Abercrombie & Fitch, once the favorite of loyal teens, is considering lowering prices on all items it sells in an effort to win them back after several years of sales declines. A&F's total sales were $6 billion last year, but they have been declining in the face of a weak economy and an intensively competitive retail environment. Price reductions are often effective in increasing sales, but marketers need to analyze how much sales must go up before a price reduction pays off and increases revenue enough to make the it worth doing. Assuming A&F's gross profit margin is 65 percent and cost of goods sold represents the only variable cost, by how much must sales increase to maintain the same gross profit margin in terms of absolute dollars if A&F lowers prices by 10 percent?
The current gross profit is $ billion. (Round to two decimal places.)
Current gross profit = gross margin x total sales
= 0.65 x 6 = $ 3.9 billion
Cost of goods sold = 6 - 3.9 = $ 2.1 billion
Cost of producing one item = 2.1 / V1
Revenue = price x sales volume
Let us assume that the initial price is P1 and current sales volume is V1
=> P1V1 = 6
=> P1 = 6 / V1
Now the current price becomes P2 and sales volume becomes V2
Revenue = P2V2
Gross profit = P2V2 - (cost of producing one item ) x V2
P2 = 0.9 P1
=> 0.9 x P1 x V2 - ( 2.1 x V2 ) /V1 = 3.9
=> ( 0.9 x 6 x V2 ) / V1 - ( 2.1 x V2 ) /V1 = 3.9
=> 5.4 V2 - 2.1 V2 = 3.9 V1
=> V2 = 3.9V1 / 3.3
=> ( V2 - V1 ) / V1 = ( 39 - 33 ) / 33
=> % increase in sales volume = 18.18%
Therefore, in order to maintain the same absolute gross margin ( $ 3.9 billion ), for a 10% price decrease, the sales volume must increase by 18.18%
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