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Answer to Question #53769 in Microeconomics for denise

Question #53769
The Stopdecay Company sells an electric toothbrush for $25. Its sales have averaged 8,000 units per month over the past year. Recently, its closest competitor, Decayfigh- ter, reduced the price of its electric toothbrush from $35 to $30. As a result, Stopde- cay’s sales declined by 1,500 units per month.

a. What is the arc cross elasticity of demand between Stopdecay’s toothbrush and Decayfighter’s toothbrush? What does this indicate about the relationship between the two products?

b. If Stopdecay knows that the arc price elasticity of demand for its toothbrush is −1.5, what price would Stopdecay have to charge to sell the same number of units as it did before the Decayfighter price cut? Assume that Decayfighter holds the price of its toothbrush constant at $30.
c. What is Stopdecay’s average monthly total revenue from the sale of electric toothbrushes before and after the price change determined in part (b)?
d. Is the result in part (c) necessarily desirable?
Expert's answer
Ps1 = $25, Qs1 = 8,000 units per month, Pd1 = $35, Pd2 = $30, Qs2 = 1,500 units.
a. The arc cross elasticity of demand between Stopdecay’s toothbrush and Decayfighter's toothbrush is:
Ec = (Qs2 - Qs1)/(Pd2 - Pd1)*(Pd1 + Pd2)/(Qs1 + Qs2) = (1500 - 8000)/(30 - 35)*(30 + 35)/(8000 + 1500) = 6500/5*65/9500 = 8.9
This indicates that the two products are substitutes with a close relationship.

b. If Stopdecay knows that the arc price elasticity of demand for its toothbrush is −1.5, it would have to charge the price of P = 25/(8000/(1500*1.5)) = 25/(8000/2250) = $7.03 to sell the same number of units as it did before the Decayfighter price cut.

c. The Stopdecay’s average monthly total revenue from the sale of electric toothbrushes before and after the price change determined in part (b) is TR1 = 25*8000 = $200,000 and TR2 = 25*1500 = $37,500.

d. The result in part (c) is not necessarily desirable

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