-1.8 = [(Total revenue-$ 490,000) / $ 490,000] / -0.02

0.036 X $ 490,000 = Total revenue-$ 490,000

$ 17,640 + $ 490,000 = Total revenue

Therefore,

Total revenue after price change (shoes) = $ 507,640

Cross elasticity of demand = (% change in quantity demanded of shoes) / (% change in quantity demanded of cleaning product)

% change in quantity demanded of cleaning product = [(new quantity-old quantity) / old quantity] X 100

Let price of cleaning product be y,

old quantity= $ 150,000 / y

new quantity= Total revenue / y

Therefore,

% change in quantity demanded of cleaning product = [(Total revenue/y)-($ 150,000 / y) / ($ 150,000 / y)] X 100

by eliminating y,

% change in quantity demanded of cleaning product = [(Total revenue - $ 150,000) / $ 150,000] X 100

By substituting values in, Cross elasticity of demand = (% change in quantity demanded of shoes) / (% change in quantity demanded of cleaning product),

-1.4 = {[($ 507,640-$ 490,000) / $ 490,000] X 100} / {[(Total revenue - $ 150,000) / $ 150,000] X 100}

-1.4 = [$ 17,640 / $ 490,000] / [(Total revenue - $ 150,000) / $ 150,000]

[(Total revenue - $ 150,000) / $ 150,000] = [$ 17,640 / $ 490,000] / (-1.4)

(Total revenue - $ 150,000) = ($ 17,640 X $ 150,000) / ($ 490,000 X -1.4)

Total revenue = [($ 2,646,000,000) / (-$ 686,000)] + $ 150,000

Total revenue = -$ 3857 + $ 150,000

Total revenue, after price change, from cleaning product = $ 146,143

Total revenue from both products after price change = $ 146,143 + $ 507,640 = $ 653,783

Therefore, if the price of shoes is decreased by 2%, the total revenue from both products would increase from $ 650,000 to $ 653,783