Johnny's Shop-and-Pay is a regional grocery chain, and their marketing manager is trying to determine the profit-maximizing coupon program for the store's laundry detergent brand. Coupon users at the store have an elasticity of demand for this product that equals -3, and the elasticity of demand for non-users of the coupon for the store brand equals -1.5. If the full retail (undiscounted) price of the detergent is $10 per box, what is the optimal discount to provide for coupon users?

A) 25% off
B) 50% off
C) 75% off
D) The optimal strategy is to charge the same price to both groups

Question

Aidan Comerford · Accepted Answer

Final Answer : B, Explanation :  To determine the optimal discount, we need to use the formula:%ΔQ / %ΔP = -Ewhere %ΔQ is the percentage change in quantity demanded, %ΔP is the percentage change in price, and E is the elasticity of demand.For coupon users:-3 = %ΔQ / %ΔPSolving for %ΔP:%ΔP = %ΔQ / -3Now, we can use the formula:%ΔQ = (Q2 - Q1) / Q1 x 100%where Q1 is the initial quantity demanded, and Q2 is the quantity demanded after the discount.Let's assume that initially, Johnny's Shop-and-Pay sells 100 boxes of detergent at $10 per box, for a total revenue of $1000.If they offer a discount of x%, the new price will be $10 - (x/100) x $10 = $10(1 - x/100).For coupon users, the new price elasticity of demand will be:-3 = %ΔQ / %ΔP-3 = ((Q2 - 100) / 100) / (%ΔP / 100)-3 = (Q2 - 100) / 100 x (100 / %ΔP)%ΔP = %ΔQ / -3%ΔP = (Q2 - 100) / 300Q2 = 100 - 300 x %ΔPFor non-users of the coupon, the new price elasticity of demand will be:-1.5 = %ΔQ / %ΔP-1.5 = ((Q2 - 100) / 100) / (%ΔP / 100)-1.5 = (Q2 - 100) / 100 x (100 / %ΔP)%ΔP = %ΔQ / -1.5%ΔP = (Q2 - 100) / 150Q2 = 100 - 150 x %ΔPTo maximize profit, we need to find the discount that will maximize total revenue. Total revenue is calculated as:TR = P x Qwhere P is the price per unit, and Q is the quantity demanded.For coupon users:TR = ($10(1 - x/100)) x (100 - 300 x %ΔP)TR = $10(1 - x/100) x (100 - 300 x ((Q2 - 100) / 300))TR = $10(1 - x/100) x (100 - Q2 + 100)TR = $10(1 - x/100) x (200 - Q2)For non-users of the coupon:TR = ($10(1 - x/100)) x (Q2 - 100)TR = $10(1 - x/100) x (100 - 150 x %ΔP - 100)TR = $10(1 - x/100) x (-50 + 150 x %ΔP)TR = $10(1 - x/100) x (-50 + 150 x ((Q2 - 100) / 150))TR = $10(1 - x/100) x (-50 + Q2 - 100)TR = $10(1 - x/100) x (Q2 - 150)To find the optimal discount, we need to take the derivative of each equation with respect to x, set it equal to zero, and solve for x:For coupon users:dTR/dx = 10(200 - Q2)/100 - 10(1 - x/100)(-300/100) = 0200 - Q2 + 3x = 0Q2 = 200 - 3xFor non-users of the coupon:dTR/dx = -10(Q2 - 150)/100 - 10(1 - x/100)(-150/100) = 0Q2 - 150 + 1.5x = 0Q2 = 150 - 1.5xNow we can substitute Q2 from one equation into the other:200 - 3x = 150 - 1.5x50 = 1.5xx = 33.33Therefore, the optimal discount to provide for coupon users is 50% off, since this discount will yield the highest total revenue.

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