Question

Old Navy sell sports shirts for men; during 2016, Old Navy sold an average of 23,000 sports shirts for an average price of $26 per shirt in its Hulen Mall store in Fort Worth, Texas. In early January 2017, Gap a major competitor cut the price of its sports shirts from $30 to $24. The orders that Old Navy received for its own sports shirts dropped sharply, from 23,000 per month to 13,000 per month because of this action.

Calculate the cross elasticity of demand between Old Navy’s sports shirts and Gap’s sports shirts during February and March. Are the two companies’ sports shirts good or poor substitutes? Why? Explain.

Suppose that the coefficient of price elasticity of demand for Old Navy’s sports shirts is -2.0 Assuming that Gap keeps its price at $24, by how much must Old Navy cut its price to build its sales of shirts back up to 23,000 per month? (Hint: Use the arc formula for price elasticity and substitute the known values into it and solve for the unknown price).

Answer 1

(a)

Cross elasticity of demand = [(Change in demand for Old Navy shirt / Average demand for Old Navy shirt)] / [(Change in price of Gap Shirt) / (Average price of Gap shirt)]

= [(13,000 - 23,000) / (13,000 + 23,000) / 2] / [($(24 - 30) / $(24 + 30) / 2]

= [- 10,000 / (36,000 / 2)] / [- $6 / ($54 / 2)]

= (- 10,000 / 18,000) / (- 6 / 27)

= 2.5

Since cross price elasticity is positive, the shirts are substitutes.

(b) Let new price to be charged by Old Navy be P.

Price elasticity of demand = [(Change in quantity demanded of Old Navy) / (Average quantity demanded of Old Navy)] / [(Change in rice by Old Navy) / (Average price by Old Navy)]

- 2 = [(23,000 - 13,000) / (23,000 + 13,000) / 2] / [$(P - 26) / $(P + 26) / 2]

- 2 = [(10,000 / 18,000)] / [$(2P - 52) / $(P + 26)]

- 2 = [10,000 x (P + 26)] / [18,000 x (2P - 52)]

- 36,000 x (2P - 52) = 10,000 x ( P + 26)

- 18 x (2P - 52) = 5 x (P + 26)

- 36P + 936 = 5P + 130

41P = 806

P = $19.66

Required decrease in price = $(26 - 19.66) = $6.34