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?

Question

Quizplus · Accepted Answer

To determine the optimal discount, we need to use the formula:

%ΔQ / %ΔP = -E

where %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 / %ΔP

Solving for %ΔP:

%ΔP = %ΔQ / -3

Now, 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) / 300
Q2 = 100 - 300 x %ΔP

For 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) / 150
Q2 = 100 - 150 x %ΔP

To maximize profit, we need to find the discount that will maximize total revenue. Total revenue is calculated as:

TR = P x Q

where 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) = 0
200 - Q2 + 3x = 0
Q2 = 200 - 3x

For non-users of the coupon:

dTR/dx = -10(Q2 - 150)/100 - 10(1 - x/100)(-150/100) = 0
Q2 - 150 + 1.5x = 0
Q2 = 150 - 1.5x

Now we can substitute Q2 from one equation into the other:

200 - 3x = 150 - 1.5x
50 = 1.5x
x = 33.33

Therefore, the optimal discount to provide for coupon users is 50% off, since this discount will yield the highest total revenue.

Johnny's Shop-And-Pay Is a Regional Grocery Chain, and Their Marketing