Section 8.1 Expanded: Constructing the nonlinear profit contribution expression

Question

Section 8.1 Expanded: Constructing the nonlinear profit contribution expression

Let PS and PD represent the prices charged for each standard golf bag and deluxe golf bag respectively. Assume that “S” and “D” are demands for standard and deluxe bags respectively.

S = 2250 – 15PS (8.1)

D = 1500 – 5PD (8.2)

Revenue generated from the sale of S number of standard bags is PS*S. Cost per unit production is $70 and the cost for producing S number of standard bags is 70*S.

So the profit for producing and selling S number of standard bags = revenue – cost = PSS – 70S (8.3)

By rearranging 8.1 we get

15PS = 2250 – S or

PS = 2250/15 – S/15 or

PS = 150 – S/15 (8.3a)

Substituting the value of PS from 8.3a in 8.3 we get the profit contribution of the standard bag:

(150 –S/15)S – 70S = 150S – S2/15 – 70S = 80S – S2/15 (8.4)

Revenue generated from the sale of D number of deluxe bags is PD*D. Cost per unit production is $150 and the cost for producing D number of deluxe bags is 150*D.

So the profit for producing and selling D number of deluxe bags = revenue – cost = PDD – 150D (8.4a)

By rearranging 8.2 we get

5PD = 1500 – D or

PD = 1500/5 – D/5 or

PD = 300 – D/5 (8.4b)

Substituting the value of PD from 8.4b in 8.4a we get the profit contribution of the deluxe bags:

(300 -D/5)D – 150D = 300D – D2/5 – 150D = 150D – D2/5 (8.4c)

By adding 8.4 and 8.4c we get the total profit contribution for selling S standard bags and D deluxe bags.

Total profit contribution = 80S –S2/15 + 150D – D2/5 (8.5)

Homework assignment:

Reconstruct new expression for 8.5 by changing “15PS” to “10PS” in 8.1, “5PD” to “7PD” in 8.2, cost per unit standard bag from 70 to 75 and cost per unit deluxe bag 150 to 140. Use up to 2 decimal points accuracy.

Explanation / Answer

Homework assignment:

Reconstruct new expression for 8.5 by changing “15PS” to “10PS” in 8.1, “5PD” to “7PD” in 8.2, cost per unit standard bag from 70 to 75 and cost per unit deluxe bag 150 to 140. Use up to 2 decimal points accuracy.

Solution: given demand:
S = 2250 – 15PS (8.1)
D = 1500 – 5PD (8.2)

After reconstruct, the equation will be as follows;
S = 2250 - 10PS
D = 1500 - 7PD

By rearranging the above equations, we get
S = 2250 - 10PS
10PS = 2250 - S
PS = 2250 /10 - S/10
PS = 225 - S/10

D = 1500 - 7PD
7PD = 1500 - D
PD = 1500 / 7 -D/7
PD = 214.29 - D/7

Revenue generated from the sale of S number of standard bags is PS*S. Cost per unit production is $75 and the cost for producing S number of standard bags is 75*S.

So the profit for producing and selling S number of standard bags = revenue – cost = PSS – 75S

AND

Revenue generated from the sale of D number of deluxe bags is PD*D. Cost per unit production is $140 and the cost for producing D number of deluxe bags is 140*D.

So the profit for producing and selling D number of deluxe bags = revenue – cost = PDD – 140D

Substituting the value of PS and PD:

First we will substitute the value of PS, we get the profit contribution of the standard bag:

(225 –S/10)S – 75S = 225S – S2/10 – 75S = 150S – S2/10

Now, we will substitute the value of PD, we get the profit contribution of the deluxe bags:

(214.29 -D/7)D – 140D = 214.29D – D2/7 – 140D = 74.29D – D2/7

Therefore, By adding the above two equations, we get the total profit contribution for selling S standard bags and D deluxe bags.

Total profit contribution = 150S – S2/10 + 74.29D – D2/7

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